Oco /7/ CORNELL UNIVERSITY LIBRARY Cornell University Library QC 171.S45 V.2 Electrodynamic wave-theory of physical f 3 1924 012 325 399 ... ^NINTCOINU.S-i The original of this book is in the Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924012325399 ELECTRODYNAMIC WAVE-THEORY OF PHYSICAL FORCES VOLUME II NEW THEORY OF THE AETHER Definitely establishing The Cause of Universal Gravitation, Magnetism, Electrodynamic Action, Molecular, Atomic and Explosive Forces, etc., including a notable improvement in the Foundations of the Wave-Theory of Light, and discovery of the Cause of Acoustic Attraction and Repulsion, which is especially suitable for illustrating the invisible Processes of Gravitational Attraction. In Seven Mathematical Memoirs Reprinted from the Astronomische — Nachrichten, 1920 1922; to which are added two Mathematical Memoirs on the Earth, and one on the Sun and Variable Stars. By T.J.J. SEE, A. M., Lt. M., Sc. M. (Missou.); A. M., Ph. D. (Berol.); Professor of Mathematics, U. S. Navy, Formerly in Charge of the 26-Inch Equatorial Telescope of the U. S. Naval Observatory, Washington, D. C, More Recently in Charge of the U. S. Naval Observatory, Mare Island, California. — O Qedg del ysw/j.^TQet Plato. 1922. Astronomische Nachrichten, Kiel. I. Hermann, Paris. Thos. P. Nichols & Son Co., Lynn, Mass., U. S. A. Wheldon & Wesley, London. IT- DEDICATED TO MY BELOVED WIFE FRANCES GRAVES SEE WHOSE STEADFAST SUPPORT MADE POSSIBLE THE COMPLETION AND SUITABLE PUBLICATION OF THESE DISCOVERIES IN THE NEW THEORY OF THE AETHER. n "Eti Toi'pvv, e'^T], Tcdfifieyd elvai avTo, xal fj^idg otxelv Tovg li'^XQi' ^HqaxXsioiv arriXmv and Wdaidog ev OjjuixQm tivI fiioqCw, coc neQ negl z^Xfia iivQfirixuq fj /SatQaxovg, tcsqI ttjv S-dXazTav olxovvrac, xal aXXovg dXXox)-i noXXovq iv noXXolci, toiovtoii; tonoig olxslv. elvav yaQ navTa^i] neql T'qv yijv noXXd xozXa xal navcodund xal Tag ld£ag xal id /j,eya^rjj «V a '^vvsQ^vrjxivai to %s vdwQ xal ttJv ofiCxXriv xal top d^qa. avTrjV Si Tijv y-ijv xa-d-aqdv ev xa-d-aqco xelad-ai Tfo ovoavw, iv w nif) Igti, rd darga, dv drj ald-fqa ovofid'Qsiv Tovg noXXovg zcSv negl rd ToiavTa siwd-OTdov X^ysiv. ov drj vnooTdd-fiiriv Tavia elvai. xal ivQ^slv del elg rd xolXa ttJc yijg. IJXaTwv, OaCdiav, 109. I believe that the earth is vety vast, and that we who dwell in the region extending from the river Phasis to the Pillars of Heracles inhabit a small portion only about the sea, like ants or frogs about a marsh, and that there are other inhabitants of many other like places; for everywhere on the face of the earth there are hollows of various forms, and sizes, into which the water and the mist and the lower air collect. But the true earth — is pure and situated in the pure heaven there are the stars also; and it is the heaven which is commonly spoken of by us as the Aether, and of which our own earth is the sediment — gathering in the hollows beneath. Plato, Phaedo, 109. Introduction. — During the past six years several of the most venerable Scientific Societies in Europe have been considerably occupied with the Theory of Relativity, without, however, taking the usual philosophic precaution to inquire whether such a theory is at all. necessary to our understanding of the Physical Universe. The introduction of unnecessary complications into our processes of Scientific Thought always has been viewed as an evil, great in proportion as it is indefensible. Thus in his Rules of Philosophy (Principia, Lib. Ill) Newton lays down the following as the First Rule: „We are to admit no more causes of natural things than such as are both true and sufficient to explain their appearances". „To this purpose the philosophers say that Nature does nothing in vain, and more is in vain when less will serve; for Nature is pleased with simplicity, and affects not the pomp of superfluous causes". — Accordingly, whilst many investigators were debating the mystical Theory of Relativity, with Four-dimensional Time-Space manifolds, Geodetic Curves, the Curvature of Space, and similar devices — for adding hopeless complexity to our geometrical and physical conceptions, I took refuge in Newton'^ rule of maximum simplicity, and developed the New Kinetic Theory of the Aether, which showed that the Theory of Relativity is entirely devoid of physical foundation. In fact, early in the year 1914, I entered upon the development of The Electrodynamic Wave- Theory of Physical Forces, in the hope of illuminating the unsolved problem of the Cause of Universal Gravitation. Now that eight years have elapsed, and the memoirs of these two volumes are published, it may interest the reader to learn that in November, 1914, when the present researches were — still in a primitive stage, I sent the first outline of them to the Royal Society, in the belief that any definite light on the Cause of Universal Gravitation, which Sir Isaac Newton had not been able to obtain, ought first to be communicated to that illustrious Society. At that time, however, the War was very disturbing to European investigators. And if my preliminary Paper was studied attentively by the Referees of the Royal Society, it is probable that they — did not understand it, possibly because several of the leading physicists in England already were proposing to do away with the Aether. Yet, whatever cause operated to obscure the start which had been made, it is a fact that fifteen months elapsed before any report from the Royal Society was made to me (May, 1916). Meanwhile my researches had been renewed and much extended, and in due time were published under the title: Electrodynamic Wave-Theory of Physical Forces, vol.1. Quarto, 171 pages, Boston, London, Paris, 1917. This was, however, only the first part of the New Theory of Physical Forces, and the subject therefore has been extended and greatly improved during the past four years. These later discoveries in the Kinetic Theory of the Aether, which the Editor of the Astron. Nachrichten has done me the honor to publish in that celebrated Journal, 1920-22, already are widely known to the Scientific Public. Perhaps it may not be inappropriate to point out also the failure of the Royal Astronomical Society and several more of the oldest Scientific Societies in Europe. Sagacious observers have regretfully remarked how they have wasted both time and precious resources in fruitless speculations on the mysticism of the Theory of Relativity, with no other result than to confuse the public mind. (v) VI In view of the definite results here brought forth, the student of sound Physical Science may — find it interesting to contrast the barren discussion of the abandoned Theory of Relativity based on the inadmissible Gerber formula, equation (1) below, now clearly shown to violate the Conservation — of Energy, with the Kinetic Theory of the Aether, which has led to the Cause of Universal Gravitation, and the Wave-Theory of the various Physical Forces. Thus it occurs to me that it would be a convenience to many investigators if these Memoirs were collected into a volume. Accordingly, with the kind permission of the learned Editor, Professor Dr. H. Kobold, I am enabled to offer to investigators the Second Volume of the Electr odynamic Wave-Theory of Physical Forces, 1922. Although these Memoirs have been published serially only a short time, it appears that they have awakened no ordinary interest among investigators who are inclined to examine the Physical Causes underlying the Phenomena of Nature. Until this fundamental work is carried much further than has yet been done, we shall not be able to make satisfactory progress in dealing with even the simpler natural phenomena. — And as for the more intricate phenomena, the methods of research heretofore in use, based so largely on undiscerning if not blind empiricism, thus utterly ignoring the physical properties which Transcendental Physics always was capable of correctly assigning to the Aether as a Monatomic Gas — 689321600000 more elastic than Air in proportion to its density, were of course hopelessly inadequate. The first prerequisite of progress was therefore a valid Kinetic Theory of the Aether, deduced di- rectly from observed phenomena, and thus capable of furnishing a secure foundation for the Science of Dynamics. The strange proposal recently made in certain quarters to do away with the Aether, is of course inadmissible and indefensible, because the elementary principles of Mechanics show us that there must be a Medium pulling towards the Sun, to overcome the centrifugal force of a planet's orbital motion, with Tension equivalent to the breaking strength of millions of immense cables of the strongest steel. Such an unauthorized proposal merely illustrates the need of profounder researches into the foundations of Natural Philosophy. The physical necessity for the Medium was so fully recognized by Newton and by Maxwell that to the competent investigator it requires no defense. In his letter to Bentley, Febr. 25, 1692-93, Sir Isaac Newton remarks: „That gravity should be innate, inherent and essential to matter, so that one body may act upon another at a distance through a vacuum, without the mediation of anything else, by and through which their action and force may be conveyed from one to another, is to me so great an absurdity, that I believe no man who has in philo- sophical matters a competent faculty of thinking can ever fall into it. Gravity must be caused by an agent acting constantly according to certain laws; but whether this agent be material or immaterial, I have left to the consideration of my readers." In Newton's discussion the Aether evidently is taken to be immaterial, which conforms to modern views in Physical Science. In his Account of Sir Isaac Newton's Philosophical Discoveries, London, 1748, p. Ill, Maclaurin says: „He (Newton) has plainly signified that he thought that those powers (of Gravitation) arose from the impulses of a subtile Aetherial Medium that is diffused over the Universe, and penetrates the pores of grosser bodies. It appears from his letters to Mr. Boyle that this was his opinion early; and if he did not publish it sooner, it proceeded from hence only, that he found he was not able, from experiment and observation, to give a satisfactory account of this medium and the manner of its operation in producing the chief phenomena of Nature." What Sir Isaac Newton ascribed to the Impulses of a subtile aetherial medium, we now define as Waves; and in the New Theory of the Aether, we make known the manner of the operation of — — these wave-impulses in producing the chief phenomena of Nature. The leading objection to the Theory of Gravitation, in Newton's time, that it introduced into philosophy occult qualities, no longer will hold in our day, because wave-impulses in the Aether are universally recognized by modern Physical Science. : vu In the Preface to the Second Edition of Newton's Principia, 1713, his celebrated pupil Coates combats the reasoning of that time as follows: „But shall gravity therefore be called an occult Cause, and thrown out of philosophy, because the Cause of Gravity is occult and not yet discovered?" . . . „Some there are who say that gravity is praeternatural, and call it a perpetual miracle. Therefore they would have it rejected, because praeter- natural causes have no place in Physics." In view of such reasoning, we can well understand the statement of Voltaire, that although the great Newton outlived the publication of the Principia by more than forty years, yet at the end of that time he had not over twenty followers outside of England. Indeed, since Newton had clearly shown the nature of the planetary forces, and the laws they obey, and the beautiful Science of Celestical — Mechanics was thus developed for two centuries only to be contradicted recently, by the strange — claim that „Gravity is not a force, but a property of Space" (De Sitter, Monthly Notices, Oct. 1916, p. 702) we may more justly regard it as a miracle that any progress can be made and sustained in Physical Science as recently cultivated. The difficulty of making progress would be much less than it is, but for the state of confusion which has arisen in Physics from certain mystical speculations, now at length recognized to be both vague and chimerical. For example, there can be no defense for a Theory based on Gerber's formula for the Potential (Zeitschrift fur Mathematische Physik, Band XLIII, 1898, p. 93-104), „ k-mm' k^mm'i 2 dr 3 /dr\' 1 ,, i_l^V r [ c dt c-\dt ( c dt) which contradicts the great principle of the Conservation of Energy. For this formula differs from the Potential of Weber's Law, long ago shown by Maxwell to be valid and conformable to the Conservation of Energy (cf. Maxwell's Treatise on Electricity and Magnetism, section 856). This Weber Potential is: and therefore essentially different from the Gerber formula. Accordingly, since it conforms to the Con- servation of Energy, the Weber Potential alpne is admissible in a valid physical theory of the Universe. Incidentally it may be noted that the Weber Potential corresponds to a wave-field, and thus points to the Electrodynamic Wave-Theory of Physical Forces. Recently the writer was asked by an astronomer how he came to take up the New Theory of the Aether. The answer was that such hopeless confusion existed in this vital subject that a valid clearing up of the foundations was necessary to our progress; and as others had not been able to carry it out, the labor finally was devolved upon the present author. It will therefore be allowable to trace a few features of this progress which appear to me somewhat remarkable. In view of the many researches heretofore carried out in the theory of wave-motion, it will always seem very extraordinary that earlier investigators were not led to the simple relationship between the mean velocity (») of the Corpuscle of a Monatomic Gas and the Velocity (V) of a wave in the Gas, AN namely (cf. 5079, p. 234) V=^/,7tV. (3) Yet in reaching this Theorem it was not sufficient merely to notice the geometrical ratio theoretically existing between the paths of oscillating particles describing a semi -circumference while the wave traverses the diameter, which is 72?^: on the contrary, it was deemed necessary, as a physical pre- caution, to confirm the ratio from the best experimental data of six actual gases, with the following results: Vlll IX there exists a greater degree of rarefaction, or the consequence of the impulsive force of some fluid medium." This is a good summary of Newton's conception that the Aether is heterogeneous, which we now establish by definite mathematical and physical demonstration, (AN 5044), showing that in tri-dimensional space waves propagated or reflected from the particles of matter necessarily produce such outward in- = = A crease of density, a vr, owing to the law of Amplitude k/r, with the central force /== k^/r^. Accordingly, the Potential of Gravitation (Oeuvres Completes de Laplace, Tome X, p. 348) y ~ rrr a dx dy dz J J J V{x- x'y+if-yy+iz-zf ~ r r radxdydz JJJ r , , varies as the reciprocal of the Aether Density, as centrally thinned out by wave-action. The Potential is thus shown to be an accumulated state of stress incident to the triple integration for the superposed Wave-Amplitudes of the various Atoms, Ai^=ki/ri. Comformably to Newton's impression, Arago points out that the Aether tends to move towards the planetary bodies, yet under the increased amplitudes of the receding waves encountered towards — move these centres, it is so churned up or thinned out, that it does not really only exerts a steady stress in that direction, thus yielding an energy flux or gravitative force proportional to the energy of = the vibrating Aetherons and therefore proportional to the square of the Amplitude A^ k^/r^. The only — way to decrease this central stress is to remove the matter of the planet on which the waves depend, the motion of the Aetherons in Collision with the Atoms in some way generating the receding waves, or renewing them from the incoming waves already existing and incessantly propagated from the other bodies of the Universe. The infinitude of vibrating Atoms in each of an infinite system of bodies renders the wave-field infinitely complex; but from any one planetary mass, the receding waves pursue paths of Least Action, and the state of the central gravitative stress therefore is perpetual. It will always appear wonderful to investigators that the brilliant Maxwell should have made the unaccountable slip of imagining Gravitation due to a pressure in the direction of the force, and an equal AN tension at right angles thereto (cf. 5048, p. 163-164). It appears that prior to the publication of these Papers, English physicists never questioned Maxwell's erroneous assumption; and thus they handed down his errors for half a century, when the truth of the matter could have been noticed and verified by any good student of Mechanics. For Maxwell's postulated stresses were dynamically impossible; and although the learned Professor Minchin of Oxford, in 1886, found that they would not explain Gravitation, he did not suspect the error underlying them, nor remove it. The learned Editors of Maxwell's Scientific Papers, in the two imposing volumes published by the University of Cambridge, 1890, equally failed to notice what was required to balance the centri- — fugal force, as simply and clearly explained by Huyghens and Newton over two centuries before. The mischief thus done came in time to be spread over the entire world, and vividly illustrates the perversion of thought which may arise from the slip of a great authority. The slowness of our progress under these circumstances is less remarkable than it might seem at first sight. Looking to the future, for valid and simple conceptions of the Cause underlying Gravitation, we consider the new explanation of Acoustic Attraction (AN 5130, p. 342) to be especially suitable for disclosing vividly the unseen wave-processes operating from star to star, in straight line minimum paths, throughout the immensity of the heavens. This wave-field of Gravitation is dealt with more fully in the Sixth Paper, and fully illustrated by plates admitting of one interpretation and only one. Thus we establish the Cause of Universal Gravitation by necessary and sufficient conditions. The proof therefore is absolute and always will remain incontestable (AN 5140, p. 95-127). It appears that the discovery of the Cause of Gravitation now rests on six classes of phenomena: 1. The Fluctuations of the Moon's Mean Motion, Dec. 10, AN 1916, (cf. 5048, p. 159). 2. The New and Direct explanation of Acoustic Attraction, in which the wave-process is rendered AN visible to the eye, 1916, (AN 5130, p. 341-42, 5140, p. 98-100, plate 7). 3. The Proof of the Cause of the Distortion of the Equipotential Surfaces, about two equal Stars, 1917, at length somewhat more fully developed in the Sixth Paper, 1921, (AN 5140, p. 95-127). 4. Majorana's Physical Experiments on the Absorption of Gravitational Wave-Action by a layer AN of Mercury, 1919, (cf. Philosophical Magazine for May, 1920, also 5079, p. 301-302). 5. The experiments by Dr. Chas. F. Brush of Cleveland, Ohio, (Proc. Am. Philos. Society, — Philadelphia, vol. LX, no. 2, Jan. 1922) showing that under conditions otherwise identical the Earth's at- traction exerts a different grip on pendulums of different metals a Bismuth, pendulum gaining rapidly on one of Zinc. In discussing the Kinetic Theory of Gravitation Dr. Brush adopts the view that the energy of the Aether is in wave- form, in other words, the Wave-Theory. 6. It has long been recognized that Earth Currents, Aurorae, etc., recur periodically with certain Solar disturbances. The writer has now (Sept., 1921) obtained a new and direct proof that Aether waves upwards of 2400 meters in length are continually received upon the Earth from commotions in the Sun. These are long enough to pass through the solid body of the Earth with but slight refraction, dispersion and absorption. And as wireless waves of corresponding length are bent around the globe by the re- AN sistance of this solid body, (cf. 5044, p. 71), we thus have observational proof that Gravitational waves, such as are modified by our globe to produce the Fluctuation of the Moon's motion, do really exist, and can be experimentally studied in Radio-telegraphy. Accordingly our. present proof that the Cause of Gravitation is to be found in Wave-Action is most ample: and we may safely predict that further investigation will only confirm the results indicated by the sextuple proof above cited. The Wave-Theory of Magnetism outlined in the Third Paper is treated with greater rigor in the Seventh Paper. The Harmonic Law there developed definitely connects the Magnetism of the Earth with Universal Gravitation. Extending Gauss' method for calculating the amount of Magnetism in the Earth, we compute the amount of Magnetism in the Sun! It appears that Magnetic Action is conveyed not in right lines, like Gravitation, but along the Curved Lines of Magnetic Force; and thus the new Law of Nature very appropriately becomes a geometric tribute to the memory of the great mathematician Gaussl AN In the Fourth Paper, 5085, will be found the Correction of a fundamental difficulty in the Wave- Theory of Light which has stood for a full century. Poisson's Geometrical Theory of the nature of the vibrations in the Aether is fully confirmed, and harmonized with the most refined optical phenomena. ^ = The longitudinal component in Light is shown to be utterly insensible to observation, I : (66420- 10*), AN (cf. 5085, p. 427-428, footnote). The removal of this longstanding difficulty in the Wave-Theory of Light, and its harmonization with the Theory of Sound, as Poisson always held should be possible, is a triumph of no ordinary character. Attention should be called to the simple explanation of the Michelson-Morley experiment of 1887, by means of the Kinetic Theory of the Aether, (AN 5048, p. 181-183). No change is required in the dimensions of moving bodies, and such assumptions as Fitzgerald's Hypothesis are shown to be un- authorized. As the Aetherons move with the velocity of 471000 kms the state of the wave-field is instantly adjusted to any state of steady motion; and thus there is no such thing as the Earth moving through the Aether. At all times the Earth carries its wave-field with it, adjusted to perfect Kinetic equilibrium; and thus the Michelson experiment is perfectly explained, without any Theory of Relativity whatever. The outstanding motion of Mercury's perihelion is explained by an absorption of wave-energy, like that noticed in Majorana's experiments, and harmonizing still better since Grossmann has shown, (AN 5115), that the outstanding motion is less than 43", between 29" and 38" per century, AN with 14:'5 still to be deduced for the propagation in time, according to Weber's Law (cf. 5048, p. 137). Accordingly, in the Second Paper, (AN 5048), we show that the whole Theory of Relativity is a foundation laid in Quicksand. A discerning investigator who has studied this new aspect of the Kinetic Theory of the Aether, with the resulting abandonment of the Theory of Relativity, could now say with Laplace, in dealing with another matter: „I have no use for this Hypothesis". XI In removing the mystery of the Michelson- Morley experiment, without the hypothesis of the Earth moving through the Aether, and therefore without Relativity, we solve at the same stroke of the pen the historical difficulty of the Aberration. The solution of the difficulty of the Aberration is simply AN the parallelogram of motions, and thus as clear as any theorem in Geometry (cf. 5048, p. 183). The problem of the density of the Aether is found to be capable of direct and simple solution by the following process. It is fully established by precise Laboratory experiments that Hydrogen propagates Sound four times faster than Oxygen, which is a gas 16 times denser. The Cause of the rapid velocity of Sound in Hydrogen is therefore the lightness and high molecular velocity of the molecules of that gas. Now the Aether propagates wave motion 217839 times faster than Hydrogen, when the latter is corrected for a Monatomic constitution. Therefore Hydrogen is (217839)^ times denser than Aether, or the Aether has 1 : 47453880000 of the density of Hydrogen, making the Aether's absolute density AN 1888.15-10-18 (cf. 5079, p. 236). The argument here developed from exact experimental data in the Theory of Sound thus settles the question, without raising any other perplexing problem. For just as the four times slower propagation of wave motion in Oxygen, compared to Hydrogen, indicates that the Oxygen is 16 times heavier; so also the Hydrogen must be held to be (217839)^ denser than Aether, which propagates waves 217839 times faster. In view of the simplicity of this reasoning, it is strange that the Aether should have been spoken of by certain electronists as 2000 million times denser than lead! No such result is authorized by the laws of experimental Physics; and all such inference is as mischievious as it is contrary to our Common Sense. Out of this New Theory of the Aether, in which each body carries with it a wave- field, requiring adjustment every time the velocity changes, has grown a new theory of Inertia, Momentum etc. The adjustment of the wave-field is treated of briefly at the end of the Third Paper, (AN 5079, p. 299); and as it explains Inertia, Momentum, etc., it is especially worthy of the attention of natural philosophers. In the Fifth and Sixth Papers we have dealt with Molecular and Atomic Forces. These Forces are traced to short waves in the Aether, by an argument from the theory of Physical Continuity which will be found difficult to evade. If wave-action be the Cause of one of these forces, it will also be found to be the active agency in the others. Thus we have been able to throw much light on the secret of Capillarity and of Vital Forces, and have worked out the source of the awful power noticed in Explosive Forces, and in the mysterious forces of Chemical Affinity and of Cohesion which bind together the Molecules of an Elastic Solid. This view is strikingly confirmed in a recent development of the New Theory of the Aether, 2"'' Postscript to the Sixth Paper, and is so notable that I forbear to enter into elaborate comment. But we may point out that the Sextuple Integral defining the Molecular Strength or power of resistance of a solid body to molecular displacement is given the form i2= CUr^-hr^+k)dr+cY^-Jds=\^-^+kr+C^^-s. (^)---(5) As the integration is to be extended from r^ to ri, and over the whole of this range the functions are both finite and continuous, we may subdivide the range into parts for the entire region of stability, r^ to r^, thus: a -r 7- -hkr-hC (6) 5 4 Now we see by the accompanying (see p.xii) Fig. a, from the 2"'' Postscript to the Sixth Paper that some of these areas are positive, as in the whole region between r^ and r^. These positive areas correspond to the accumulation of attractive forces. When the molecular distances accord with this region, and the oscillations do not carry the particles beyond the range r^-r^, the wave action only binds the molecules more solidly together. This is a state of entire stability, as in typical elastic solids such as Stone, Steel, Diamond, etc. Xll But if the parts of the molecules come into close contact and so oscillate as to range from r4 to r^, repulsive forces begin to assert themselves quite powerfully; yet the stability may be secure, at least until — a distance r, smaller than r^ is approached, at which the repulsive forces rapidly become infinite. This region of excessively close contact, r^-r-^, is the danger zone, because the repulsive forces in- crease asymptotically. Thus when the molecule has its parts suddenly rearranged, and they come into such close contact, the stability it dissolved and the reaction gives a wide oscillation beyond r^, so that evaporation or an explosion may follow. Such sudden outbursts may occur from waves of Heat, or the waves of an electric current, or when the Atomic Structure of cer- Fig. a. Illustration of the curve of molecular forces 8 Wldr=f, the unessential parts outside the limits fi-r^ being indicated by pointed lines. — tain molecules is geometrically rearranged, thus breaking down in becoming more compact. The reaction incident to this sudden exertion of repulsive forces yields of course a tendency to a violent explosion the degree of violence depending on the closeness of the contact in the molecular rearrangement. As the explosive force increases asymptotically at small distance, we see that the most terrific concentration of power resides in certain atomic and molecular structures. This power comes from the Aether itself, as already explained in paragraph (iii) of Section 5, of the Sixth Paper. Yet the evaluation of the Sextuple Integral for i2, in the Z""* Postscript, has given us a better grasp of the extreme power of Molecular and Atomic Forces, because we see from the Curves why the integration, giving the ex- plosive action, rapidly becomes infinite. Viewing the Aethereal Medium in its larger aspects, chiefly as the vehicle for propagation of waves, it appears surprising that heretofore only three authors have investigated the elastic constant: 1. Newton, Optics, 1721, p. 326 e= 490000000000. = 2. Sir John Herschel, Familiar Lectures, 1867, p. 282 e 1148000000000. AN 3. See, 5044, p. 62 £= 689321600000. In this Elastic Constant of the Aether rests the power of the stresses exerted through this medium in the form of Physical Forces; and the interactions of the waves in traversing the various bodies give them their molecular and other physical properties. Maxwell had developed the Theory of the Medium to an enormous extent in electrical and magnetic phenomena; and he even concluded that the forces observed in Nature are due to stresses in the Aether. But owing to his premature death at 48 years of age he had not formulated any modus operandi as to how such stresses could arise in the Medium, = nor studied the Elastic Constant of the Aether, s 689321600000. Accordingly we have labored to extend and to improve the work of Newton and of Maxwell, and endeavored to give a working theory of the chief Forces observed in Nature. In conclusion it only remains to call attention to the two Memoirs on the Earth, and especially the Memoir on the Sun and Variable Stars. This latter investigation is so very remarkable that it can scarcely fail to be of the widest interest. It is not often that one can bring to light the true physical cause of so many great mysteries of three centuries since the age of Kepler and Galileol The author's most grateful acknowledgements are due above all to Mrs. See, for the loyal support of an unfailing faith in the outcome of this very extensive investigation ; to Professor Dr. H. Kobold, for his indefatigable labor and care in supervising the publication, admidst many difficulties ; to Mr. W. S. Trankle, who has aided so greatly in completing the work, between the numerous engagements of the public service. Starlight on Loutre, Montgomery City, Missouri, 1922 May 8. T.J.J. See. : Abdruck aus den Astr. Nachr. Nr. 5044 — (Band 2H April 1920) New Theory of the Aether By T. y. y. See. (First Paper.) I. The Medium of the Aether is necessary for Jeonveying Physical Action .across Space. A superfine medium associated' with the stars and with the light of day, known as the Aether [Aidt'jQ), has been universally recognized since the time oi Homer (Iliad, XV, 20, and XVI. 365). During the last three centuries the greatest natural philosophers and mathematicians, from Huyghens, JSfewton, and Euler to Maxwell, Lord Kelvin and Poincari, have regarded this aetherial medium as a necessary condition for the action of physical forces across space. In his Me''.eanique Celeste 4-54 1> 1896, 7zwcra»(/ expresses the general opinion thus: »Les theories les plus recentes de la physique donnent -lieu de croixe que les attractions des corps celestes ne peuvent Be transmettre a distance que par I'intermediaire d'un milieu, sans doute I'ether. Mais on ne connait rien encore sur ce mode de transmission. II parait probable que le meme milieu sert de vehicule a des actions electriques ou electromagnetiques«. Notwithstanding the very secure foundation for a valid theory of the aether erected by the labors of the most eminent iSeometers and natural philosophers since the age of Newtqn, a strange tendency has arisen within recent years, for abandoning the aether as an unnecessary hypothesis. Whether this reactionary tendency is based upon adequate grasp of the geornetrical ^nd physical considerations involved may be doubted by the more experienced natural philosophers of today. At any rate we leave this to the judgement of those investigators who follow the argument here developed. In their treatise on Magnetism and Electricity, London, igi2. Brooks and Poyser, who were inspired by the electronic Aleories emanating from Cambridge, express themselves thus »In this book, we have implicitly assumed the existence rf a medium, which is the seat of the phenomena denoted by the terms electric and magnetic lines of force. It may, however, be mentioned that at the present moment the various questions associated with the ether give rise to problems of great complexity and difficulty. The experimental knowledge acquired during the last twenty years, taken in conjunction 'with recently acquired knowledge regarding the .electron' and the constitution of matter, leads to apparently irrecon- — cilable results, and the real nature of the ether if it — exists at all in the old sense of the word must be regarded :as absolutely unknown. For instance, if the ether is in- compressible, as it is usually assumed to be, we are driven, by one line of argument, to the conclusion that it is 2000 jmillion times denser ^) than lead and possesses enormous energy of internal motion. On the other hand, if it is compressible, it may be much rarer than the rarest gas. There I is no intrinsic difficulty in either view, but at present no method is known by which we may hope to discriminate between them. The whole subject of the ether is in that state of uncertainty and apparent confusion, which in other branches of science has usually preceded some great advance in knowledge*. Such an attitude as the above, by physicists of recognized authoritative connections, is confusing enough; but an even more bewildering doctrine has been put forth by Eiitstem, and quite widely adopted in England, though it generally is rejected in America. The english observers of; the total solar eclipse of May 29, 1919, found some evidei^ces of a deflection of the light of stars by the field of the sun, but it was by no means conclusive, and the weakness of the whole Theory of Relativity was impressively pointed out by 'Dr. Silberstein, (Observatory, November 1919, p. 396-7), who showed that Einstein'?, theory will not account for the refinement of moving perihelia, and would even permit a planet or comet to move in a straight line, under the gravitative action of the sun. In view of these facts Dr. Silberstein justly says that the Einstein theory stands or falls by the Evershed and St John spectral observations, which are ample, yet do not confirm the theory. In an interview at Chicago, Dec. 19, 1919, Professor A. A. Michelson, the eminent authority on light, openly rejects Einstein's, theory, because it does away with the idea of light traveling by means of vibrations in the aether which is supposed to fill all space, t Einstein thinks there is no such thing as aether«, remarked Michelson. »He does not attempt to account for the transmission of light, but holds that the aether should be thrown overboard* In view of the confusion of thought introduced by the electronists, on the one hand, and by the Einstein pure — mathematicians, on the other, both extremes leading to ideas not appropriate to the facts, which Dr. Whewell, History — of the Inductive Sciences, 1847, I.8i, showed was the cause of the failure of the physical sciences among the greeks it seems highly important to enter upon an account of certain unpublished researches on the aether made by the present writer during the past six years, omitting so far as possible the results already available in volume I of the Electrodynamic Wave-Theory of Physical Forces, Boston, London and Paris, 1917. And first we shall show that the aether is necessary for holding the planets in their orbits, from the established law of the centrifugal force. This centrifugal motion must be counteracted, otherwise a planet can not be made to curve the path at every point and thus revolve in a Keplerian ellipse with the sun in the focus. ^) In a future paper a conclusive criterion will be given for rejecting this claim of a large density for tlie aether. : 5044 52 It is well known that the centrifugal force is given by the expression, f =z mv^lo (i) m where is the revolving mass, v is the instantaneous velocity, and q the radius of curvature of the orbit. As the planetary orbits and the orbit of the moon are not far from circular, we may with sufficient approximation calculate the centrifugal force for circular orbits. In the case of the earth's attraction for the moon, it suffices to take the earth's weight in metric tons, the moon's mass == 1/81.45, and the distance of the rtioon 60 terrestrial radii, so that the weight at the earth's surface is to be reduced by the divisor 3600. Then, as gravity balances this centrifugal force, we have for the at- traction of the earth on the moon /= (s.9S6292Xio")/(8i. 45x3600) = 20.3137 X lo^'' metric tons. (2) This enormous tension would require for its support the full breaking strength of a weightless solid circular column of = steel 645 kms in diameter, when the steel has the tensile strength of over 30 metric tons to the square inch 6.4 sq. cms, and such a small bar of steel would thus about lift a modern battleship of the largest type. The tensile strength of the above single column, 645 kms in diameter, would be equivalent to about 5000000000000 columns of such = weightless steel, each of one square foot cross section, g2 2 sq. cms, or about one such column to each area 16x16 256 sq. feet of a hemispherical cross section of the earth. So much for the stresses which control the moon's motion. But the gravitational attraction of the sun upon the earth is very much more powerful than that of the moon. , The attraction of the sun upon the earth is of course equal to that of the earth upon the sun, which is easily seen to be /= 332750/(23445)^x5. 956292X10" 3.60572X10^^ metric tons (3) where the number 332750 represents the sun's mass, in units of the earth's mass, and 23445 is the sun's mean distance, in units of the earths radius. This attraction of the sun on the earth is equivalent to the tensile strength of 1 000000000000 weightless cir- cular pillars of steel, like that discussed above, but each having a diameter of 30 feet, about 9 metres. This is equivalent to the tensile strength of a forest of weightless steel pillars, each 11 inches or 28 cms in diameter, on each square foot of a hemispherical cross section of the earth; so that the surface of the globe would be almost covered with these cables of steel. Such calculations of the enormous gravitative power of the heavenly bodies were first brought to my attention by Professor Joseph Ficklin, of the University of Missouri, about },?> years ago, and have never been overlooked in my subsequent studies of the cause of gravitation. Now with these concrete figures before us, we see that the cause as- signed for gravitation must be adequate to sustain these tre- mendous forces, miraculously pulling like stupendous cables of steel,, imagined as weightless as spider webs, yet stretched to the utmost limits of their tensile strength across the celestial spaces, for holding the planets in their orbits. Accordingly Einsteini, proposal to do away with the aether is chiefly remarkable for the lack of understanding of the physical universe which it displays. Sir /s-ff^t -^' "'j' himself denounced those who believed action could oc(^ across empty space as not having a competent faculty | thinking in philosophical matters. In his letter to Benn 1692-3, Febr. 25, he says: »That gravity should be innate, mherent and essentj to matter, so that one body may act upon another at| distance through a vacuum, without the mediation of anythj] else, by and through which their action and force may conveyed from one to another, is to me so great an absurd that I believe no man who has in philosophical matter competent faculty of thinking, can ever fall into it. Gra must be caused by an agent acting constantly according certain laws; but whether this agent be material or immateri: I have left to the consideration of my readers*. In a paragraph cited below, Maclaurin tells us t Newto?! held gravitation to be due to impulses of the aethi but could not make out exactly how they arose; and f passage shows that Newton did not regard this medium aj'; ordinary material. a) It is shown below that the elasticity of the aeth is 689321600000 times greater than that of our air in p portion to its density: it has therefore enormous power contraction, if any natural process be at work to cause to collapse. b) It is shown in the Electrodynamic Wave-Theory I Phys. Fore. I, 19 17, that between any two sources, as t sun and earth, the waves so interpenetrate, with rotatio: in opposite directions, as to decrease the stress and cau; collapse of the medium between the sun and the earth; a this therefore develops an enormous tension, with njaximu] stress in the right line between the bodies, while beyo: them there is corresponding increase of stress and thus external pressure also overcoming the effects of the centi fugal force, and compelling the pldnet to follow the Kepleri ellipse about the sun in the focus. c) It is shown in section 7 below, that the potential is simply an expression for the total accumulated stresS(*l-1 due to the waves from all the individual atoms of a body, ^ each wave following the law of amplitude, ; = A klr (I and giving an element of force, as in gravitatipn, , }. Accordingly we see that Laplace's, definition of the potential, 1782, points directly to the wave-theory: d) Therefore it is natural to hold that gravitation a wave phenomenon in the. aether, and to dismiss all oth I hypotheses as not fulfilling conditions essentia] to a tri physical cause. This wave-theory oi gravitation will give new ground for the deflection oi the light of stars when tl paths, of their rays pass through the gravitational field the sun, as indicated in the eclipse of May 29, and reporte at the meetings of the Royal Society and. Royal Astronomic; Society, Nov. 6, 1919, e) It will be shown below that both the density an rigidity of the aether increases as we go outward from th sun, according to the laws 53 5044 54 = D vr E=.v'r. ' ,(7) Accordingly the velocity of the waves remains approximately constant, (Electrodynamic Wave -Theory of Physical Forces ;l.'i4-i57, 1917) V= = CV.{ElD) CV[v'rlvr). (8) But experience alone can determine whether this condition holds with geometrical rigor, or whether along the actual path, containing :dififuse coronal matter, the stationary condition, = (5jd. o (9) may not lead to a sm^H deflection of the original path of light. f) Such an increase of density in the aether, as we recede from the sun was suspected by Newton in , 172 1, (3"* edition of Optics, p. 325). It is of authentic record that Newton hAi&yeA gravitation arises from the impulses of a subtile aethereal medium, but he»was not able, from ex- periment and observation, to give a satisfactory account of this mediiAn, and the manner of its operation, in producing the chief phenomena of nature*, [Maclaurin, Account of TV^z^z/wz's ^.Philqsophical Discoveries, London, 1748, p. in), and thus he left the problem of the. cause of gravitation to future investigators. g) The observed deflection of the rays of stars passing near the sun, amounting to about i''7 5, may be most naturally explained by the action of the gravitational and mag^ netic wave -fields, under the influence of coronal matter, varying as the inverse fourth power of distance, and the arrangement of the density and rigidity of the aether, near the sun. An arc of i" at the sun's mean distance corresponds to an absolute space of 725 kms, i''7S to 1269 krns. In the presence of the sun's strong gravitational and magnetic fields, and the magnetized faint coronal matter pervading that wave- agitated region, - it is probable that a central refraction or deflection of the light, of this magnitude, somewhat analogous to an unsymmetrical Z^ifwa«-effect, may be. anticipated. ' The rotation of .the beam of polarized light by magnetism, in Faraday's experiment of 1845, would lead us to expect some action in the sun's coronal wave-field. h) As Einstein's predicted displacement of the spectral lines towards the red coiild not be confirmed by Evershed and Stjohn, who had ample telescopic pbwer to make this shift-effect at least 50 times the probable error of their measures, it cannot be presumed that the deflection of starlight passing near the sun is a confirmation of a purely mathematical theory. The deflection of the light must rather be explained by the physical propsrties of the aether, interspersed with faint coronal matter, varying as the inverse fourth power of the distance, in the region of intense wave- agitation about the sun. -i) At the joint meeting of the Royal Society and Royal Astronomical Society, Nov. 6, 1919, no one attempted to answer the weighty objections brought forward .by Dr. Silberstiin, who had made a careful study of Einstein's theory, and thus pointed out the bizarre conclusions drawn by some pure mathematicians who are prone to forget that the de- , flection of starlight near the sun is as purely a physical problem as the refraction of light in the earth's atmosphere. Now the sun's deflection of light is similar to refraction, but — very miriute, half of it being o"87S, as against 2000" in our atmosphere, or about 2300 times smaller. j) Since, according to the report of the observers of ; the eclipse of May 29, 1919, this minute deflection disappears, when the sun moves out of the path of the light from the stars lying behind it, such a temporary, effect cannot properly be attributed to » a warp of space*, but only to the refractive action of the sun's envelope.. When Newton observed the refraction of light by a prism' he had no thought of attributing the effect tp »a warp of space«; and one cannot but reflec^ how fortunate it is that the physical theory of astronomical refraction was perfected by Newton, Laplace and Bessel before such confusing terms as »fourth- dimension -time -spice- manifolds*, were introduced into science. . k) It cannot be held that Einstein's theory enlightens us on the motion of mercury's perihelion, because af least half a dozen explanations, some of them approved by Newton, Hall, Newcomb and Seeliger, are already known; and another simple one, involving no mysticisrtf and no rash assumptions, but following from definitely established physical laws, will be brought oufc in the present investigation. 2. New Law of the Density and Rigidity of theAether. To deduce the law of the wave amplitude (4) in tri- dimensional space, we proceed as follows. The displacement of any particle of a medium due -to wave motion, of a given wave length, is independent of the periodic time, and since the oscillatory orbits of the particles are described in equal times, under continuous' flow of the waves, these "orbits will be proportional to the displacements or other homologous, lines pertaining to the. periodic paths of the particles. Let m the velocities of the' moving particles be v, and their mass; then their kinptic energies will be represented by ^l^m v^. In the spherical expansion. of the aether waves, there will be no loss of energy in free space ; hence on two successive sphere surfaces of thickness Ar^ the energies are equal,' so. that we have: 47rr' V. ^, .1 \nr- ^]iin v''^ (ip); The kinetic energy of the vibratirig molecules varies inversely as the square of the, distance. But the velocity varies also as .the ampUtude, in simple harmonic motion: therefore, for the, amplitudes A' and A"j corresponding to the radii r' and r", we have by taking the sqtiare root in equation (10) / == A.', m, -.A" ' r",r , : = = A" A'r'lr" //'>" I \ , (11) ,(12) Accordingly.' the amplitude or side displacement becomes, = A klr. And V = Mir = (13) = + lll{alV\[x-x'Y^[y-y'Y [z-z'Y\\iixAy^z (14) which is the law of the potential first used by Laplace in ,1782. Thus it appears that if there be aether waves propagated Outwardly from any molecule of matter, the amplitude, or.maxirtium displacement of the oscillating particles of the aether, will vary inversely as the radius of the spherical wave-surface. ! « 55 5044 56 A partial development somewhat like this is given in certain treatises on physics, such as Wullner's, Experimental Physik, 1.784, and Mitchie's, Elements of Wave Motion, p. 11, but no importance is attached to the result, as in my Electrodynamic Wave-Theory of Physical Forces, 1.14-157, 1Q17. A = So accurately is this true, that when I brought this simple formula for the wave amplitude, kjr, before the Aca- demy of Sciences of St. Louis, in a public address, Sept. 21, 1 9 17, great surprise at the simplicity of the formula was expressed by such experienced investigators as Professor F. E. Nipher, and President E. A. Engler. Thus it is necessary to develop the subject a little more fully in the present paper, since no adequate discussion of the problem appears to be available in existing works on physical science. Let us now consider the arrangement of the density of the aether about the sun. i) Suppose we consider carefully the amplitude of the waves from the sun in any solar spectral line, such as that of sodium, V- It is evident that if we disregard all other radiations, and fix attention upon this sodium light alone, then as the wave amplitude varies inversely as the distance from the sun's centre, this amplitude of our vibrations con- stituting sodium light will be 219 times greater at the sun's — surface than at the surface of the earth since the earth's mean distance is 219 solar radii. 2) Similar reasoning will hold for the waves of light of the spectrum of such elements as strontium, barium, boron, calcium, hydrogen, carbon, iron, nickel, cobalt, copper, ti- A = tanium, etc. Thus all the light waves of all elements conform to the law : kjr. 3) All these chemical elements also radiate heat waves which follow the same law of amplitude. And for both light and heat the above law holds rigorously true. If there be any other type of waves in the aether, the same law will hold for these undulations also. 4) Now magnetism and gravitation have been referred to electrodynamic waves, in the author's work on physical A = forces, 19 1 7. If these waves exist, they also will follow the same law kjr; and that they do exist is shown by a variety of phenomena, which admit of no other interpretation. For example, the electrodynamic action of a current of elec- tricity is due to waves: thus arise electrical forces: also mag- netic forces, gravitational forces, etc. 5) Gravitation admits of no other explanation, while on this explanation we have an immediate insight into the fluctuations of the moon's mean motion, which so long proved utterly bewildering to astronomers. And there must be not only a cause of gravitation, but a simple one, harmonizing with electrodynamic action, in the generation of electrical forces, magnetic forces, etc. The electrodynamic wave-theory alone fulfills this necessary and sufficient condition, for the following special reason. 6) The aether is shown to have an elastic power 689 321 600000 times greater than that of our air in proportion to its density. Hence it will have practically, unlimited power of contraction, and thus be able to generate the stupendous forces required for holding the planets and stars in their orbits. 7) But this will be possible only if the aether is arranged according to the law of density (S= vr; which A = in turn will follow if electrodynamic waves recede from the sun, having amplitudes k/r. For the amplitudes in- creasing towards the sun's centre insures a decrease of density of the aether about that centre, owing to the increasing wave- agitation near the sun's surface. 8) Now all these mutual arrangements, favorable to the wave-theory, would not exist, unless that theory repre- sented a law of nature. Because not only are all facts of the aether harmonized, but also all the forces brought under the principles of the conservation of energy,' and of least action. Thus nature not only acts simply, but also by the most uniform processes throughout all space. It is not there- fore admissible to hold any theory of the aether other than that it is an infinite aeolotropic elastic solid, with the density arranged about the heavenly bodies to increase directly with A = the distance. And the wave amplitudes varying inversely as the radius, kjr, supports this theory, by geometrical considerations, which exclude every other theory of the medium for the interpretation of the forces operating through- out the physical universe. g) In the course of the article Aether (Encyclopedia = Britannica, Qth. ed., 1877), Maxwell calculates the density as ^ 1.07X10""'^'', thus implying homogeneity, and speaks of this medium as »a vast homogeneous expanse of isotropic matter. But it is obvious on reflection that this medium cannot be homogeneous ^) ; for in that case there would be no stresses in the medium for generating the forces which govern the mutual interaction of bodies throughout space. The mutual actions between bodies is an observed fact. In motion the bodies are everywhere found to describe ellipses, parabolas or hyperbolas about one another. Nothing but forces, due to tension between the bodies, and increase of pressure beyond them, could possibly produce this remarkable power for holding the planets in their orbits. 10) Thus forces imply waves, and waves lead to forces, when the mutually interpenetrating waves are so directed as to undo one another, and cause the collapse of the medium in the right line between the bodies. As the gravitational forces are of enormous intensity, it follows that the elastic power of the aether has to be tremendous, in order to generate the forces actually observed. 1 1) Accordii)gly, the existence of forces implies stresses in the aether : the stresses imply waves : the waves imply = heterogeneous density in the medium, which must vary with ' the radius from any mass according to the law a vr. There is no other view of the aether which can be held. Homogeneity of density would imply no stresses ; no stresses would imply no forces; no forces would imply an inert universe; which is contrary to observation and thus wholly inadmissible. ') In the Baltimore Lectures, 1904, p. 265, under date of Nov. 16, 1899, Lord Kelvin says: »We have strong reason to believe that the density of ether is constant throughout interplanetary and interstellar space'. This error is very widespread, and its persistence shipwrecks physical research 57 5044 58 12) The aether is therefore arranged about the sun with = the density foirowing the law, a vr, which results from A = wave-agitations having amplitudes, kjr The energy of the forces generated by these waves is proportional to the square of the amplitude, and therefore we have for the force, /-^V (is) which explains all the pbserved effects of gravitation, mag- netism, etc. 13) Now quite aside from the simplicity and continuity of the process of reasoning here outlined, it remains a fact that the wave-theory is adequate to explain all the observed phenomena of nature. The simple law of density of the aether here imagined may therefore be admitted to really pervade the universe. So far from being homogeneous, the aether is really very heterogeneous. Indeed, it is a gas, — behaving as an elastic solid an infinite aeolotropic elastic — = solid fulfilling the law of density, d vr, and of wave A = amplitude, kjr, and therefore yielding forces following f ^ the law, k'^jr^, as required by Newton in 172 1, for explaining the cause of universal gravitation. At the earth the density of the aether is 2ig times what it is at the sun's surface, because the earth's mean distance is 219 times the solar radius. But Newton s, formula for the velocity, v= CV\EJD] (r6) would give a change of velocity if the density alone increased, E while the elasticity remained constant. Now the v^ocity of light across the planetary spaces was originally found by Rdmer, 1675, from the eclipses of Jupiter's satellites, and subsequently confirmed by the elaborate researches of Delambre, on the motions of these satellites (cf. C. d. T. 1788, and Astronomic Theorique et Pratique, 1814). By discussing a thousand eclipses of the i^' satellite Delambre fixed the constant of aberration at 20^255, while Michelsons velocity of light, near 300 000 kms., and the solar parallax SfSo makes the aberration about 20^48. V Thus is about the same for the aether acros^ the diameter of the earth's orbit, and for the aether of the terres- trial atmosphere, in which the velocity has been investigated experimentally by Cornu, Michelson, Newcomb and others. Fig. I. Diagram showing graphically the decrease of the density of the aether towards the sun, o\ving to the asymptotic increase in wave amplitude. Accordingly, this observational fact requires us to hold E that increases in about the same ratio a's D, so that our V law of for the heavenly spaces becomes, V= CV[v'rlvr) (17) E and therefore ^= v' r. Thus both the elasticity and rigidity of the aether increase directly as the radius from the sun, or other heavenly bodies. The reason for this remarkable law is this: namely, the viscosity of a gas depends upon th^ friction of the mole- cules projected from one layer of gas into the adjacent layer, and vice versa. In the case of the aether the viscosity becomes rigidity. And with the increase of >the density of the aether particles there should be more molecules projected into the adjacent layers mutually, by the ordinary kinetic exchange, in strict proportion to the density. Thus the rigidity of the aether increases directly as the density, as in the stbove formula. It may be noted that by the formula of Newton, an increase of the density by the factor 2ig, without change in E, would lead to a reduced velocity of only about V15''' of the original. No such enormous difference, in the velpcity of light as determined by observations of Jupiter's satellites, and that found by terrestrial experiments, is admissible; and thus the above law of rigidity of the aether is approximately verified by the comparison of celestial and terrestrial obser- vations. But a more exact test of the value of V, from eclipse observations of Jupiter's satellites, taken as directly as possible across the diameter of the earth's orbit, for com- parison with the experimental value found by Michelson, is highly desirable. 3. The Relation between the Mean Molecular Velocity of a Gas and that of a Wave transmitted in such a Medium. The Philosophical Magazine for June and September, 1877, contains two important articles on the theory of gases by Dr. 5. Tolver Preston, and also notes on the conclusions then reached by the celebrated Professor y. Clerk Maxwell, with whom Preston was in correspondence. In the first of these papers, p. 452, § ig, Preston reaches- the following remarkable conclusion : »That the velocity of propagation of a wave (such as a wave of sound) in a gas is solely determined by, and proportional to, the velocity of the molecules of the gas; that this velocity of propagation of the wave is not affected by density, pressure, or by the specific gravity of a gasj or by anything else excepting the velocity of its molecules*. In the second Postscript, p. 453, Preston states Maxwell's conclusion as follows: » Professor Clerk Maxwell, to whom this paper was communicated, and who has taken a kindly interest in the subject, has worked out mathematically the velocity for a wave or impulse propagated by a system of particles moving among each other according to the conditions of equilibrium in- — vestigated in the first part of this paper the diameter of the particles being assumed so small as to be negligible compared with their mean distance, and the particles being further assumed spherical, so that there is no movement of rotation developed- at the encounters (which would involve loss of velocity)*. 59 5044 6o » Under these premises, the velocity of the wave was found to be Ys 1^5 (or 0.745) into the mean velocity of the particles. In most gases the velocity of sound is slightly less than this. This is referable to the movements of rotation developed at the encounters of the molecules (which cal- culably would delay the wave to a certain extent). In vapour of mercury, according to the determinations of Kundt and Warburg, the velocity of sound is exactly ^/g "I/5 into the molecular velocity*. , According to these announcements, the corpuscles of = the aether, viewed as a monatomic gas, should have a mean F molecular velocity of 3/ v's • 1.34 F, where F =: A 3Xioi''cms, the velocity of light. conclusicfti of such great importance, which received the approval of the lumi- nous mind 6f Maxwell, is entitled to profound attention. Thus I have had it before me for some five years, but only undertook the mathematical verification and physical test of this Preston- Maxwell theorem quite recently'; and, as my results differ slightly from those of Freston and Maxwell, I will give the process of test and verification employed. In order to confirm this theory I have compared the observed velocity of sound for the four leading gases which are best determined, with their mean molecular velocities, and firjd the following indications of experiment, without regard to the Preston- Maxwell theory. In the experimental data there remains a little uncertainty. For the older values of V and k^ the table yields for the corrected ratio a mean of 1.64, which is 0.07 above the theoretical value of 1.57. The newer data, preferred by Jeans, Dynamical Theory of Gases, 2°*^ edition 1916, p. 9-13 1, give a- mean value of 1.57, though the discordance between the results for the individual gases is somewhat increased. Gas : : 6i 5044 62 operations of the physical universe. Moreover, since light, heat, chemical affinity, etc., have long been referred to such waves in the aether, the more general electrodynamic wavetheory thus gives complete continuity to our theories of physics, thereby confirming the: correlation of all natural forces, and giving new physical grounds for the doctrine of the conservation of energy. In the closing paragraph to his celebrated Treatise on Electricity and Magnetism, 1873, Maxwell justly says that »'whenever energy is transmitted from one body to another in time, ther^ must be a medium or substance in which the energy exists after it leaves one body and before it reaches the other*. This also points to wave action, such as Gauss was considering in 1835, and of which Weber gave the fundamental law in 1846, Newton's \a.^ of 1686 beiftg a special case corresponding to circular orbits. In the Principia, Lib. 2, Prop. 48, Sir Isaac Newton deduces the formula for the velocity of waves or pulses propagated in an elastic medium, such as waves of sound in the air, y ^= CV{£/£>) . This is now written F= V{{i^/ia/n)-ii+at)} _= 33 1-76 m 1/(1-1-0.003665/) (18) , where t is the temperature; o: is a coefficient, 0.003665; = = ^ acceleration of. gravity, 98 1 cm; A normal barometric = D pressure, ,76 cm; cr 13.6, density of Mercury; .=== the density of air, 0.001293; ^"d A ^1.4050 (cf: Wiillner's Experimental Physik, S-SS^) is the ratio of the specific heat of air under constant pressure to that under constant volume,, introduced^ by> Laplace for harmonizing Newton s, theoretical formula with the observed velocity of sound in air. In many investigations it is "possible to determine the velocity with which waves are propagated, but it is not always" possible to determine independently the elasticity or density of the medium -^ we can only find the ratio EJD. This is partly true of thp aether, for-example, which transmits light waves or electrodynamic waves with the speed of 300000 kms -per second, but gives no process of fixitig the elasticity of this medium except by an independent calculatiori of the density, which, however; may be made by the process first used by Lord Kelvin in 1854, (Baltimore Lectures, 1904, p. 261-263), and afterwards adopted hy Maxwell, Scientific Papers, 2.767. In section 5 below we find, by the process here de- = scribed,, that at the sun's surface the density of the aether is ^ ^Xio'"-^* and the rigidity 1800. Using these con- stants 'm_ Newton's formula, we tnay verify the observed velocity of wave propagation V^ == 'l^{n/D) l/{ 1800/(2 X io~^^)} ^= 30 000 000 000 cms' 3X,io^°, the velocity of light. To compare a perfect monatomic gas like the aether with diatomic gases like the air, we use the formula for the velocity of sound V= = V{{s-/tG/B)-k-{i-i-at)] = l/[(9. 808X0. 76X 13. 59/0. ooi293).(i. 405) (i-Ha/)] = 331.8m 1/(1-1-0.003665/) at /° C. (19) This shows that the velocity of light is 904268 times swifter than sound. Squaring this number, and dividing, the result by . 1. 666/1. 405 =^ 1. 18624 we, get the immense number 689 32 1 6000Q0; which shows how much the elasticity of the aether, regarded as a monatomic gas, exceeds that of the air in proportion to its density^). In the Optics, 3"'^ edition, 17 21, p. 326, Newton makes this number 490000000000, which is 7'i per cent correct. In view of this excessive elasticity of the aether, in proportion to its very small density, compared to that of air, we can understand the almost inconceivable velocity of light. It is also necessary to bear in mind this enormous elasticity in order to understand why the aether is practically incom- pressible. When a wave begins to be generated, the distur- bance is propagated away so rapidly that the wave amplitude necessarily is small compared to the wave length. In the calculations of section 5 we have taken the wave length as 101.23 times its amplitude, which Maxwell, Lord Kelvin and Larmor consider a safe basis in all numerical determinations. The incompressibility of the aether is due to the very high mean velocity of the aether corpuscles, 47 12 39 kms per second, and their enormously long free path, 572959 kms: which makes the medium behave as an elastic solid for quick acting forces, but enables the corpuscles to hiove out of the way of the swiftest planets with a 10000-fold, greater speed. Owing to its enormous elasticity, the aether instantly adjusts itself to any state of steady motion, and thus this medium offers no resistance whatever to uniform celestial motions.. This circumstance fully explains a grave difficulty which has been felt from the age of Newton, and hitherto appeared utterly bewildering to natural philosophers. In connection with such extraordinary physical conditions in the medium, it may^ be useful to recall an account of the interior con- stitution of the sun given by Professor Newcomb in the En- cyclopedia Americana, 1904: »Yet another unknown factor is the temperature of the - interior, . . it may be 1 000000 degrees. As the highest temperature which it is possible to produce artificially pro- bably does not amount to 10 000 degrees, it is impossible to say what effect such a temperature would have upon matter. Thus we have two opposing causes, the one an inconceivable degree of heat, such that were matter exposed to it on the surface of the earth, it would explode with a power to which nothing within our experience can be compared, and a pressure thousands of times any we can produce, tending to condense and solidify this-intensely heated matter. One thing which we can say with confidence as to the effect of these causes is that no chemical combinations , can take place in matter so circumstanced. The distinction between liquid and gaseous matter is lost' under such conditions. 'Whether, the central portions are compressed into a solid, or remain liquid;, it is impossible to say.«' '-) In his thoughtful Familiar Lectures, on Scientific .Subjects, 1867, p. 282, Sir John Herschel- ^-/)'+(^-^')']}d^cbdz. (25) This expression has come into the most extensive use in all the physical sciences, and been of the highest service in the mathematical theory of gravitational attraction, magnetism, electrodynamic action, and also in theory of static electricity. But it is very remarkable that up .to the present time an expression of such universal use has not been given a clear geometrical or physical interpretation. The difficult} doubtless arose originally from beliefs like that expressed b} Laplace, in the opening paragraph of the Mdc. eel. I, 1799 that the »nature of force is now and always will be unknown* *) jfoule's value of molecular velocity of Hydrogen, which makes the aetheron perhaps a little too large. : : : 69 5044 JO = In the state of darkness, relative to the invisible aethereal since the element of mass dm adxdydz can be made so ' medium, existing at the close of the iS'*" century, Laplace doubtless considered it sufficient to deal with expressions small as to apply to every single particle or atom. V At first sight the mere fact that the potential as thus which give the forces acting on the planets, without inquiring defitied follows the law of wave amplitude in tridimensional into, the geometrical nature and physical mechanism involved space strikingly suggests that the wave-theory represents the in the generation of these forces, which were then believed order of nature. To find out by exact calculation what is to lie beyond the reach of the investigator. the probability of such a coincidence occurring by mere After the development of Faraday's Experimental Researches in Electricity, and Maxwell's mathematical interpretation of these results, very- different views came to be entertained by geometers and natural philosophers. Yet it chance, we may proceed as follows. Taking the expressions for two independent the amplitude and the potential, we have A=y = klx, = = V y M/x. curves, (27) was only the developments brought out in the »Electrod. Wave-Theory of Phys. Forc.<<, which seemed to justify definite expectations of. forming clear geometrical and physical' conceptions of the mechanism involved in the action of the magnetic and the planetary forces across space. Recently It will be noticed that they belong to the same geometrical — species both being rectangular hyperbolas referred to their — asymptote.s and can be made identical throughout,, from' = ^ == o to X 00, by introducing a summation 2, such — that 2^ M. these conceptions have been verified and extended, and Accordingly it appears that by the mere variation of = ^ therefore we shall here attempt to give a geometripaL-and a parameter the curves are made to coincide rigorously, point physical, interpretation of the potential which so long proved by point, from x ,0 to x c)o. Therefore the chances bewildering to the physical mathematician. In the »Electrod. Wave-Theory of Phys. Forc.«, 1Q17, p. 134, it is pointed- out that if waves be the basis of physical action across space, then the amplitude of such waves when propagated spherically and without resistance, in tridimensional = = against such a rigorous coincidence accidentally occurring , throughout infinite space, x o to x 00, becomes in- finity to one, or, 00 C=Jdx = oo (28) o space, will be given by the equation = A k\r . and thus its actual occurrence points unmistakably to. a true (26) law of nature. In an address to the Academy of Sciences of St. Louis, Sept.- 2 1, 1917, I gave this simple formula and pointed out its geometrical and physical significance. Professors F. E. Nipher, E. A. Engler and other physicists were present and showed great interest in the results announced, from which it would appear that this law had Jargely or entirely escaped the notice of earlier investigators. Now by comparing this expression (26) with that in (25) above, we notice that the wave amplitude has the same form as the potential defined by Laplace in 1782. The question thus arises : Can the coincidence in form be due to chance, or is the potential in fact an analytical expression for the total aether stress due to the superposition of waves from all the atoms, each of the waves being of the average wave amplitude, appropriate to the coordina.tes in the field of force about an attracting mass? To get at the truth in this interesting inquiry, we notice that Laplace's formula of 1782 integrates the mass of every particle of the attracting body, divided by its distance, which corresponds to a summation of the effects due to the superposed wave amplitudes and thus increases directly as the mass, each set of waves superposed from the atoms in any element a Ax&y Azlr,^ being independent - of all the rest, but the triple integral including the accumulated wave action of the whole mass = V Mjr == It seems therefore certain and incontestible that the potential represents geometrically and physically the totalaccumulated stress due to the whole mass under the average wave amplitude of the field about the attmcting body in question. It is to be noticed also that physically our definition of the potential confirms this conclusion'. In free space there is no cause to alter the spherical distribution of the waves, as they expand with 'increase of r. But in or near the shadows , of the earth, as shown in the »Electrod. Wave-Theory of Phys. Forc.«, a circular refraction of .the sun's waves will necessarily occur. The sun's potential varies, even at a constant distance, near the shadow of the earth; and owing to -this refraction, fluctuations of the moon's motion should arise near the time of, lunar eclipses, as fully explained in this work of 19 17. This circular refraction of the electrodynamic wayes in passihg t'hrough the earth's mass changes the potential or total accumulated stress due to the integration of the waves from all the atoms, under the average w^ve amplitude and distribution of the waves in the space near the shadow of the earth: and therefore also the sun's forces acting on the moon. Partially released from the sun's control, by the interposition of the body of the earth, with its refractions of the sun's wave-field, the moon tends to fly the tangent while traversing the region of the shadow cone, and thus arise the fluctuations of the moon's mean motion, connected with ^lll{alV{[x-x'Y^[y-y'Y+[z~z'Y\]AxAy^z. (25) lunar eclipses, which long perplexed Laplace, Hansen, Newcomb, The elements under the integral signs represent the ' individual potentials of every partick, and thus the potential increases directly as the mass whose wave-effects are integrated. This conforms rigorously to our conceptions of the Newtomaxi law of attraction, and involves no approximation, Hill, Brown and other astronomers. 8. Explanation of the Propagation of the Wireless Waves around the Earth. In the unpublished manuscript sent by the writer to the Royal Society in November, 19 14, which was the first 71 5044 72 outline of the »Electrod. Wave-Theory of Phys. Forc.« the following figure was used to illustrate the propagation of wireless waves around the earth. Fig. z. Illustration of the refraction of 'the wireless wave about the earth, and of light in a prism, owing to slower propagation of waves in dense masses. It is a sufficient explanation of this figure to say that it corresponds exactly with' the propagation of light through a glass prism, as shown in the figure of the prism above. The wireless waves travel faster in air than through the solid earth. The enormous elasticity of the aether, as set forth in section 4, prevents bodily rupture of the medium; and this secures continuity of the wave front, by beriding the surface backward near the globe, to correspond to the slower propagation in that dense mass. The retardation of the waves propagated straight through the earth causes the wave front to be bent and held back near the curved surface of the earth, and thus the wireless wave is refracted around the earth by the much greater resistance encountered in that solid mass. The correct theory of the bending of the wireless wave about the globe is thus the same as that of a ray of light by a prism, as sHown in the accompanying figure. The speed in the air is 4, but in the glass only 3, and thus there is a bending of the wave front through the angle. 6 when the light enters the glass, and also when it leaves the glass, as long recognized by physical investigators. The explanation of the refraction of light in a prism is directly confirmed by Foucaulf% celebrated experiment on the relative velocity of light in air and in water, (Annales de Chim. et de Phys. Ser. 3, t. 41, 1854), which has always been recognized as a crucial test of the wave theory of light, and which finally led to the total rejection of the emission theory. The simplicity of the above explanation of the propa- gation of wireless waves about the globe is thus remarkable. But it is also confirmed experimentally by observations made by officers of the American Navy, upon wireless waves sent from Mare Island to San Diego, California, and received by submarines lying on the bed of the sea, through a depth of some 30 metres of sea water. In some experiments with the receiving apparatus underground the same effect was observed. It appears that the earth also conducts the signals, so that wireless apparatus may be installed and used in deep mines, which would enormously increase the power of signalling in case of accidents interrupting communication by the shafts and tunnels, It is probable, however, that the irregularity in the structure and conducting power of the earth's strata would, somewhat handicap such iinderground signalling, yet not prevent the successful development of the method of signalling through the earth to the limited depths at which miners work. - The problem of explaining the propagation of wireless waves about the earth has hitherto challenged the ingenuity of the foremost mathematicians. It has been unsuccessfully attacked by Professor H. M. Mac'donald (Proc. Roy. Soc. 1903 and Phil. Trans, igio). Lord Rayleigh and Prof. H. Poincaiii\ (Proc. Roy. Soc. 1903). See aXso.Foincard's Lectures of 1908 (La Lumiere filectrique, vol. 4, 2"^ series, Nov. 28, Dec. 5, 12, 19, 1908, especially p. 323). Professor A. Sommerfdd (Ann. der Phys., vol. 28, p. 665, 1909) has shown that a surface wave should exist; and Professor J. W. Nicholson (in the Phil. Mag., March, April, May, 1910) has dealt with certain problems of the exponential factor of the wave amplitude, but none of these eminent mathematicians arrived at any satisfactory theory of wave propagation about the globe. In his well known work on the Principles of Electric'; Wave Telegraphy and Telephony, London, 3"^ editiori, 19 16, p. 826-851, Professor J. A. Fleming gives a full and accurate account of the difficulty experienced by these and other mathematicians. In this revised edition of 1916, Fleming gives the following: » General conclusions as to the mode of propagation of long electric waves round the earth «. »Summing up the conclusions so far reached by radiotelegraphists we may say tfiat the effect produced by a radiotelegraphic transmitter at a great distance, say 2000 or 6000 miles over the surface of, the earth, is a complex one in which several different actions play a part«. » There is, first, a propagation through the aether of a true space electromagnetic wave which is difiracted round the earth. The extent to which this contributes to the whole effect is,' perhaps, greater than was formerly supposed, but is yet an undetermined quantity. Some mathematicians are now inclined to attribute to it the major portion of the transmission by day«. »Then in the next place there is undoubtedly a contribution made to the effect by waves which have suffered a refraction equivalent to a reflection by ionized air at high altitudes, and a very small effect due to the decrease in refractive index of air as we ascend upwards*. » These causes tend to make the ray follow round the curvature of the earth and so assist as it were diffraction. It is to this variable ionic refraction that we must attribute the diurnal and annual variations in signal strength, and also the greater signalling distance by night as well as the irregularities attending the transition times of sunrise and sunset«. »Then in addition we may inquire how far any contribution is made by a surface wave of the type investigated by Sommerfeld, which is equivalent to an electric wave pro- pagated through or along the earth «. ' S044 74 »It has been definitely proved that we can receive signals from stations hundreds of miles away without any high receiving aerial, but merely by connecting one terminal of the receiving circuit to earth, and the other terminal to any large well -insulated mass of metal, whether inside or outside of a house does not matter*. If I understand the difficulties so lucidly outlined by Fleming, they will be found to have proceeded from the inadequate theory of • the aether heretofore in use, the dis- cussion being based upon diffraction around the earth, instead of upon refraction and dispersion within the denser mass of the earth, and thus a bending of the wave front. This will sufficiently justify this quotation, since it is essential that the difficulties heretofore encountered should be autho- ritatively described. The reader can then judge aS to whether a simpler and more practicable solution of this problem has been obtained. As to the feebleness of wireless transmission by day, I have reached the settled conviction that it results from the magnetic wave field of the sun. When this storm of waves fills our air by day, the wireless waves have great — difficulty in getting through, just as any system of regular water waves in a lake, used for signalling across it, would be almost lost in distinctness, owing to the surface chui;ning of the lake under the violence of a wind storm. The trans- mission is more difficult with the distance, and, after a certain distance, entirely fails. At night the sun's magnetic wave field is largely absent, and thus wireless transmission is much better. It only remains to add that the celebrated argument of Cauchy, to the effect that refractive' dispersion of light necessarily implies a granular structure in the transparent matter, is equally valid for showing that the aethereal mediurn itself is corpuscular. In his Popular Lectures and Addresses 1. 190, Lord Kelvin has modified Cauchy 1. theory of refractive dispersiori in his usual lucid manner. It is believed that the considerations adduced in this paper will render the arguments of both Cauchy and Kelvin sornewhat more definite and interesting. When the aetheron is so small, and moving so rapidly, the generation and propagation of waves in the aether is intelligible. The refractive dispersion, by the resistance to the waves from the much larger molecules of ordinary matter, is easily understood; and thus refractive dispersion implies in common matter, coarser granules than those of the mediurn itself, but yet points to the moving aetherons as easily deranged by the resistance of the waves dispersed. ~ It will be shown hereafter that resistance soon changes the form of the wave, and causes it to break up into two distinct parts, the larger having increased amplitude, and shorter length, hence encountering more resistance than the original wave. It is certain therefore that we not only have retardation in the propagation through the earth, but also dispersion of the fragmentary waves, and absorption of some of their .energy as heat. 9. Outlines of the Wave-Theory of Magnetism, with explanation ofthe mechanism of Attraction and Repulsion. For the sake of completeness the present investigation requires a brief notice ofthe cause of attraction and repulsion in magnets, and in electrodynamic action, as first outlined in the vol. i, Electrod. Wave-Theory of Phys. Fore, 1917. Accordingly we begin with magnetism, which the celebrated English physicist Maxwell had been so long engaged upon, but had--failed to solve at the timeof his death 40 years ago. The accompanying figure from the work of 1917 will illustrate to the eye, the essential character of a magnet, as A conceived in the wave -theory, of physical' forcfes. large magnet A is exhibited in the same field with two smaller magnets, B. In the first case unlike poles are presented, and we have mutual attraction. In the second case the poles presented are like, with the well known result of mutual repulsion. But how does this attraction and repulsion come about?. What mechanism is involved, and in what medium . does it work? Obviously the medium is the aether, because an electric current produces a magnet from a piece of steel wound in a solenoid, and because also the electrodynamic action of a current travels with the ve- locity of light, as was first inferred by Maxwell, and afterwards proved by ex- periment. a) In the case of attraction, it will o -o.--T> /<> Q '/ Q'- . .,jj. be seen that the waves from the small B '"^f^gvi'''' have the elements ofthe aether rotating in the opposite direction to the l»»TBI.i.|»». i. ijg„-., rotations in the more fully outlined waves from the magnet A. The plane waves A from are to be imagined, for the sake of simplicity, in the central plarje, or equator, and travelling away with the ve- — locity of light, for the reason just as- signed in electrodynamic action, by which magnets are produced. f^^iS^Srfi^ '/ Plane W«vo Morton I(/, As shown graphically by the curve traced just above the heavy waves in the figure, the amplitude of these receding theory of magnetic attraction and repulsion. waves decreases according to the law: . ,' 75 5044 76 = A kjr (29) and as the force due to -svave action is- siiown, in works on physics, to be proportional to the square of the ampHtude, ^72/2 we have for the force': f=Plr^ 1 \' (30) which is the form of law for gravitation, magnetism, and' all similar forces of nature obeying the law of the inverse squares. B Now let the waves from magnet interpenetrate the waves from magnet A, It will be seen that at every point of space the rotations of the elements of the two sets of waves are exactly opposite: the result is that the rotations B from magnet undo as far as possible the opposite rotatit)ns from. magnet A. Accordingly the stresses in the medium due A to rotations of the aether, in the field between and B, and also beyond A and B, are reduced : the medium is thus everywhere less agitated than before, and shrinks, so as to A collapse or contract between and B. But a collapse of the aether is equivalent to a contraction, and thus the two bodies attract as if held together by a stretched mass of India rubber. This is a simple and direct explanation of attraction. Nothing is postulated except waves like those known to exist in light and heaj, but here seen to be ex- actly parallel and somewhat differently 'directed from those of light and heat, which usually have their planes tilted in haphazard fashion. B) The cause of' repulsion is similar to that of at- traction, but in this case the poles presented are like; and if we examine the above diagram, we discover that when the waves from magnet B, 2°'' case, interpenetrate the waves from magnet A, the rotations at every point will be con- formable and in the same direction. The medium therefore at every point is more agitated than before. The amplitudes of the disturbed waves are thereby increased, and hence there is an increase of stress; ' and under the elasticity of the aether the result, is an expansion, of the medium, which . gives a miitual repulsion of the two bodies. ' This is a simple explanation of repulsion, and it had never been worked out prior to the researches published by the writer in, igiy. Maxwell waS unable to conceive of any mechanism for the explanation of attraction and repulsion - of' magnets, though he found that mathematical stresses of a certain type, yielding tension along the lines of force and pressure at right angles thereto, thus dynamically equivalent to, those outlined above, would account for the phenomena of magnetism. It is true that Maxwell believed that there are rotations around the Faraday lines of force, as Lord Kelvin had also rendered probable as early as 1856; but neither A'^&m nor Maxwell had seen that this would arise from the type of waves here outlined, though Faraday'^ experiment of 1845, on the rotation of the plane of a beam of polarized light, — when passed along the line of force, through a dense, — medium such as lead glass, should have suggested the 'correct theory of the magnetic, waves to Kelvin and Maxwell, as it did to me in ig 16. As Maxwell was unable to unlock the Secret of mag- netism, with both attraction and repulsion, it will not greatly surprise us to learn that he was utterly bewildered by the mystery of gravitation, and could not make a successful attack upon this most difficult problem. In fact no considerable progress as to the cause of gravitation has been made by other investigators since the time of Newton. As the subject of 'gravitation is immense, we must not enter upon it here, except to say that the evidence is most conclusive that it is a wave-phenomenon, closely allied to that of magrietism-, but differing from magnetism which - has a parallel arrangement of the atoms and what Airy calls (Treatise on Magnetism, 1870, p. 10) a duaHty of powers — two poles while gravitation is a central action only, owing to the haphazard arrangement of the planes of the atoms. It 'is well known that about 1822 Ampere first made electro-magnets out of common steel, by means of an electric current sent through a solenoid. The way in which the wire IS wound about the bar' being magnetized suggests, and, in fact, proves that the wire bearing the current has a wave-field about it. There is proof that the waves are flat in the planes through the axis of the wire: this conception harmonizes' all.,;: the known phenomena of magnetism, in relation to electro- dynamic action, and also harmonizes Ampire'% theory of ele- mentary electric currents about the atoms with the waVe-theory of magnetism above set forth. ; The wave-theory of magnetism explains all the phe^ •'; nomena of terrestrial magnetism, in relation to the periodic influences of ,the sun and moon, such as magnetic storms, earth. currents, the auror-a, and the semi-diurnal magnetic tide depending on the moon, of which ho other explanation is known. For the dependence of magnetic storms on sunspots consult a paper by the author, in the Bulletin Society Astr. de France, November, igi8. There has been such' a bewildering confusion of thought connected with the whole subject of physical action across , • space that it is necessary to bear ii> mind clearly the fun- damental principles of natural philosophy. In the well knoWn article on attraction, (Scientific Papers, vol. 2.487), Maxwell points out that in the Optical Que'ries included in the third edition of the Optics, 17 21, Newton shows that if the pressure of the aether^al medium is less in the neighborhood of dense bodies than at great distances from them dense bodies will be drawn towards each other, and if the diminution of pressure is inversely as the distance from the dense body, the law will be that of gravitation. Maxwell considers that , Newton's conception rests largely on the idea of hydrostatic pressure, as in incompressible liquids. But we have shown that the' amplitude of the waves, A =:kjr, with forces f^k^lr^, fulfills the condition which ^Newton held to be essential. 10. Integration of the General Differential Equations of an Elastic Solid, *hich applies to the Aether, when this Medium is viewed as an Infinite Aeolotropic Elastic Solid propagating Waves.; m As is usual in the theory of an elastic solid, let denote a function ' of the bulk modulus k, and of the rigidity «, «"=1^ that + ni^k ^l^n., (31) — Then k =^ m '^j^n, and this bulk modulus measures the elastic force called out by, or .the elastic resistance against, change of volume. On the other hand the »compressibility« ^ is measured by = i/^ il{m-^Un) (3 = : . : n 5044 78 Let «, /J, ;' be the component displacements experienced by a particle, so that when undisturbed the coordinates are x, y, z, and when disturbed x-^a, y-^fi, z-hy. Then a strain of any magnitude is specified by six elements: A B = (|)'*(|+0'*(87)' da 8a Vy Oz M \^ d^y-^' ):dz 8^/9^ dy \dz _~da(t-n) o ISoJ or the density is defined by the expression: = Q jlUn{m+n)]-{dXldx^dYldy^dZldz) . (51) This specifies the density throughout space of the infinite isotropic solid, that of the finite solid body in (41) being unity per unit of volume. R To reach Lord Kelvins result most directly, we let = denote the resultant of the forces, X, Y, ,Z, at any point [x, y, z], at the distance r V[x^-^y^-\-z^) from the oriigin, whether discontinuous and vanishing in all points outside some finite closed surface, or continuous and vanishing, at all infinitely distant points with sufficient convergency to make Rr converge to o as r increases to 00. Then the con- vergency of Xr, Yr, Zr to zero, when r is infinite, clearly; F=o makes for all infinitely distant points. Accordingly,- if 5 be any closed surface round the origin of coordinates, ' everywhere infinitely distant from it, the function ((5— V] is = = V zero for all points of it, and satisfies the equation V" (d— V). o for all points within it. Therefore d throughout , the infinite isotropic solid. Z Now let X', Y', Z' denote the values of X, Y, at any point [x,y, ^), and by a triple integration throughout all space, V we shall have for the potential or dilatation 6: -+-OD -HOO -4-00 = — (J i/[47i:(ot-i-«)],-J — —J 00 00 ^{dX'ldx^-^Ay'ldy'-^AZ'lAz')lV[[x-x'Y-^{y-y'Y-h{z-z'Y]-dx'dy'ds'. 00 (-52) For the element of the mass is p =i= i/[47r (ot-h«)] • (dX/d^'-t-dy/dy-HdZ'/dz') (53) and- the mutual distances of the elements of mass filling the element of space dx dy dz is r^V[[x-x'Y+[y~yY+{^-z'Y]- " •,,(54)' These expressions may be, rendered more convenient by integrating by parts, and noticing the prescribed con- dition of convergence, according to which when x' is infinite, ' , -+-00 -4-00 = J ^X'jV[[x-x'Y+[y-y'Y+{^-z'Y]-Ay'dz' o 155) — -00 CX3 And, therefore, for the three components of finite value, resolved along the coordinate axes, ' and integrated throughout ' all space, we have: -t- 00 -t- 00 -1-00 ' d=ilUn[m+n)]-^ J .^ [X {x-x')+ Y' {y-y')+Z' [z-z')]l V[[x-x'YHy-y'Y'+{^- e'Y]-Ax' dy' dz' . (56) — -co cso ^00 U = = w-¥W We may integrate each of the equations (38) in the same way, for a, /J, y respectively. displacements is: a ^= u-\- • ^ v-hV y The result for these W where u, v, w, U, V, (57) denote the potentials at [x, y, z) of distributions of matter through all space of densities respectively (m/4TTn]Sd/dx {m/47tn)dd/dy {m/4nn)dd/dz Xj^nn Yj^nn Zj^nn. (58) In other words the functions are such that throughout all space ...; :^<: = = = = ^^u-h{m/fi) dS/dx o '^/'U-hX/n o V^v-h{m/n] dd/dy o ^^V^YJn o = = \'-w^{mln) dd/dz o V" W-hZjn o . (S9) Z Accordingly, if X", Y", Z" denote the values oi X, Y, for a point [x", y", z"), we find -hoo H-CO -hoo = + a (i/47r«)J —-CXD —J00 ^ {m-dd"ldxf' 00 X")IV[[x-x"Y+{y-y"Y+{z-z"Y]-dx/' dy" dz" -f- 00 -h 00 -f- 00 = + + fi [ilAnn)l J l[fn-^d"ldy"+Y")IV[[x-x"Y {y~y"Y [z-z"Y]-dx"dy"dz" (60) 00 00 00 -hoo -+-00 -t-oo = r (i/4/r«)J -00 —J00 ^[m-dS"ldz"+Z")IV[[x-x"Y+[y-y"Y+[z~z"Y]-dx"dy"dz" 00 : : . 8i 5044 82 By substituting for = F1+F2COS/. (77) ;') It is only between the bodies in the line S-E that cos/= I, and the velocity of the relative interpenetration /^. = is a maximum. When the radii vectores meet at right angles, the angle 90°, and cos/= o, so that i3; V^, only, or V2 only, as the case may be. If the angle / exceeds qo°, the addition in (77) becomes negative, and the value of Hi is less than V^, or V^ separately. And when cos/= 180°, the addition gives O- =: Vi— V^ ^^ o . (78) d) From this reasoning it follows .that: i) If the medium contracts owing to the mutual interpene- tration of waves, the contraction will be a maximum in SE the right line where i2,- is a maximum. 1= b) The contraction will be zero when 180°, and thus the tension in the medium is wholly between the bodies, or on' either side of the line connecting them. It is zero 5£ in the line prolonged, but as the waves here superpose,, the pressure or stress will increase externally. e) This accords with experience in gravitational, mag- netic, and electrodynamic forces, etc. And.^as. the theoryof least action is recognized to hold generally in nature, this , geometrical plan of the contraction of the medium, under mutual wave interpenetration, must be held to conform to the rigorous criterion of least action or maximum wave inter-; penetration. This function attains the maximum with least action of the forces thereby developed ; and according to the geometrical methods of Weierstrass, this points to a rigorous mathematical law. f) It is not accidental that the mutual wave inter- penetration should be a maximum between the bodies, in the SE \\n& SE, a minimum in the line prolonged. For as the aether is under an elastic power of 698321600000 times greater than that of air in proportion to its density, the medium will always contract to the maximum extent possible, and thus pull in the right line connecting the' two bodies SE. Hence if the postulated waves exist, the waves superposed being accumulated with the mass, they will fully explain the stupendous gravitational forces which govern the motions of the planets. ri) That the waves exist is obvious from several con- siderations : a) Forces can only result from maximum tension in the line SE, and this implies interpenetration of waves ; for no other cause could produce this effect, whereas waves certainly would do so. b) Waves also make gravitational forces conform to other physical forces, according to the recognized law of con- servation of energy, c) When a kriown general cause exists, it must be held to be the true cause, in default of any other known cause. d) The probability is infinity to one that no cause other than a true one could fulfill all the geometrical conditions of gravitational forces without resting on the true laws of nature. 6) As the wave-theory harmonized all gravitational phenomena under the recognized criteria of least action, and without the introduction of any mystical hypothesis, it must, from geometrical and physical laws, be held to represent the true order of nature. In closing this first paper it remains to add that the second paper will deal with, the Fourier solutions of the j;ele- brated- equation oi Poisson (Traite de Mdcanique, 2.697, 1833). = d^ (DlAf a- V'® O [x, y, z, t) ® where is a scalar quantity, and a Ihe velocity of wave propagation. This applies to wave motion in normal gases, the aether, and an elastic solid. As the aether is a gas, but under such elastic forces, that it behaves as an elastic solid for quick acting forces, and is of infinite extent, while on the other hand Lord Kelvin's, integration of the general equa- tions of an infinite elastic solid likewise confirms this con-' elusion, the deductions thus brought out will establish the wave-theory with the required geometrical rigor. The out- standing motion of the perihelion of Mercury, and of the lunar perigee, together with the lunar fluctuations, under the Newtonian law, as generalized by Weber in 1846,; will har- monize every known celestial phenomena without the intro- duction of any mystical hypothesis. In the third and fourth papers I hope to give a simplified view of certain outstanding electrical problems and of Michelson and Lodger's experiments, and throw a very unexpected, but searching light on the nature of molecular forces. Thus the several fields covered will lead us to apply the wave-theory to such varied phenomena of nature, that it may not be without interest to both the geometer and the natural philosopher. I am indebted to Mr. W. S. Trankle, for efficient aid in completing these researches. Starlight on Loutre, Montgomery City, Missouri 19^0 Jan. 14. T. y. y. ^if^. Am Zusatz. 25. April sandte Herr Prof. See telegraphisch folgende Nachricht: »Have discovered from wave theory new method for determining density of aether, only advance since Lord Kelvin's method 1854. Now find density 472X10^-'^ against 438X10"^^ by Kelvin's, method. See.'-'- • Abdruck aus den Asir. Nachr. Nr. 5048 — (Band 211. Juni 1Q20.) I. Gravitational Action propagated with the Velocity of Light. In the first paper on the New Theory of .the Aether, AN 5044, we have showii that the existence of this medium is a necessary condition for conveying physical action from one body to another across the celestial spaces, and have given the elements of the kinetic theory of the aether -gas as the subtile vehicle of energy. Maxwell had a very clear conception- of this medium 47 years ago, when he pointed out, in the closing paragraph of the celebrated Treatise on Electricity and Magnetism, 1873, vol.11, p. 493, that * whenever energy is transmitted from one body to another in time there must be a medium or substance in which the energy exists after it leaves one body and before it reaches the other«. No better description can be given of the aether, as the vehicle of energy, than that just quoted.. And since Maxivell says that the energy must exist in the medium, after it has left one bo.dy, but before it has reached the other, owing to the propagation in time, we see that this energy obviously must be conveyed through the agency of waves travelling with the velocity of light, just as radiant heat from the sun and electrodynamic action travel with the same velocity, 300000 kms per second. From the celebrated letter oi Gauss \o Weber, March 19, 1845, [.Gauss, Werke 5-629) we learn that as early as 1835 Gaztss looked upon physical action across space as conveyed ^in time, and was trying to formulate a law of this action, but put it aside temporarily, .and only recurred to it when Weber had formulated his fundamental electrodynamical law, published in 1846: = + / [mm'lr'^) { r -(i^^) (dr/d/)- (2;-/^2) dV/d^^} ^ (j) The first term of this formula is Newton's law of gravitation, 1686, whilst the other terms take account of the effects of m induction in the relative motion of the two bodies and m'. The minor terms thus give the energy effects of the velocity and acceleration or change of velocity, under wave action, in the direction of the radius vector, as required by the present author's Electrodynamic Wave-Theory of Phys. Fore, vol. I, 1917. In the work here cited (p. 143-149) I have calculated the eff'ects of Weber's law upon the progressive movement of the perihelia, periplaneta, and periastra of the best known bodies of the solar system and of the sidereal universe. The 6^ tabulated is the progression of the orbital perihelia in a Julian century, owing to the propagation of gravitation with the velocity of light. Progression of Perihelia in a Julian Century, Weber's Law. Planets Mercury i4"Sii Satellites V Jupiter: 4*233655 Venus 2.9125 I 1. 8212385 The Earth 1.2964 II 1-14345° Mars 0.45619 III 0-715544 Jupiter 0.02 104 IV 0.40508 Saturn 0.0046 13 VI 0.068685 Uranus 0.00080395 VII 0.064658 Neptune Satellites 0.00026 I 5 VIII IX 0.034681 0.034128 The iSIoon (Earth) 0.00637 Saturn : Mimas 1.2403 Phobos (Mars) o,.o2 65 I Enceladus 0.966394 Deimos » 0.0 1 1098 Tethys 0.78066 Ariel (Uranus) 0.18439 Dione Umbriel »' Titariia » 0-13 23 s 0.080504 Oberon - » 0.060339 : : « 139 5048 140 above effect of Weber's law removes i4"5 of the total amount^), leaving outstanding about 2%"i instead of the 43" assumed in Einstein's, Theory of Relativity. The outstanding 2 8"5 can be explained by the transformation and absorption of wave energy from the atoms on the opposite side of the sun, yielding a law of attraction of the very form approved by Newton in the Principia, 1687: / == ;;i,„'/^20000001046 _ (2) This explanation of the motion of Mercury's perihelion is more fully discussed below. Such a result was long ago anticipated by Newton, and in 1894 carefully examined and proposed by ffall, and subsequently used by Newcomb and^ Seeliger. It therefore has the sanction of the most eminent astronomers, and as it rests upon a known physical cause, it involves no vague and chimerical reasoning such as underlies Einstein's mystical Theory of Relativity. Towards the end of this paper, we develop a new view of the experiments of Michelson and Morley, 1887, and of Sir Oliver Lodge, 1891-97, which results from the kinetic theory of the aether, originally outlined by Newton, 1721, approved \yY Maxwell &r\AS. Tolver Preston, 1877, and recently developed by the present writer, as shown in the first paper. This new view of the chief physical experiments on which the theory of relativity so largely rests may well claim the attention of natural philosophers. As bearing on the same question we treat carefully of the outstanding rnotions of the perihelion of Mercury and of the lunar perigee; and show that neither phenomena lends the slightest support to non- Newtonian mechanics. In fact, although the theory of relativity has occupied much space in scientific literature, and many treatises, memoirs, and other papers have appeared on the subject, it is impossible for a careful observer to escape the conviction that the whole development heretofore brought out is false and misleading, — — a veritable foundation laid on quicksand and that some day philosophers will wonder that such an improvised ab- surdity ever became current among men. Among the most pernicious of these temporary doctrines is FitzGerald's hypo- thesis, which under the kinetic theory of the aether is wholly untenable. A considerable number of persons are much impressed .with the admissibility of any doctrine which becomes current among contemporaries, yet. the study of the history of science shows that truth is neither dependent upon popularity, nor discovered by majorities, but by the few individuals who think carefully and frequently in complete isolation, and who thus attain superior vision into the deeper mysteries of nature. In promulgating his new System of the World, 1543, Copernicus describes his reasoning in daring to depart from the opinion of the majority: »Though I know«, he says, »that the thoughts of a philosopher do not depend on the judgment of the many, his study being to seek out truth in all things as far as that is permitted by God to human reason: yet when I considered*. he adds, »how absurd my doctrine would appear, I long hesitated whether I should publish my book, or whether it were not better to follow the example of the Pythagoreans and others, who delivered their doctrines only by tradition and to friends*. 2. The Effect of Resistance is to break up Long Waves into Shorter Ones and actually to increase the Amplitude of the Principal Cotnponent, as noticed in Breakers at the Sea Shore. In his celebrated work on Tides and Waves, Ency- clopedia Metropolitana, 1845, Sir George Airy obtained one of the most comprehensive and useful theories of wave motion ever developed. Airy's theory has the advantage of being intensely practical, because it applies to wave motion in a canal, water being the chief liquid found upon the earth, and nearly incompressible. The formula for the periodic time of the ^ waves is ^, (2^;^/^) (^4"-^/^+ i)/(^4'i^A_ i) . (3) It may be shown analytically that when the wave length is shortened, as by resistance to the movement of the fluid, the exponential expression ^4'^'^' increases, and thus the amplitude increases^). This change has been much discussed in various treatises and memoirs, and we shall not attempt to add to it here, except in the practical application of the result to physical problems. Now Airy finds (art. 201-2 10). the following theoretical curves for the breaking up of water waves in rivers, considered as straight canals, with smooth banks. After explaining his analysis of these theoretical waves in water. Airy interprets the results as follows »(2oi). To represent to the eye the form of the wave produced by the combination of the two terms, we have constructed the curve in figure q. The horizontal line represents the level line of the mean height of water: the elevation or depression of the curve represents (on an enormously exaggerated scale) the elevation or depression above the mean height, given by the expression above. The value of x' is supposed to increase from the left to the right : on which supposition the quantity mvt—mx', representing the phase of the wave, diminishes from the left to the right [fnvt being constant).* »(202.) To exhibit to the eye the law of the ascent and descent of the surface of the water at different points of the canal, the figures 10, 11, 12, and 13 are constructed. The first of these is intended for the point where the canal communicates with the sea : the others for points successively more and more distant from the sea. The horizontal line — is used as a measure of time, or rather of phase mvt mx': in which, for each station, x' is constant: the elevation or depression of the corresponding point of the curve represents the corresponding elevation or depression of the water above its mean height, as given by the expression above. »An inspection of these diagrams will suggest the, following remarks — ') In the Monthly Notices for April, 1917, p. 504, Dr. Silberstein treats at some length of the Einstein calculations, based on Gerber's formula (Zeitschr. Math. Phys. 43.93-104, 1898) in which for the Newtonian potential Mir is put Jflr{i i/c-drldtY, and concludes: "As far as T can understand from fefrey's investigation, (MN 77.1 12- 11 8), it would rather alleviate the astronomer's difficulties if the sun by itself gave only a part of these 43 seconds." Accordingly this is all the more reason for adopting Weber's law, though I reached it from a different point of view. *) This increase of amplitude will prove of high importance in the new theory of molecular forces, to be dealt with in a future paper. « 141 5048 142 9. -13. 14. 15. Airy's graphical illustration of the breaking up of waves under resistance. The canals considered are connected with the sea and of uniform width. = Theoretical form of tide-wave in a shallow river, to second approximation, mx = ^ ^ ^ mx 471 second station, mx 8t third station. o, first station, the sea; = ^ Theoretical tidal curves for different stations on the river. lo = ^ 12 third, 13 fourth station. first station at mouth of river, 11 second, Theoretical form of tide- wave in a shallow river to third approximation with large tide. The same with small tide. »(2 03.) When the wave leaves the open sea, its front slope and its rear slope are equal in leligth, and similar in form. But as it advances in the canal, its front slope becomes short and steep, and its rear slope becomes long and gentle. In advancing still further, this remarkable change takes place in the rear slope : it is not so steep in the middle as in the upper and the lower parts: at length it becomes horizontal at the middle : and, finally, slopes the opposite way, forming in fact two waves (figure g).* »(204.) At the station near the sea (see figure 10), the time occupied by the rise of the water is equal to the time occupied by the descent: at a station more removed from the sea (figure 11) the rise occupies a shorter time than the descent: the rise is steady and rapid throughout, but the descent begins rapid, then becomes more gentle, then becomes rapid again: at stations still farther from the sea (figures 12 and 13) the descent, after having begun rapid, is absolutely checked, or is even changed for a rise, to ^hich another rapid descent succeeds : in this case there will be at that station two unequal tides corresponding to one tide at the mouth of the canal. This numerical and practical discussion by Airy, with curves for illustrating the results is more satisfactory than any purely theoretical analysis of the effects of resistance, and thus all we need to do is to point out, that, just as water waves in canals degenerate and break up into partial waves, under the action of a variable resistance, depending on the depth of the water, and its distance up the river from the sea : so also in the aether, the long waves encounter resistance which progressively is more and more disintegrating on their existence, kinetic stability, and continuity. Ac- cordingly we may be sure that long waves in the aether will undergo corresponding changes by disintegration into shorter waves, and that the chief component will have increased amplitude. There are various physical illustrations of this effect which may be cited, as when the sun's radiation impinges on the earth, and the longer invisible infra-red rays, so much studied hy Langky, pass into heat waves of shorter wave length. Again, in our electric stoves and heaters, the electric current, made up of very long waves, first develops heat, so that the resisting wire acquires a dull glow, then a red heat, and finally becomes incandescent, with light of shorter and shorter wave length the longer the action continues. The transition here sketched is therefore known to be a reality in dealing with the transformation of electric energy into heat and light, under conditions observed daily 'in every part of the world. The analogies here cited are so obvious and familiar to us in the changes noticed when waves pass into breakers at the sea shore, that it seems impossible to deny the validity of the conclusion above drawn from every day experience, and fortified by the profound researches on tides and waves produced by one of the greatest mathematicians and natural philosophers of the past age. To those who hesitate at the contrast between water and aether, we point out that it is true that water is heavy and inert, and sluggish in its movements, whereas the aether is excessively rare, with density at the earth's mean distance equal to 438X10"^*, and having an enormous elastic power, 68g 321 600000 times greater than that of our air in proportion to its density. Thus the light and electric waves in the aether travel 902000 times faster than sound waves in the air, and about 200000 times faster than sound in water at 30° C, which travels 4.54 times faster than in air, owing to the high incompressibility of the water. There is thus tnuch diff'erence between the speed of waves in the aether and in water, even if the dense water, like the rare aether, be highly incompressible. But notwithstanding this difference, due chiefly to the extreme rarity of the aether, water being in comparison, with aether 2 2 8Xio-'-^ times denser, there is a substantial physical basis for comparison of the actions in the two media. Our reasoning therefore is not speculative or hypothetical, but purely practical, since it rests upon facts definitely determined by experience, anH verified by careful observations of recognized phenomena of the physical universe. : s : 143 5048 144 In order to bring out the practical bearing of the wave- theory upon the motion of the perihelion of Mercury, and the lunar fluctuations, discussed below, we notice that as long ago as 1901, Professor Planck of Berlin supposed that in all matter there were a great number of »resonators« of every possible period (Ann. d. Physik, 4.556, 1901). Thus matter would receive and emit vibrations of all possible periods, as postulated in the Electr. Wave-Theory of Phys. Fore. 1.85-88, 9 1 7 1 . The lunar fluctuations occur where the sun's gravita- tional waves have to traverse the solid mass of the earth, and thus the action on the moon is decreased near the time of lunar eclipses; and the moon partially released from the sun's control, thus tends to fly the tangent. This gives rise to disturbances in the mean motion which Neivcomb declared to be the most enigmatical phenomenon presented by the celestial motions. Now the lunar theorists were unable to find the perio- dicities required to explain the lunar fluctuations, until I discovered the obstructing cause at work, near the shadow of the earth, to modify the sun's gravitative action on the moon. If this explanation of the fluctuations of the moon be conceded, a similar cause will have to be admitted to act on the planet Mercury, which renders our sun gravitationally unsymmetrical or lopsided, as if a small part of the matter on the opposite side of the sun were removed, or ineffective, owing to the interposition of the sun's huge globe in the path of gravitational action. In other words, owing to re- fraction, dispersion, absorption, large masses of matter exercise a slight screening effect. Jl Fig. 2. Illustrating the absorption and circular refraction of some of the waves from part of the matter in the side of. the sun opposite to Mercury, as if parts of the Sun's niass had been removed, and the globe thus rendered slightly lopsided. Compare also Fig. 3, in section 5 below. Mercury therefore is less attracted than if the strict law of inverse squares established by Newton held, and thus we have the feebler law of force explained below: / = =3 ,„,„'/^2.ooooooio4.i i/^o.ooooooioiej [nim'lr'') ( (^^ whence arises the hitherto unexplained progression of Mer- cury's perihelion, by 28''44 per century, which has proved so bewildering to geoiiieters and astronomers ever since Leverrier discovered the difference in 1859. This explanation is very much simpler than any heretofore offered, and as it harmonizes the motion of Mercury with the motion of the moon, under well established physical laws, without introducing any vague and chimerical hypotheses, it would seem difficult to deny its essential physical truth. 3. Explanation of the outstanding Motion of the Perihelion of Mercury, based on the Electrodynamic Wave-Th"eory of Physical Forces. Aside from the investigation of the amount of the outstanding motion of Mercury's perihelion, by Leveiricr, 1859, and by Newcomb, 188 1, duly noted below, we cite the following researches as offering various explanations of the phenomenon : 1. Untersuchungen iiber die Bewegung des Planeten Merkur, and other notices of researches by Dr. Jl Bauschinger, AN 109.32. 2. tJber die Bewegung des Merkurperihels, by P. Harzer, AN 127.81, 1891. Harzer investigates the effects of unequal moments of inertia of the sun about polar and equatorial axes, and of the matter in the corona, and finds these hypotheses admissible. 3. A Suggestion in the Theory of Mercury, by A. Hall, AJ 14.49, 1894. Hall adopts the suggestion oi Newton that the law is not exactly that of the inverse squares, and puts /=;„;,//^2.00000016^ (5) 4. Hypothesis, that gravitation towards the sun is not exactly as the inverse square of the distance, Astronomical Constants, p. ii8, by S. Newcomb, 1895. Newcomb adopts Hall's, hypothesis, with very slight modification /=,«^2'/r2<'000001574, (g) 5. Uber die empirischen Glieder in der Theorie der Bewegung der Planeten Merkur, Venus, Erde und Mars. VJS 41.234-240, by H. Seeliger. Das Zodiakallicht und die empirischen Glieder in der Bewegung der inneren Planeten. Sitz.-Ber. d. Kgl. Akad. d. Wiss. zu Miinchen, 36.595-622, by H. Seeliger. Seeliger assumes the . matter of the zodiacal light to be distributed in two ellipsoids, an outer one and an inner one, which will effect Mercury's perihelion, as observed, without disturbing the other planets. He gets a very perfect agreement with observations, fully as good as that supplied by Einstein fi theory, without the vagueness of relativity. Seeliger'?, chief results are Seeliger' = Mercury e dn = sinzdft = d/ -i-8"64 H-o.6i -1-0.38 Seeliger'i theory applies equally well to Venus, the Earth and Mars. 6. A Memoir on the outstanding anomalies of the celestial motions, by Professor E. W. Brown, Amer. Journ. of Science, 29, in which various hypotheses, including the effects of the magnetic fields of the earth, sun and moon, are examined and rejected. See also Report of British Association for 19 14, for Prof. Brown's Address to Section A, p. 31 1-32 1. 7. Einstein's General Theory of Relativity, 1916, in which this author uses the value (JcT =: -1-43", and deduces the term of Gerber'% formula: V=[Mlr)[i-xlc-ArlAt)-'- (7) required to be added to the law of gravitation to make this = difference between theory and observation disappear. By using the value 6^ +43" per century, and deducing a very exact agreement based on this difference, instead of the « : : 145 5048 146 difference 28*44, which results from Weber's, law, Einstein adds to the improbability of his theorj'. It has long been remarked that among the outstanding motions of the solar system recognized by astronomers during the past sixty years, and of which geometers have sought a valid explanation, none is more justly celebrated than the excessive/progression of the perihelion of Mercury, announced bv Leverrier to the Paris Academy of Sciences, Sept. 12, 1859, (CR 49.379)- Leverrier' s, announcement of an outstanding motion of 38" per century in Mercury's perihelion seemed to find almost immediate confirmation in Dr. Lescarbaulfs supposed observation of an intra-mercurial planet named Vulcan; and this anomaly therefore was made the basis for the provisional elements assigned to the new planet. If on the one hand later observational researches, during many total solar ecHpses, have shown no signs of an intramercurial planet, it may be noticed, on the other, that the fullest confirmation of Leverrier's analysis of the planetary motions, 1859, has been obtained by later investigators, especially by Newcomb, who used all the observations of the transit of Mercury from 1677 to 1881, and deduced an outstanding motion in excess of that found by Leverrier, namely about 43" per century. (Astron. Pap. of the Amer. Ephem.,' 1.367-484, 1881.) Accordingly, Leverrier spoke conservatively in the original announcement of his discovery, when he said »The necessity of an increase iri the secular motion of the perihelion of Mercury results exclusively from the transits of the planet over the disc of the sun. The exactitude of these observations is beyond doubt. The anomalous motion of Mercury's perihelion thus established by Leverrier and Xewcomb, has been widely discussed in natural philosophy, and in fact combined with the Michehon- Morley experiment of 1887, for laying the foundation of a Theory of Relativity, on which already many treatises have appeared, without, however, thus constituting a simple and consistent physical doctrine which commands universal assent. There are, I think, grave reasons for doubting the whole Theory of Relativity, as now developed, on grounds which will be more fully outlined in treating of the Michehon- Morley experiment. For the present it must suffice to allude to the unsatisfactory theory resulting from Leverrier^ discovery of an outstanding motion in Mercury's perihelion, and the growth in natural philosophy of a doctrine which many regard as both vague and chimerical. In 1894, Prof Asaph Hall of Washington outlined a new view of the anomalous motion of Mercury's perihelion (at 14.49), based on the hypothesis that for some unknown reason the Newtonian law of the inverse squares might not be strictly correct. Already in 1686, while preparing the Principia, (Lib. I, sect. IX), Sir Isaac Newton had considered such a possible modification of the law of attraction; and even included some computations, in which he assumes that the central force departs a little from the inverse square of the distances. Newton found that the perihelia would move forward under such a modification of the law of attraction (Lib. I, sect. IX, Prop. XLV, Prop. XXXI, cor. I), but considered the observed approximate fixity of the planetary perihelia a strong proof of the accuracy of the law of the inverse squares. His final view evidently is expressed in the General Scholium to the Principia, 17 13, where he says that in receding from the sun gravitation » decreases accurately in the duplicate pro- portion of the distances as far as the orb of Saturn, as evidently appears from the quiescence of the aphelia of the planets; nay, and even to the remotest aphelia of the comets, if these aphelia also are quiescent*. In the Mecanique Celeste, 1799, Laplace likewise con- cluded that the law of gravitation holds accurately for the satellites as well as for the planets, (Liv. II, ch. I, § 6). In Liv. XVI, chap. IV, however, Laplace investigated more fully the effect on certain terms of the flioon's motion of some assumed changes in the Newtonian law of attraction, but from his remarks it is evident' that he did not consider it probable that there is a departure from the strict law of the inverse squares. Thus, up to the time of Leverrier''!, researches on the motion of Mercury, 1859, there were no well established deviations from the Newtonian law which might be made the basis of observational inquiry, so as to serve as a crucial test of the accuracy of that law. In his paper of 1894, however. Professor Asaph Hall sagaciously remarks »If the Newtonian law of attraction is not a rigorous law of nature, or if it is modified slightly under certain conditions, probably this lack of rigor would become apparent first among the swiftly moving bodies of our solar system, such as our moon and the planet Mercury* (AJ 14.49). Our moon indeed does not move so swiftly, but owing to its great proximity to the earth and the eclipse records' extending over nearly 3000 years, the motion is very ac- — curately known, both by observation and by theoretical — research and calculation, so that the smallest disturbances may become sensible to observation (cf. Electr. Wave-Theory of Phys. Fore, 1. 113, 1917), which doubtless is the chief point Prof Hall had in view. That Leverrier'% researches on the motion of Mercury, 1859, set in motion several hew lines of inquiry of great theoretical importance is shown by two investigations de- veloped within the next fifteen years. 1. The researches of Tisserand on the motion of a planet under fF^^^r's electrodynamic law, communicated to the Paris Academy of Sciences, Sept. 30, 1872, by the eminent geometer Bertrand, who had inspired these investigations. 2. The problem proposed in 1873 by Bertrand to the Paris Academy of Sciences, (CR 84), to find the closed curve R = O described by a planet when the forces have the form of an unknown function {x,y) of two independent variables X and y, and the differential equations of motion are = = - m-A^-xJAt' -Ji-xjr m-A'^ylAt"^ R-yjr (8) R it being required to find the function whatever be the initial values of the coordinates x^, y^, and of the components of the velocity = x^' =- (d^/d/)o ' j'o' (dj)'/d/)o . (9) The solution of this problem showed that this function : « . : : : 147 5048 148 R ^ always takes the form m r", where m is the mass of the planet, and r the radius vector. It was Bertrand's theoretical improvement in the treatment of Newton's problem of a moving perihelion which led to Hall's hypothesis of 1894, for explaining the excess in the motion of the periheHon of Mercury. Since Hall's hypo- thesis has been further developed by the writer's recent researches in the Electr. Wave -Theory of Phys. Fore, it is necessary to treat bf these successive steps for attaining an Electrodynamic Theory of the motion of Mercury's perihelion. (i). Bertrand' s so\n\.\on oi Newton's "pxdhltm of finding the central force for a moving perihelion. As propoS)ed to the Academy of Sciences, in 1873, Bertrand's prdble.xn reads (CR 77) »We consider a planet attracted by the sun under a force of which the intensity depends only on the distance. We suppose known this one fact: that the planet describes a closed curve, whatever be the magnitude and direction of We its velocity. "have to find the law of attraction from this single datum. Bertrand remarks that as the force is central, the motion takes place in a plane through the centre of the sun, and Kepler's law of equal areas in equal times holds true. If ^ the force have the form ^^ ffir" (10) it is found that there result just two formulae i?2 =^ '"/''^ Rx= mr (11) (12) And these are the only two laws of attraction which permit a planet to describe a closed curve, whatever be the initial data (the velocity being nevertheless below a certain limit). And if we suppose the attraction zero at an infinite distance, there remains only one formula (11), or the law oi Newton, which could thus be deduced from the sole fact of obser- vation : that any planet whatever describes a closed curve, without our being able to know the nature of this curve (cf. Tisserand's Mecanique Celeste, 1. 48, 1889). Resuming Newtoris problem of a moving perihelion, © Bertrand derives a perfectly general formula 'for the arc swept over by the planetary radius vector between the mini- mum value ifx) and the maximum value (^2) 0= [7r/l/(«-H3)]x [l+V24(«-l)(«+2)[('-2-n)/(^2+'-l) (13) — He remarks that when r^ r^ tends towards zero, we have in the limit the Theorem of Newton, 1686: = Lim© 7r/"l/(«-i-3) (14) which applies to an orbit almost circular described by a planet under the influence of a central force proportional to a power of the distance. If for the motion of a planet around the sun, we take n= — R = with Newton, 2, mjr^, the relation (14) gives Q = n, which is rigorous. Thus it only remains to find — what will happen when we modify slightly the exponent 2 = — in the Newtonian law of gravitation. If, for example, we supposed n 2.001, it follows that we should have: lim ^ == jtj V[i —0.00 1 ) n (i-f- Y2°-°°i -<-•••) 180" 24" s' USJ or a progression of the apsis line at each revolution of 10 48 , which is so large a quantity as to be totally inconsistent with observation. Without further examination of the effects of changing the exponent in Newton's law (cf. Principia, Lib. I, Prop. XLV), we recognize that the change in the exponent must be extremely small. This case has been considered by Prof. Asaph Hall, who has applied the hypothesis to the motions of the planets and of our moon. f ^ (2) Hall's hypothesis of 1894, that the law of attrac- tion may he =^ mm'lr'^'^'' , where j' 0.00000016. In — A.J. No. 319, June 3, 1894, Prof. Asaph Hall remarks that on applying Bertrand's formula to the case of Mercury with Newcomb's value of the outstanding motion of the peri- — heHon, or 43" per century he finds that the perihelion would move as the observations indicate by taking «= — 2.00000016 (16) the difference of the expo"hent from the law of Newton being = V 0.00000016. ' The change in the law of attraction required for pro- ducing this progression of the line of apsides is therefore very minute. If we use Weber's law, as in the author's Elect. Wave-Theory of Phys. Fore, and Newcomb's value of the out- = standing motion of Mercury's perihelion (Astr. Pap. of the Amer. Ephem. 1. 473); namely, dc7 42^95, we shall obtain an outstanding motion of 2 8''44 per century, which is to be accounted for by modification of the exponent in the law of attraction. (3) Law of attraction indicated by the outstanding motion of Mercury's perihelion. As the motion of Mercury's perihelion offers the principal difficulty in modern celestial mechanics, we take the law of attraction to have the form f=mm'lr^+' = [fJe^loo +28^44 (i?) and determine v by the condition that the outstanding cen- tennial motion of the perihelion shall be -H 28^44. If the perihelion shifts 28"44 in 100 years, it will shift 0^2844 in one year; and as there are 4.1521 revolutions of this planet in a year, the shift will be o;'o684956 in a single revolution, and therefore, ©"0342 47 8 in a half re- volution of Mercury. By Bertrand's formula (13) above, we notice that when the orbit is considerably eccentric, as in the case of the planet Mercury, the term depending on [{r2---ri)j[r2-^r-i)Y == e'^ becomes sensible. In fact in this formula depends on the products of two series as follows = n/V{n-i-^)x + {i+V24(«-i)(« 2)[(r2-ri)/(r2+ri)]2H = Tt/V(i-v)-{i-h^/2i(3-^v)ve^-\ } } =^ n{i+y,p+%v'+- }{i-^y^ve^+^/,,v'e'} (18) = + 7r{ 1+^(1/2 V8^^)- (19) Accordingly, our equation of condition is: + [l+»^(V2 V8^^)+---} = 180° o' 0^0342478 =^ 648000^0342478. (20) As the coefficient of the term involving p in the case + of Mercury becomes (V2 V8f^) =0.5052839, we find from (20) by calculation that = v 0.0000001045977 (21) : :, 149 5048 ISO And the modified Newtonian law becomes: / = ,;2;«'/^2.0000001046_ , J^^) Applying this law of attraction (22) to the eight prin- cipal planets of the solar system we have the following table of centennial progressions for their perihelia: Mercury Venus The Earth Mars 28*44 11. 1341 6.8496 3-6418 Jupiter Saturn Uranus Neptune 0^577448 0.2325307 0.0815288 0.0415681 The progression of the perihelia here calculated from the modified Newtonian law are not contradicted by any known phenomena. The exact position of the periheHon of Venus is not well defined by observations, owing to the great circularity of the orbit; and some slight uncertainty also attaches to the position of the perihelia of the earth and of Mars. It will be seen that the change made in the Newtonian law is exceedingly minute. For the change in the exponent the ratio is ^Uv -= 1046/20000000060= 1/19120459 (23) a little less than one nineteen-millionth of the whole. Such an infinitesimal alteration in the resulting attractive force would give no sensible effect in a single revolution, but as the change 6^ accumulates with the.lapse of time, it finally becomes very sensible, and we are obliged to take account of the secular progression of the perihelion. This cumulative effect is very similar to the alteration in the moon's mean longitude which results from the secular acceleration of the moon's mean motion, first explained by Laplace in 1787, under forces which are insensible for short intervals, but by continuing for long ages in the same direction, finally become sensible and have to be calculated in the formation of tables of the moon designed for use over many centuries. 4. The Modification of the Newtonian Law indicated by the outstanding Difference between the observed and Calculated Motions of the Lunar Perigee. Just as the motion of Mercury's perihelion is the chief means for. throwing light on the form of the law of attraction for the planets of the solar system, so also the motion of the lunar perigee affords the best criterion for the form of the law of attraction operating on the motion of the satellites. As the subject has been but little discussed heretofore, we shall briefly outline the results of astronomical research on this interesting problem. In the Monthly Notices of the Royal Astronomical Society 74-396, 19 14, Prof. E. W. Brown gives the anriual motion of the lunar perigee depending on the ellipticity of the earth as follows: ^ {diH/dt)^ -l-6;'4i, for an oblateness of 1:296.3. (24) He adds that for an oblateness of 1:297, the value would be reduced by the factor — (1/297 0.00 1 7 34): (1/296. 3 -0.001734) (25) and become: {da/dt)i =: -i-6:'38 . (26) From these data it follows that the annual motion of = the lunar perigee for an oblateness of i :.2g8.3 would be (8cT/8/), -t-6:'32. (27) The above values by Brown, as thus reduced to an oblateness of i : 298.3, are confirmed by the part of the motion of the lunar perigee depending on the ellipticity of the earth's figure calculated by Dr. Hill, in his supplement = io Delaunay's Theory of the Moon's Motion, Astron. Pap. 3.334, namely: (8oj/9/), -h6:'82 . (28) This value, however, refers to Hill's oblateness of I : 287.71, and must be reduced to correspond to the ob- lateness of I : 298.3; which leads to a result differing only o'oi from that found by Brown and cited above. Hill's value for this reduced elHpticity of the earth therefore is = (8c7/8/), +6:'33. (29) Hence we conclude that this value of the annual per- turbation of the lunar perigee depending on the eUipticity of the figure of the earth is very accurately known. The difference in these two authorities would be only 0^0124 per annum, or i"24 in a century, which is below the limit of determination in the present state of science. Prof. E.W.Brown also gives data to show (MN 75.514), that when the theoretical secular acceleration of the perigee is determined with the highest accuracy, it is 16" per century smaller than the observed centennial motion of the perigee. This is for an ellipticity of the earth of i : 297. By changing the ellipticity to i : 294 Broivn reduces this value from 16" to 3'; and by taking an ellipticity of i : 293.7, the outstanding difference entirely vanishes. Such a large value of the oblateness, however, seems to be quite inadmissible; and thus on calculating the excess in the actual motion of the perigee over the theoretical motion, for an oblateness of 1:298.3, I'find it to be 2if9, or say 22" per century. If we admit this ellipticity of the earth ^), — — which is decisively indicated by the four best methods namely i) Pendulum observations of gravity, as discussed by Helmert and the U. S. Coast Survey, 2) Geodetic measurements of arcs on the earth's surface, 3) The lunar inequality in latitude, 4) The fluid-theory of the earth, isostasy and Laplace & law of density; then it will follow incontestibly that the moon has an outstanding motion of its perigee of about 22" per century, almost exactly one half the outstanding motion observed in the perihelion of Mercury. To form a better idea ofthe accuracy heretofore attained in these calculations, of the centennial motions of the lunar perigee, we recall the results of Hansen and Brown: Observed — O Calculated Diff. C Authority [dn7/d4o= 14643560" 14643404" +I56'^T81'4!';.'"34T'' + [dsT/d4o= 14643520" 14643504" ,e"^'-'"""'^^fl^^' , ') In the writer's "Determination of the oblateness of the terrestrial spheroid", begun in 1904, but not yet published, this question las been carefully examined, and the value I : 298.3 shown to be the most probable of the various values heretofore proposed. 151 5048 152 As above pointed out, the difiference of 16° per century here indicated by J^rown's calculation of the theoretical motion of the perigee becomes 22" when the elKpticity of the earth is reduced to i : 298.3. It is also to be noticed that the observed centennial motion of the lunar perigee used by Hansen is 40" larger than that used by Brown. It would seem that very little doubt could attach to the increased accuracy of Brown's observed motion, though owing to the fluctuations in the mean longitude the value 14643520" for the observed cen- tennial motion of the perigee may yet admit of some im- provement, if any of the observational equations should prove to be vitiated by this troublesome cause. Indeed, it is a little difficult to understand why feo considerable a difference as 40" per century should exist in the observed centennial motion of the perigee used by two such very modern authorities as Hansen and Broivn. For the position of the perigee is given with considerable accuracy from the eclipse records of the Greeks, and the calculations of Hipparchis and Ptolemy; and as about 226 revolutions of the perigee would occur in 2000 years, the motion of the perigee ought to be quite accurately fixed by the eclipse records of the Greek astronomers. The above difference of 40" per century, increasing as the square of the time, in 20 centuries would accumulate to 16000", nearly four and a half degrees, or about nine times the diameter of the moon. The difference of 100'' between the above calculated centennial motions of the perigee is less striking than it otherwise would appear, but such differences warn us not to overrate the accuracy attained. It seems remarkable that the eclipse records of the Greeks would leave the position of the perigee open to so much uncertainty. Besides, in the modern observations of the moon since 1750,- which are quite accurate, an uncertainty of even 20" per century, or an accumulated difference of 5 7"8, in the interval of 170 years, ought not to exist. Still more intolerable is the difference of iisl'6, based on the difference of 40" per century! But Hansen was unaware of the fluctuations in the moon's mean longitude; and as the fluctuations affect the node as well as the longitude, it may also have vitiated sensibly his calculation of the observed centennial motion of the perigee. It is worthy of notice that Hansen's outstanding diffe- rence between the observed and calculated centennial motion — = O of the lunar perigee is C -(-156"; while Brown's values — = make this difference O C +22". The mean of these — = O two values is C +89". Now, in default of definite knowledge it is not quite safe to assume that Hansen's values are wholly wrong, and Brown's entirely right, notwithstanding the preeminence of the latter's exhaustive researches in the lunar theory. Both investigators may be somewhat in error, for one reason or another, or for several reasons combined. Thus, apparently the safest thing is to assume that the truth lies between -Hi 5 6", as found by Hansen, and -1-22", which results from Brown's calculations. And as we do not know what weights should be assigned to these extreme- values, we can only take the = simple mean of the two outstanding motions of the perigee, and thus we have: [(807/8/)^]^^ +89". (30) It is to be observed also that in our researches on the outstanding motions of Mercury's perihelion, we found the exponent of Newton's law should be modified from 2 to = 2-i->', where x' 0.0000001046. To calculate the resulting outstanding motion for the lunar perigee we notice, in the' first place, that the effect of the time of propagation of gravitation by Weber's law, as = shown in the table of section I above, is almost insensible, (3c7 0*00637 per century. Thus we need consider only the effect of the exponential change for a body having a mean motion 3.219763 times greater than that of Mercury. And since the unexplained motion of Mercury's perihelion is 2 8"44, we get for the corresponding motion of the lunar perigee = [(8WH}oo=+28:'44X3.2i9763 +9i-S7. (0--C) (31) This calculated value is so very near the mean of the values found by Hansen and Broiun as to appear worthy of — = O attention. If for example, Hansen's value C -+-156"' 0= — were 65" too large, leaving -+-91", while Brown's — 0= were as much too small, yielding O 2 2"-t-65"== -f-87", the two values would be quite reconciled. And since Hansen and Brown disagree as to the value of the observed centennial motion of the perigee to the astonishing extent of 40", the possibility of such unknown errors in their several results is not to be wholly excluded. Accordingly, for some hitherto unsuspected reason, Hansen's value of the observed centennial motion perigee may be substantially correct, namely: = [(8n7/8/),]oo -+-14643560". of the (32) In this case, it would suffice to assume an error of 18" per century in Brown's calculated motion of the perigee. Unfortunately Prof. Brown even proposed to adopt an oblateness of the earth of i : 293.7, as if to avoid a modification of the form of the Newtonian law ^) ; and hence it seems not wholly improbable that an error of 18" per century in the calculated centennial motion of the perigee may have been introduced, through some step based upon the tacit assumption of the strict rigor of the Newtonian law. Under the circumstances, since Hansen's value of the outstanding residual in the centennial motion of the perigee apparently was obtained without prejudice, it should not be rejected, till Brown's values are independently tested and found to be not only the more accurate, but also wholly free from possible prejudice due to assumed rigor in the Newtonian law, or other systematic cause which might thus unexpectedly creep in. Under the present circumstances, it follows that if the = outstanding residual in the centennial motion of the perigee be [(5cj/8^)j]p(, -)-pi»j7 the exponent of the law of attrac- tion for the moon would be the same as that for the planet Mercury, namely: f ^^ jj^jj^'l^i.wowQ\^\& ( \ = ') In his address to the British Association in Australia, 1914 p. 316, Brown estimates that the exponent in the Newtonian law does not differ from 2 by a fraction greater than i : 400000000 0.0000000025 ; but the present discussion shows that this prediction probably overrates the accuracy we are justified in claiming, from 10 to 42 times. : 153 5048 154 In conclusion, it would appear from this investigation that the change in the exponent for the law of attraction may be the same for the moon and for Mercury. But if future researches should develop a smaller difference in the observed and calculated centennial motions of the lunar perigee, such as 2 2" per century, which seems to be the minimnm value now admissible; then there would be a smaller value of v in the exponent of the modified law of Newton. The value 22' per century leads to a value about one-fourth of that found for the planet Mercury, as may be seen from the following considerations. = The moon makes 1336. 85126 revolutions in a century, and therefore: i I'/iSSS.Ss 126 o!'oo82283 ig the amount of this secular progression of the perigee in half a lunation. The equation of condition; ^ B "= n{\ -i-Ya'^"^ • • • } JT {1-1-0.0082283/648000} therefore gives v == 0.000000025396. (34) But although there can be no assurance that this modi- fication of the exponent for the earth would be the same as for — the sun the earth being so different in density, size, and — physical constitution from the sun yet at present apparently we are not justified in using this smaller value, because in the existing state of our knowledge there are no definite grounds to authorize it. Accordingly for the sake of simplicity and uniformity the value of v applied to the motion of the perihelion of Mercury is preferable also for the motion of the lunar perigee. 5. Outline of the Cause of the Fluctuations of the Moon's Mean Motion. In the Electrodynamic Wave-Theory of Physical Forces, vol. I., 19 1 7, it is shown that the previously unexplained fluctuations of the moon's mean motion, discovered by Ne.tOcomb in 1909, after a study of the moon's motion extending over more than forty years, (1867— 1909), is due to the refraction, dispersion, and perhaps absorption of the sun's gravitational waves in passing through the solid globe of the earth. The result is a slight decrease in the sun's gravitative action upon the moon when near the shadow of our globe in space, by which, near the time of Lunar eclipses, the moon is slightly released from the sun's control, and in the tendency to »fiy the tangent*, has certain long period disturbances introduced into its mean motion. An attempt to find such disturbances in the motion of the, moon depending on the 18 -year period, had been made by Dr. K. F. Bottlinger, in a crowned prize Inaugural Dissertation, at the University of Munich, Die Gravitations^ theorie und die Bewegung des Mondes, (Freiburg i.B., 1912). Bottlinger .deduced some evidence of an 18-year period, but in the case of the longer disturbances (61.7006 years, and 2 77-59 years respectively) he was not able to find the slightest indications of the required periods; so that in his address on the moon's motion at the meeting of the British Association in Australia, 1914, p.319, Prof. .£. ^. j9r(7z^« spoke as follows »The shading of gravitation by interposing matter, e. g. at the time of eclipses, has been examined by Bottlinger. For one reason alone, I believe this is very doubtful. It is difficult to see how new periodicities can be produced; the periods should be combinations of those already present in the moon's motion. The sixty to seventy years fluctuation stands out in this respect, because its period is not anywhere near any period present in the moon's motion or any probable com- bination of the moon's periods. Indeed Dr. Bottlinger'' 1, curve shows this: there is no trace of the fluctuation*. From this citation it is evident that Bottlinger not only had not convinced Brown of the reality of the fluctuations depending on the interposition of our globe in the path of the sun's gravitative action, but also that Brown felt that an explanation of the 60-year and 275-year periods in the observed fluctuation could not be based on the theory of gravitational disturbances depending on the known cycles of tlie moon's motion, in relation to the eclipse periodicities. Notwithstanding this confidence of Professor Brown, resulting from his great experience in the lunar theory, I was fortunate enough to discover such long period inequalities in the moon's motion, bearing the closest analogy to. the forces acting in the great inequality of Jupiter and Saturn, of which the physical cause was discovered by Laplace in — 1785, after Euler and Lagrange had searched in vain for the mystery underlying the celebrated 900-year inequality of these great planets. Without attempting to give a detailed account of these researches in the lunar theory, we shall endeavor to outline briefly the leading points, because this advance of 1917 bears very directly, on the wave-theory, above applied to the motion of the perihelion of Mercury and of the lunar perigee. It is shown from an extension of Maxwell's, theory of circular refraction in the eye of a fish (Cambridge and Dublin Math. Journal, vol. XI), that a similar circular refraction of gravitational waves occurs when the path of these waves is through the solid mass of the earth. For in the earth, as in the eye of the fish, the concentric shells are each of uniform density, but with the density increasing from layer to layer towards the centre. Thus a circular refraction of the sun's gravitational waves will occur in propagation through the globe of the earth, and also of the moon's gravitational waves in passing through the same globe, owing to the concentric layers of which it is made up. The accompanying figure 3 (pag. 156) illustrates the refraction of the sun's waves in passing through the earth. By virtue of this circular refraction of the gravitational waves in passing through the globe of the earth, it, follows that^ the mutual interpenetration of the waves from the sun and moon are not the same when the earth interposes its solid mass in their path of action. The result is a weakening of the sun's gravitative action on the moon; and, when our satellite is thus partially released from the sun's control, it tends to »fly the tangent«, as near the time of lunar eclipses. The outcome is a series of disturbances in the moon's mean longitude depending on the motions of the perigee and node of the lunar orbit, with respect to the Saros or eclipse cycle. The principal eclipse cycles, incessantly repeated in the theory of the moon's motion, are the following^): ') Cf. Electrod. Wave-Theory of Phys. Fore, l.ibi-102. 155 5048 156 = 1. The Saros, made up of 223 synodic months 6585.32 days, discovered by the Chaldeans and used at Babylon for predicting the return of eclipses, in con- junction with the eclipse year of 346.62 days. 2. The eclipse year of 346.62 days, the average time of the sun in passing around the heavens from the moon's node and returning to the same node again as it retrogrades under the sun's disturbing action in 18.6 years. Nineteen of these eclipse years make 6585.78 days, almost exactly equal to the cycle of the Saros given above, which is 6585.32 days. The difference in these two periods is only 0.46 of a day, and therefore after 18 Julian years 10.82 days (o'?46 less than ig eclipse years) the Saros of eclipses is very nearly repeated, except that the location on the terrestrial globe is o"?32 =^ 7''4o"48^ further west in longitude. = 3. The nodical or draconitic month made up of 2jl2i2-22: and thus 242 X27'?2i222 6585'?357. "Tliis again is of almost the same length as the 223 synodic months and ig eclipse years defined in paragraphs I and 2 above. Fig. 3 = 4. The anomalistic month made up of 2 7'?5 546o; and thus 23qX27'?5546o 6585'?549. Accordingly, after 223 months the moon not only returns very closely to its original Refraction of the sun's gravitational waves in passing through . the earth's 'mass, by which the moon is slightly released from the sun's control near the time of lunar eclipses. position in respect to the sun and node, but also in respect to the line of apsides of the moon's orbit; so that the pertur- — = = bations near perigee, during the interval of the difference in these two cycles, 6585'?549 6585432 0422Q 5''2g"?8 are so small as to modify but very slightly the return of the cycle of eclipses composing the Saros. Accordingly, these four fundamental lunar cycles recur in the following periods: 1. The Saros ===223 synodic months 2. 19 eclipse years of 3 2 6'?62 each ^ 6585'?3 2 = 6585.78 3. 4. 242 nodical or draconitic months of = 27'?2i222 each 6585.357 = 239 anomalistic months of 27^55460 each 6585.549 = Now the Saros 6585'?32 :^ 18 Julian years 10.82 days, or 18.0293 sidereal years of 365'?2563582 {ifansen). And according to JVezson the period of the circulation of the lunar perigee is 8.855 years. In the 10''' edition of his (3ut- ^oM , lines of Astronomy, 1869, p. 472, S\t Ifersc/ie/ uses the = period 3232^575343 8.85031 Juhan years,. which is only slightly -different from the value cited above. Accordingly, the forward motion of the perigee will carry it twice around the heavens in 17.71 years, while the node revolves in the retrograde direction in 18.6 years. Thus ^ if we call Q, the yearly motion of the node, and the cor- responding motion of the perigee, we have = = j; -i9?35484 = ta= -H4o?655o 360718.6 , 36078.855. l3SJ From the above data, it follows that the node will retro- = grade through 360° in 18.6 years, but in the same time the lunar perigee will progress through an angle of 7 56? 183 7 2o''-i-36?i83; so that after an interval of 18.6 years the perigee is displaced forward by 36? 183 in respect to the restored node. Fig. 4. -7 — <^o Illustration of the progress of the moon's perigee in respect to the node, in the 61.7-year fluctuation. (i) Determination of the period of the 60 -year ^Vlc- tuation. It is very easily shown that owing to the relative magnitudes of these direct and retrograde revolutions the angular conjunctions will tend to recur in the regions of 360°, 240°, 120°, like the actual conjunctions of the planets Jupiter and Saturn in the theory of the celebrated 900-year ine- quality which was first theoretically explained by Laplace in the year 1785. Here, too, as in the theory of Jupiter and Saturn, the conjunction lines move forward. The amount of = the displacement is 36?! 83 in 18.6 years; and in 3.31648 such periods, 3.3 1648 X r8. 6 years 61.7006 years, the angular conjunction which started out at the angle 360° will revolve forward through 120°, and the cycle of angular con- junctions at all three points will begin over again, exactly as in the great inequality of Jupiter and Saturn. This leads 157 5048 58 at once to the second long inequality in tHe moon's mean motion, which, without suspecting the cause, Newcomb estimated at » 60 years,, more or less«. His judgment of the period was surprisingly accurate; and as he concluded that the coefficient might be about 3^0, here again his value could be adopted. (2) Determination of the period of the great fluctuation in 277.590 years. In the case of the great fluctuation in the moon's mean motion, of which Newcotnb estimated the period at about 275 years, the calculation of the period is somewhat similar to that just cited, but also somewhat different. It is physically obvious that the modification of the sun's gravitation in passing through the body of the earth will depend on the relative shifting of the line of angular conjunctions node-perigee. Now it is easily found by calculation that the angles of the node-perigee are in angular conjunction, on a line ii?67o in advance of the original conjunction, after an interval of 17.9971 years. For in this time the perigee pro- gresses over an- arc of 47r-l- 1 i?67o, and the nqde retrogrades over an arc of 2 71— i i?67o, and meet exactly at the conjunction line specified. The problem thus arises to find the Interval in which this secular displacement of the angular conjunction line will complete the cycle in the moon's motion due to the reduction of gravitation near the shadow of the earth. In each period of 17.9971 years, the node retrogrades through the angle in in respect to the shifting mean position of the perigee, and' in the same interval the perigee progresses through the double of this angle, 45T, in respect to the retrograding mean node; so that on the average their opposite motions amount to iin in IT .qg-j i years. As the physical effect of the reduction of gravity near the shadow of the earth is the same whether the shifting conjunction line node-perigee refer to ascending or descending node, we perceive that this advancing conjunction line need only sweep over the angle tt to give the required interval for completing the cycle due to the changes of gravitation near the shadow of the earth. = Now i8o°/ii?67o 15.422, = interval of 15.422 X 17.997 1 years and _ therefore in an 277.590 years, the cycle of- the changes of gravitation near the shadow of the earth will be complete. This is the period of the great fluctuation in the moon's mean longitude which Newcomb estimated at 275 years, from the modern observations, and used in calculating the secular acceleration from the eclipse records extending over 2 600 years since the era of the Babylonians. The diagram in Fig. 5 presents to the eye a continuous representation of the changes in node (outside circle) and perigee (inside circle) during 18 years. At the end of 18 years they both are in conjunction at i, near the original line of conjunction, 360°, but ii?67o further forward. In each of these periods of 18 years the nodes turn to every part of the heavens, so that eclipses occur all around the earth's orbit, with the earth and moon at all possible distances from the sun. In this interval the lunar perigee revolves twice, and the node once; so that the effectof the progression of the perigee goes through symmetrical phases in respect to the earth's orbit in 18 years, as shown by the above diagram. Fig. 5. Illustration of the progress of both node and perigee for producing the moon's great fluctuation in 277. 59. years. This diagram also illustrates the secular progress of the line node-perigee, the restoration to parallelism in this conjunction line, advancing by ii?67o every 17.9971 years, and requiring 277.590 years for completing the full cycle of a semi-circumference. We may express this result also by observing that physically the decrease of gravitation near the shadow of the earth will take place with equal effect whether the eclipse be near the ascending or the descending node; and this decrease will always correspondingly affect the moon's mean = longitude. Therefore, the 18-year movement of node-perigee conjunction line over the arcs i, 2, 3 • • •«, where n 15.422' at 180°, will comprise all possible combinations of the con- junction line node-perigee for modification of the sun's gravity on the moon when near the shadow of the earth. (3) Determination of the 18-year period of the Saros cycle. The Saros cycle is so well known that we need scarcely = add that a minor disturbance in the moon's mean longitude will recur in this period of 6585.32 days 18.0293 years. In this period the symmetrical eclipse cycle of 223 lunations is complete and the eclipses begin to repeat themselves, with the moon very near the same relative position with respect to the sun and node, and also with respect to the line of apsides or perigee. This Saros cycle of the Chaldeans gives rise to a minor disturbance in the moon's mean longitude, with period of 18.0293 years, and a coefficient of about I'o. It is the smallest of the moon's sensible fluctuations, yet indicated by the researches of Newcomb and Bottlinger, and illustrated graphically by the accompanying Fig. 6 (p. 159). 159 5048 i6o i6i 5048 162 in addition to the mystical one offered by Einstein, which is devoid of physical basis ^and finally the natural and simple explanation based on the wave -theory, and outlined above in section 3. On the other hand, the lunar fluctuations, which are vastly more complicated than the motion of Mercury's perihelion, admit of but -a single known explanation, namely, that discovered by the present writer in 19 16. It is therefore with some reason that the, most experienced physical mathematician at Cambridge wrote me, Jan. 28, 1917: »1 wish the perihelion of Mercury could be resolved similarly (to the new work on the lunar fluctuations). Otherwise we have an unlimited number of ingenious kinds of relativity on our hands; which will be remarkable for self- contradiction of the principle that everything is relative*. It is just such confusioti as this that I have labored to get rid of, and now my theory of the motion of Mercury's perihelion is found to conform to the wave-theory, and to correspond to the ideas of Newton, 1686, that the law of gravitation in certain cases differs a little from the exact law — of the inverse squares the difference being explained by the wave-theory, and the nature of the aether. 6. Gravitational Action is propagated by Stresses due to Waves in the Aether, but MaxweU's conception that the Stress is based on Pressure in the Direction of the Line of Force and on an equal Tension in all directions at tight angles thereto is not admissible. From the electrod. wave-theory of gravitation, outlined in the writer's work of 1Q17, it follows that gravitation is propagated by stresses in the aether due to the interpene- tration of waves, and the action across space therefore travels with the velocity of light. This mode of action is already outlined also in the first paper on the new theory of the AN aether, 5044. Forty-seven years ago in the celebrated Treatise on Electricity and Magnetism, 1873, '^ol. i. Chap. V, §§ 103-116, Maxwell gave a remarkable theorem for the stresses between two electrically charged material systems, as producible by a distribution of stress over closed surfaces about these systems. He takes two electrical systems, namely, Ei, with volume density q^, of the element whose coordinates are *i> yii ^1 ; s^nd similarly for the other system, E^, Q^, Xo, y^, «2- Then the x-oomponent of the force acting on the element of .£1, owing to the repulsion of the element of E^, will be: dX — =^ Qi Q2 {xi X2)/r^ • d*i dji d^ij dx^ dj/2 d^g ^2= (.^l-*2)^-|-(j>'l-J)'2)^-|-(%'-22)^ (36) ^^^ J J JJ J J(^i —^2)/f^'QiQ2dxi d ri dzidx2 dj/j dz^ This is as in the theory of action at a distance, and the integrals will not be altered by extending the limits from —00 to -4- 00. Maxwell then proceeds to remark (§ 105) that if the action of E^ on E^ is effected, not by direct action at a distance, but by means of a distribution of stress in a medium extending continuously from E^, to E^, it is manifest that if we knew the stress at every point of any closed surface s which completely separates E^ from E^, we shall be able to determine completely the mechanical action of E^, on E-^^. Accordingly, he concludes that if it is possible to account for the action of E^, on E^ by means of a distribution of stress in the intervening medium, it must be possible to express this action in the form of surface integrals extending over the surface s, which completely separates one system from the other. Maxwell then develops the solution at some length, and after obtaining the required mathematical expressions, (§§ 105-110), remarks (§ in): »I have not been able to make the next step, namely to account by mechanical con- siderations for .these stresses in the dielectric. I therefore leave the theory at this point.* It can be shown that the action of waves, flat in planes normal to the Hnes of force will explain the mechanical difficulties here noted by Maxwell. For in his work on Matter, Aether and Motion, Boston, 1894, Prof. A. E. Dolbear describes an experiment of the following kind : »If a dozen disks five or six inches in diameter are set loosely an inch apart upon a spindle a foot long, so that they may be rotated fast, yet left free to move longitudinally upon the spindle, they will all crowd up close together, as the pressure is less between them than outside. If one can imagine the spindle to be flexible and the ends brought opposite each other while rotating, it will be seen that the ends would exhibit an apparent attraction for each other, and if free to approach, would close up, thus making a vortex ring with the sections of disks. If the axis of the disks were shrinkable, the whole thing would contract to a minimum size that would ' be determined by the rapidity of the rotary movement, in which case not only would it be plain why the ring form was maintained, but why the diameter of the ring as a whole should shrink. So long as it rotated it would keep up a stress in the air about it. So far as the experimental evidence goes, it appears that a vortex ring in the air exhibits the phenomenon in question.* The behavior of the flexible spindle in this experiment is analogous to that of the lines of force, which Faraday long ago observed had a notable tendency to shorten themselves. The gaseous medium of the air between the disks is thinned out, by the effect of the centrifugal force, just as the aether itself is near a magnet, owing to the rotations ') of the wave elements about the lines of force. Hence the lines of force tend to shorten themselves, as Faraday observed in his experiments with magnets and electric currents. In view of this experiment it is not remarkable therefore that the lapse of time has confirmed MaxwelV's, stresses We ') hold the lines of force to be filaments of the aethereal vortices, due to rotations of the wave elements, as the waves recede from a magnet. If Am be the element of aethereal mass in rotation, and the z-axis coincide with the axis of the magnet, the angular momentum = — •of an element in the plane of the magnetic equator will be: .4 ^&m{yAxlAt x-iyjAi). This momentum of masses of aether S dw, about the axis of the line of force, tends to beat back the aether in the equatorial plane, and causes it to press in on the two ends, parallel to the z-axis. Hence we see the inevitable tendency of the lines of force to shorten themselves. Cf. Maxwell, On Physical Lines of Force, 1862, Scientific Papers, Vol. i, p. 508. : : i63 5048 164 for electrical action, yet shown on the other hand that the stresses conceived by him for gravitation are invalid, because in this latter case he conceived the pressure to be in the direction of the lines of force. MaxweU'i, conclusion as to gravitation is announced in the article Attraction (Scientific papers, vol. 2, p. 489): »To account for such a force (gravitation) by means of stress in an intervening medium, on the plan adopted for electric and magnetic forces, we must assume a stress of an opposite kind from that already mentioned. We must suppose that there is a pressure in the direction of the lines of force, combined with a tension in all directions at right angles to the lines of force. Such a state of stress would, no doubt, account We for the observed effects of gravitation. have not, however, been able hitherto to imagine any physical cause for such a state of stress.* It seems remarkable that Maxwell himself should not have seen the error underlying this reasoning. When we whirl a stone by a string, it is the tension of the cord which holds the stone in its circular path, thus overcoming the centrifugal force. If the string breaks, the stone goes flying away, alonpthe tangent to the instantaneous path at the moment \^'eTi the tension of the string is released. Innumerable examples of this central tension or pulling, necessary to overcome centrifugal force, should have occurred to Maxwell, as perfectly analogous to the forces which hold the planets in their orbits. It was seven years after the death of Maxwell (i St g) before the mathematical test required to overthrow the validity of his 'gravitational stresses was given by Prof. George M. Minchin in his Treatise on Statics, Oxford, 1886, Vol. II, pp. 448-455. Minchin calculates the Maxwellian gravita- P tional stress intensities at any point and finds the com- ponents to be = = C= A -R^ISny B E^jSny Ji'^IZny (37) E where is the resultant force intensity, and y t^^ gravitation constant. These expressions show that the three principal A stresses are equal. The component along the line of force, is, by Maxwell's hypothesis, a pressure, and the other two components are tensions. Apparently Prof. Minchin never seriously suspected the fallacy underlying Maxwell's assumption, that pressure in the medium along the radius vector of a planet could make its orbit curve about the sun, where in fact a tension, corresponding to the full breaking strength of stupendous cables of steel, is required to be exerted for holding a planet in its elliptical path. The nature of the curvature of the elliptic orbit was established by Kepler from the observations of Tycho, 1609, and first explained by Newton from the law of gravitation, 1687. After a very learned discussion, Prof. Minchin only reaches the conclusion that since on trial, the mathematical conditions specified by the stress analysis are not fulfilled, — »either gravitation is not propagated by the Maxwellian stress, or the aether is not of the nature of a solid body.« • This is a good historical example of a false premise, on which much ingenious mathematical effort was spent,, without detecting the physical error underlying the hypothesis. It will forcibly remind natural philosophers of Einstein'^ bizarre proposal to do away with the aether, without sub- stituting any medium or substance in the planetary spaces' which might exert contractile power for holding the planets, and stars in their orbits. It is scarcely necessary to add that if the signs of Maxwell's stresses given above be changed, sO' as to give a component of tension in the line of force, and two equal pressures at right angles thereto, thus = = A B -^Ji^lSny -Ji^Sur C= -R^ISny (38) gravitational phenomena would be explained. Fig. 8. Illustration of the development of stress between the sun and earth, owing to the interpenetration of the waves, rotating in opposite directions, from these two independent wave-fields, thus causing a tendency to collapse, in the medium between the two bodies, which furnishes the tension required to hold the planets in their orbits. In the Electrod. wave -theory of Phys. Forces, 1917, pp. 131-133, will be found an explanation of why the aether tends to contract between any two bodies, as the sun and earth. This may be made a little more obvious by the following diagram, in which each body is shown surrounded by a wavefield, the aether near either -body being so agitated by the waves from its own atoms as to be of less density towards either centre than in the remoter spaces between the masses. We are to conceive the waves from either centre, by interpenetrating with those from the other centre, undoing the wave stress, depending on the other mass, and thus causing a constant tendency of the aether to collapse, which results in pulling with maximum tension along the right line connecting the two bodies. : : : :: 165 5048 i66 This gives us a very simple and direct grasp of the mechanism underlying the planetary forces, which is not very different from those operative in electricity and magnetism, except for the essentially haphazard arrangement of the planes of the atoms in the heavenly bodies. These bodies are only — slightly magnetic, this power depending on the lining up of a srriall fraction of their atoms, in planes which are mutually parallel, as in common, magnets; while the great mass of the atoms are tilted haphazard. The resulting action yields the central force called gravity, instead of the duality of powers noted by Airy (Treatise on Magnetism, 1870, p. 10) for the magnetic attraction directed towards two poles. = d® dO/Sx-dx-hdW/dydy-hdW/ds-dz ^^^' d0 being an exact differential, to which Poisson (Traite de Mecanique, 1833, Tome II, p. 697) and Cauchy have given so much attention, in the period immediately preceding and following the development oi Fourier' & analysis, (1807-182 i). This method finally appeared in the celebrated Theorie Ana- lytique de la Chaleur, 182 i. Besides the above reference to Foisson's Mechanics, we cite the important memoirs indicated below ^). + = Foisson usually treats his differential equation in the form 820/8^2_^2(92Q)/a^2^_g20/9^2 922>/a^2) ^ 7. Sextuple Integration, under Fourier's Theorem, for solving Foisson's Partial Differential Equation 82(p/8^2-_^2y2Q) fo rthevelocity- potential, in amedium like the aether, capable of freely propagating waves. We consider the partial differential equation for the velocity-potential in wave motion Thus three O is any solution of the equation (39), which involves variable coordinates, x, y, z, atid the time, /. By a well known form of Fourier's theorem we have: -f-00 -l-OO = P.{x) i/2n-^ J/^-^)^V(-i).^(g).d5dA. (40) — 00 — 00 — And as this will apply to the several variables, we get by three successive integrations between the limits 00 and -1-00 = + 0^n{x,y,z,i)=^ ( i/87r3) JJJJJJ,^l/(-0.i3( J, ^,C,^)-d§di?dUAd;(td^. A (^-x) X {^-y) (.^i^-z) v. (41) If now we substitute the derivatives of this result in (39), observing by the form of A, in (41), that we have upon actual derivation: = (32/8^2^.82^9^2+82/8^2) ,^l/(-i) ,^l/(-i) (_p_^2_„2) (^3^ we have for the solution of the original equation involving the four variables = d^(D/dt^-aHd^, i) is determined so as to satisfy the equation: + = ^•'9.{t,riX,t)l-dfi+a^[l''+l>,'' v'')9.(t,7iX,t) 0. (44) We therefore integrate this differential equation, and in place of arbitrary constants, we introduce arbitrary func- tions y^i and i//i of J, ri, "Q- Accordingly our solutions yield the following particular integrals = 9.[-%,riX,t]^eB^'^^-'^'vA%,ri,l) S2(l V. ^, ^) ^~^''^^~'^ ^Piil V. i) = F {P+f^'+v'Y" . (45) If now we substitute the first of these in (41), and include the integration factor i/Stt^ in the arbitrary function, we have (limits of integration —00 and +00): = ® =- n{x,y;z,t) JJJJJp(^+^^')^(- ')^Pi(g,^,C)-d?d^dCdAdf.d,^. (46) This is a particular integral of equation (41), and the second value in (45) would lead to an identical result, as may be proved by actual substitution. Thus it only remains to complete the solution from Let = dydx^-hd^dy^-i-dydz^ y/ =t e^ so as to reduce the given equation to the symbolical form: ®— = — ® [^/D(D i)] «^^- o = where 8/89 D. Then the transformation: ^ = O ^— 6.8j(/8e S^/S/ such particular solutions. (47) (48) (49) will give: — / [^/D(D- .i)],29.^ (S°) © which is of the same form as the equation for in (48). ') I. Fourier. Oeuvres de Fourier, Tomes I et II, publiees suos les auspices du Ministere de I'lnstruction Publique par les soins de Gaston Darboiix, Paris, i888.' 2. Poisson: a) Memoire sur la Theorie des Ondes, Dec. 18, 1815; M^m. de I'Acad., T. I. b) Memoire sur I'lntegration de quelques equations lineaires aux- differences partielles, e;t particulierement de I'equation generale du mouvement des fluides ^lastiques. Juill. 19, 1819, Mem. de I'Acad., T. III. c) Memoire sur le Mouvement de Deux Fluides l^lastiques Superposes. Mars 24, 1823, Mem. de I'Acad., T. X. - d) Memoire sur I'^^quilibre et le Mouvement des Corps ijlastiques. Avril 14, 1828, Mem. de I'Acad., T. VIII. e) Memoire sur I'Equilibre des Fluides. Nov. 24, 1828, Mem. de I'Acad., T. IX. f) Memoire sur la Propagation du Mouvement dans les Milieux Elastiques. Oct. 11, 1830, M^m. de I'Acad., T. X, g) Memoire sur I'J^quilibre et le Mouvement des Corps Crystallises. Oct. 28, 1839, M^m. de I'Acad.j T. XVIII. 3. Cauchy: a) Theorie de la Propagation des Ondes a la surface d'un Fluid Pesant d'une Profondeur Indefinie, 1815. b) Sur I'lntegration d'Equations Lineaires. Exercises d' Analyse et de Physique Math^matique, T. I, p. 53. ci Sur la Transformation et la Reduction des Integrales Generales d'un Systeme d'Equations Lineaires aux differences <''• partielles, ibid. p. 178. : : 167 5048 i68 It thus follows that / admits of expression in the form (46), and therefore by merely changing the arbitrary —00 function, we have (limits of integration and -hoo): = - X Sr{x,y, z, f) 8/8/ SSS^S^ ,(^+^/'') Vi- . y/, (l^, C) • d? di} d^ d^ d^ d^ 151) To get the complete integral from these independent particular integrals (46) and (51), we add the two solutions multiplied by arbitrary constants, (cf. Hattendorffi edition of Riemann% Partielle Differentialgleichungen, 1882, p. 100), —00 +00 which may be included under the sextuple integral signs (limits of integration and : -C2I I52i = mill Z^^^'^')!^^-')- y^i(?, n, n-d?d, dU^d/*dr +8/8/ llllll /^+^^') 1^(- ') • nh [I, n> d? dri dC d;i dft d^ . These sextuple integrals admit of reduction to double integrals leading to a form of solution originally obtained by Poisson\ but Cauchy has made this reduction by means Accordingly, at the time 4 there /=oo are 1=1 (53) :oo of of a trigonometrical transformation. The only essential precaution to be taken is to avoid processes by which the functions to be integrated become infinite within the limits. The above equation belongs to the general form = ^© 92(D/8/2 (54) ^ where is a function of the derivatives with respect to the these concentric wave surfaces, all moving with the velocity c, which is the velocity of light. But the time / also flows on, = and if there be i intervals, the summation ^, i 00 will 2=1 yield for the double integration of intervals and wave?: coordinates 9/8jc, 8/8j)/, 9/8z. For all such equations the method above outlined furnishes directly a solution expressed by sextuple integrals, which are reducible to the Poisson- A Cauchy double integrals, if is homogeneous and of the z= I /^ I which corresponds to all the points in an infinite plane. second degree, as in the case of a sphere surface, with radius increasing uniformly with the time = ^2-i-y-(-^2 f2/2 (ss) where c is the parameter representing the velocity of light. As was long ago pointed out by Fourier, Poisson and Cauchy, integrals of this type are peculiarly appropriate for Imagine another system of coordinates (?/, rji, ^i), with its origin at the centre of gravity of mi {^htTju, ^h,^i), to which the moving waves are referred at i times, so that for the n bodies we have: For the Bodies. For the Waves emitted. Wl (xi,J'i,Zi,/i) the expression of those disturbances involving the transmission of energy in a medium, as in the steady flow of waves, whether W22 [x2,y-i,Z2,ti) m^ (••«3iJ>'3,^S,4) m — ['ihi— h.i, rihi—rihi, ^hi t,hi, h) (56) of sound, light, heat or electrodynamic action. These wave disturbances are propagated through the medium in question mn [Xn,yn, Zn, h) — mn {^I„i—^hi, 'TjI^i ^Iu, ^I„i—^hi, ti) . with a finite velocity, and unless the waves are regularly renewed the original disturbance leaves no trace behind when it has passed by; so that the upkeep of the energy flow involves periodic renewal of disturbances for maintaining the steady flow of waves. In his Theorie Analytique de la Chaleur, 182 I, ./^(?z^rzVr continually emphasizes the incessant movement of heat. Solution of Poissons equation for the velocity- potential O in wave motion from n bodies. Let there be n bodies emitting waves : wzi with coordinates [xi, }\, Zu t-j) surrounded at the instant t^ by an infinite series of wave surfaces, which for simplicity we may suppose to be spherical x-^^-^yi'+Zi_^-Ci^ti^= o — = Xi'^-\-yx 0/q) dpjdy = {i/Q)Sp/dz dFldz dpjdy = + + y-j^ dO/dt-^y,[{dW/dx)^ {d(D/dyY {d(D/dzY] . And the equation of continuity + = + dQ/dt^d/dx[Q-d0/dx) d/dy(Q-da)/dy) d/dz{Q-d0/ds) o. 60 (61) (62) For an incompressible fluid the second expression in (62) vanishes: + = d^CD/dx^+d^e/dy^ d^Wfdz^ o . (63) But the aether is not incompressible, and this equation there- fore does not apply to any gaseous medium. In general the exact form of the wave surface cannot be defined, owing to changes in the density and elasticity of the bodies penetrated by. the advance of the wave front. If the medium 'be symmetrical in, respect to three axes at right angles, as in the case of certain Crystals, then the wave surface,, from a disturbance at the centre of such a mass, will pass, from the spherical form: = x^-hy^^z^-c^f' o (64) and take the form of an ellipsoid Of three unequal axes x^/a^-hy^/fi^-^zyr' c' f (65) where the axes a, /J, ;' denote the conductivities along the ct= axes of the ellipsoid, and i, at any stage of the progress with the -wave surface in the form of the ellipsoid: x^/a^+yy/S^-hzyr^ == I . (66) It follows therefore that the problem of wave motion involves the solution of Poisson % equation = Z-^Ol^f «-''(82a)/8^2+8-'®/8/-H82(Z)/8^2) (67)" where a is the velocity of the wave propagation (cf. Poisson, Traite de Mecanique, 1833, tome II, p. 663—720; or Lord Rayleigh'i Theory of Sound, vol. II, chapter XIII). w Let u, V, be the component velocities parallel to the axes Ox, Oy, Oz of an element of mass dm, at the in- stant t, so that, = X— a:' J«d/ y-y' ^= ^'I'dl ^ — z s' ^wdt. (68) = ') If for any part of an elastic fluid mass A^ == uAx-^vdy+wiz o be a perfect differential at. one moment, it will remain so = for all subsequent time. When <& is siiigle valued, the integral round any closed circuit vanishes, I d o. This is- the irrotational condition =0 of hydrodynamics. Hence, with condensations and rarefactions alternating, and of equal intensity, in wave motion, the above condition I d"!" * = is met by the plane wave /4 cos[2ic/X-(x—a^)], which is typical of the velocity-potential in general. : . . : : : 171 5048 172 If we neglect the squares of the velocities dW/dx, = O dojdy, dO/dz, and put z/ o, w =^ o, will become a function of x and t only = 820/8^2 ^2 820/8^2 = = O ii[x,t) Acoi[2nlX-[x-at)\^). The solution obviously is an' undulation of flat wavelets parallel to the axis of x, traveling with velocity a. Let f be the velocity in the direction vector, so that the resultant = t, V{u^-^V^-i-W^) of the radius (70) then since for spherical disturbances = x^-hy'^-\-z^ r'^ xdx-hydy-^zdz == rdr = = = u t, x/r V Zy/r w t, z/r ^~w£—g£i- = = udx-\-vdy-\-wdz l^dr ^ dO/dr = d(D/dx ==d(D/dr-x/r d(D/dy d(D/dr-y/r = dojdz dO/dr-z/r. (70 (72) (73) Differentiating a second time, we have = d^W/dx^ d^0/dr^-x''/r^-hd0/dr-{y^-hz^)/r^ d^ai/dy-i =d^0/dr'^-yyr^-i-d(D/dr-(z^-i-x^)/r'* = 82(0/822 d^OJdr^- z'^/>''^-^d0/dr-(x^-i-y'')/r'^ . (74) By means of these values, Poisson's equation, = 82a)/8/2 a^d-^'m/dx^-^-d^O/dy^-hd^O/dz'^) becomes = d^cp/df^ aHdi0/dr^-h2/r-dW/dr) (75) This is the same as = d^rO/dt^ a^d^rW/dr^) (76) the complete integral of which is r(D =/{y-hat)-hF(r — ai) / where and J^ are two arbitrary functions. (77) = By extending his analysis (Traite de Mdcanique, 1833, X vol. II, p. 706) Poisson shows that since dOjdr, we have = I ilr.f[at-r)-hilr'-f{at-r) = s ijar-f'[at—r) (78) Accordingly, Poisson concludes that at a great distance from the centre of this disturbance we may neglect the second terms of the values of t^, which are divided by r\ in comparison with the first, which are divided by r. Thus for the whole duration of the movement we get for the conden- sation or dilatation 5 = ^ 'Qja . (79) By equation (78), therefore, the velocity of the mole" cules in a gaseous medium decreases inversely as r, just as in the amplitudes of the waves postulated in the kinetic theory of the aether. The condensation or dilatation .y varies as the velocity in the direction of the radius vector, which itself varies inversely as r; and also inversely as a, the velocity of wave propagation. Accordingly, for a highly elastic medium, s is small, and decreases very rapidly; which confirms our view that the amplitudes of the aether waves are very minute, and decrease inversely as r in receding from the sun. In finishing this paper, Febr. 19, 1920, I am surprised to notice Poisson % sagacious remark (p. 706): »La vitesse propre des molecules d'air decroitra alors en raison inverse A = de /-« : which affords an unexpected verification of the writer's formula for the amplitudes of the aether waves, kjr, also derived from the kinetic theory, but by a different process. It thus appears that Poisson had such a result for the waves of sound 87 years ago, and its neglect for nearly a century is remarkable. As Lord Rayleigh points out in his Theory of Sound, 2°'' edition, 1896, vol. II, p. 16: the rate at which energy is transmitted across unit area of a plane parallel to the front of a progressive wave may be regarded as the mechanical measure of the intensity of the radi'ktion. This is the basis of Lord Kelvin's celebrated paper of 1854, »On the possible density of the luminiferous medium, and on the mechanical value of a cubic mile of sunlight*, (Trans. Roy. Soc, Edinburgh, 1854), which we have used, in our first paper on the new theory of the aether, for calculating the density of this medium. The energy transmitted, in the direction of the three coordinate O axes, being taken successively as a function of x (and t), = = y (and t), z (and /) only, is given by the approximate equations: 320/8^2 ^2 . 820/8^2 82®/8/2 «2 . 82(J,/8j,2 = 82a)/8/2 «2. 82(0/822 (8c) which are expressed in (75) above. In case the gravitational wave transmission occurs within = a mass of density q, we have Poisson's equation for the potential d^^VJdx^-^'d^Vldy-'+'d^Vl^z^+ATt Q o (81) instead of the equation of Laplace = d-^V/dx^-hd^V/dy^-i-d^V/clz^ o. (82) And thus within an elastic solid the equation (80) would = become 82®/8^2 a^.[^^(D/^x^-^-^^0/^y^-^-^^0/^z^-^-4^TQ) (83) which is of the form adopted by Riemann, for the induction of electric currents, in the memoir presented to the Royal Society of Gottingen in 1858, but subsequently withdrawn, and after the death of the author, published in Poggendorff's Annalen I31. 237-263, 1867. This investigation o{ Riemann was examined by Clausius (Poggendorff^'s Annalen 135.612) who doubts the validity of the mathematical processes for the phenomenon of electric induction, chiefly on the ground that the hypothesis that potential is propagated like light, does not lead either to the law of Weber or to the other laws of electrodynamics. In our Electrod. Wave-Theory of Phys. Fore, however, V = it is not held that potential is propagated like light; on the contrary that the potential is a function f[x,y,z,^, is fixed in space, yet depends on the total accumulated stress due to wave amplitudes of all the matter involved. Hence this criticism is not valid against the wave-theory here dealt with. = Moreover, we use Poisson's equation for the potential, S7^V-^4TCQ o, only within solid masses, Laplace's equation V^^^ o applying to all free space. Thus we adopt a transition between^ these two equations at the boundary of any mass of matter, as long recognized by geometers and natural philosophers. The physical meaning of the transition is the sharp difference in velocity of propagation for all aether waves at ') Lord Rayleigh, Theory of Sound, vol. II, p. 15-16, a""* edition, 1896. : 173 5048 174 the boundary of a mass of matter; and moreover the decrease in total accurmilated stress due to the aether waves from all the atoms, as the moving point J>(x,y,z) enters the body of — density q, and leaves behind a part of the mass, the aether waves coming from the atoms of this shell from all directions just balancing in a homogeneous sphere. But whatever the law of density or form of the body, there is a change in the sum of the second differentials of the potential at the boundary of the body, from Laplace's to Foisson's, equation Fig. 9. Curve of the potential function V, showing its asymptotic decrease with the distance, and the tendency to an asymptotic increase towards the centre; but owing to finite dimensions of the mass, a gradual decline to zero. This difference between Laplace's equation of the po- tential for free space, and Foisson's corresponding equation for space filled with matter of density q, owing to the inter- vention of boundary conditions, is distinctly favorable to the We wave-theory of physical forces. therefore presented the ^ treatment of the wave equation of Foisson S'^0/di'^ (jr^yag) for free space, by the general method of integration based on Fourier's theorem. This solution will hold for waves of any initial wave length, propagated with the velocity of light, from n bodies, in all parts of space, and everywhere mutually interpenetrating so as to generate maximum tension in the right lines con- necting the n bodies in pairs, in accordance with the observed phenomena of universal gravitation. If the solution will hold for separate bodies, from which spherical waves are emitted, it obviously will hold also for separate vibrating particles, within a single body; but here the mathematical difficulty is increased, by virtue of the unequal conductivity which heterogeneous solid bodies offer to wave propagation; so that the expression of the effects of the waves from the atoms would be infinitely complex. Yet the above equation (57) gives the approximate representation of the propagation of wave energy from atoms, which may be useful in certain problems of molecular physics. The solution in (57) already involves an infinitely complex integration, repeated «-times for the n bodies of the universe. To include the initial waves of all possible lengths, = = we should have to integrate this complex expression for (P between the limitsi A o; A co, involving all possible ^ periodicities, the number of which is: « [^/^]>.=o • Now, according to the researches of Prof. Flanck on E thermodynamic radiation, the energy of wave length X is given by the rather complex expression = Ex^l {klFTl)l{e^l^^'^'^-i)-?.nFTl-^Al (84) which admits of integration within certain limits. F T In this formula, and are the gas -constant and = absolute temperature, k V hV, being the velocity of light, = and h is Flanck's new constant, h 6.55 X io~^' ergs sees, = so that if the wave frequency be j', A Vjv and = = x klRTl hvJFT. (85) And Flanck's fundamental equation for the quantum = of energy of v frequency is e fiv . (86) = By the use of Planck's formula therefore ^^dA 87rJ?rA-* [*/(«*- i)]dA. (87) This integration, to take account of the various wave lengths, could be carried out, but the subject is iii too primitive a condition to be undertaken at present. 9. A Definite Criterion for deciding between the Great and Small Densities claimed for the Aether. In Section I of the first paper on the new theory of the aether, we have cited the claim put forward by certain electronists, that, on the hypothesis of incompressibility^ the aether has a density 2000 million times that of lead. In his Aether of Space, 1909, p. 91-105, Sir Oliver Lodge finds froni electrical theory that the density of the aether is 10^^, a million million times that of water. It is only fair to point out that as the aether transmits waves, as in light, heat, magnetism, electrodynamic action, and radio telegraphy, of the most varied length, and of various amplitudes, it is not conceivable that it should be incompressible, so that the dilatation is zero in the equation: = = 8 da/dx-hdji/dy-i-dyjdz o ' (88) where a, fi, y, are the displacements, and by equation (63). For this would make the wave velocity infinite, which is contrary to observation. Accordingly,_whilst the aether is highly incompressible, owing to the enormous velocity of the aetherons, and the resulting kinetic elasticity, this medium certainly is not incompressible. In the article Aether, Encyclopedia Britannica, ii'^'ed., igii, Prof. Sir Joseph Larmor is more poised and cautious than the writers previously cited, but his faith in the older theories is so shaken, that he intimates that the ratio of the amplitude of the waves to the wave length,' taken by Maxwell and Kelvin at about io~^, may be enormously overestimated. Larmor adds: »It is not impossible that the coefficient of ultimate inertia of the aether is greater than the coefficient of inertia (of a different kind) of any existing substance*; which shows his tendency to an abandonment of the older theory, under the teachings of the electron theorists. It thus appears that the excessively small density, found by Kelvin and Maxwell, namely, about lo""^*, or my own value at the earth's mean distance 438X10^^^ is opposed by the modern teaching in favor of an enormous density, about 10^'^, as stated by Sir Oliver Lodge. The difference 175 5048 I 76 between the two results presents an en6rmous contrast, name!)' the almost unlimited factor: F- 10 with the value of Kelvin and Maxwell; , , == 0.0023 X ,80'- 10 wi--t.h. .-. See'i value. (89) Accordingly, progress is nearly impossible with this irre- concilable difference of opinion among the learned. Brooks and Poyser, as representatives of the opinion of the electronists, state: »There is no intrinsic difficulty in either view, but at present (19 12) no method is known by which we may hope to discriminate between them.« The present writer has therefore labored to develop a criterion for the rejection of one of tTiese competing values, which would leave the other in possession of the field. Besides the ''above criticism, that the finite velocity of wave propagation excludes the incoropressibility of the medium, I have given in the Observatory, Nov. 19 18, p. 411— 412, a brief discussion of the consequence of the intolerable disagreement in the values of the •aether density. A simple calculation has enabled me to exclude Lodge'i density as wholly inadmissible, because if true the energy of the waves from the sun'falling upon a single square centimetre of the earth's surface would be able to yaporize the entire terrestrial globe in less than one minute of time, when ' we use Bigeloiu'% value of the constant of solar radiation, arid Kelvin and Maxwell's density. The mass of the earth is 5956292000000000000000 metric tons. If we take the average specific heat of the globe at 0.2, and the vaporizing point of its average matter at 3000° C, the total amount of heat required to reduce it to — — vapour the interior being assumed to be without heat Now if any good ground can be adduced for decreasing the ratio of the amplitude to the wave-length, I am willing — — to consider such a modification in the belief of the most eminent physicists, such as Kelvin, Maxwell, Larmor but it should be pointed out that to make the reconciliation of the extreme values complfete, the ratio of the amplitude to the wave- length will have to be lowered by the enormous factor ' p^ ,0^30 (g2) so that AlX now taken at io~^, would become = ^'A Io-3^ (93) The difficulty of this extreme step is so great that I dismiss it as quite inadmissible. Until new evidence, resting on ground more secure than mere assumption, is available it must be held that Sir Oliver Lodge's attempt to reply to this criticism completely breaks down. For even if we took = All 10-=, or .4/1= 10-'' (94) — which are values looo or loooo times more extreme than appealed to the experienced judgements of Lord Kelvin, Max- — ivell and Lannor, the required factor would scarcely be reduced in a sensible degree; and practical experience in physical science certainly would not justify us in exceeding the limit of lo"'' As a final argument against the electrical theory, assigning the aether a density of 2000 million times that of = lead (namely: i 1.352 X 2000000000 22704000000000 times that of water!), we may recall the fainiliar experience of a man swimming in water. Here the swimmer is immersed in an inert liquid of about the same density as his body; would be H = 5.g56292X lo--'- (0.2 X 3000) X 1 000000 calories, = = 6X 5.956292 X 10^" 3.6 X 10^", nearly. (90) yet to move about a strong exertion is required of the most, powerful muscles, completely under the control of the will. If the liquid had the density of quicksilver, the swim- Now Bigelow\\'^\\\t of the solar constant is 3.98 cal. mer would scarcely sink down to his boot-tops, and his per minute, or 0.0663 cal. per second; and, as Lodge's value muscles would be altogether too feeble to displace such an of the density of the aether is about 10^° that above cited from Kelvin sxiA. Maxwell, and 0.0023X10'*'' times my own value,^ we have for the effect of such an increase in density the raising of the solar radiation by the factor lo'"': = H=^ 0.0663X 10'" 6.63X 10^^, Kelvin 2.ud Maxwell or ZT^^ (0.0663X0. oo23)x 10^", with Sees value (9^ inert and heavy liquid, if he were required to move through it: yet he could walk over such a magma, by great effort, analogous to that required when we walk in very yielding volcanic ashes. Now the density of mercury (13.6) is a little greater than that of lead (11.352), but the moment we consider an The first of these values would vaporize the earth in' aether 2000000000 times denser than lead, we perceive the 54 seconds of time, the second in 0.277 of a day. But in nature this vaporization does not occur, and thus we conclude that the density of the aether stands at a value near that fixed by Kelvin and Maxwell many years ago, but slightly improved in the writer's new theory of the aether. In the Observatory, for Dec, igi8, p. 446, Sir Oliver Lodge has attempted to reply to my criticism by pointing out that the energy of the solar radiation depends on the amplitude of the wave, compared to the wave length, which with Kelvin and Maxwell I took at io~^, a value pronounced by Sir Joseph Larmor (in the article Aether, p. 292) »a very safe limit*. Lodge also adds: »many facts have suggested that the amplitude of the most brilliant light is exceedingly small compared with its wave length*. culmination of absurdity 1 Even if it penetrated all bodies quite perfectly, and gave equal pressure on all sides, still some displacement of the particles would be required when we move about in it, as in the case of water displaced by a swimn^ier. Obviously no living physical body would be capable of displacing such a dense medium; and we see that even the strongest stars, planets and comets would be dispersed to atoms under the changing resistance such a medium would interpose to their variously accelerated motions. The electrical theory assigning the aether a density 22704000000000 greater than that of water is therefore the best possible illustration of a physical Reductio ad Ab- surdum, and we know that either some premise or some : ; 177 5048 178 link in the chain of reasoning eventually will not bear in- vestigation "). In the article Aether, Encyclopedia Britannica, ii'*'ed., 19 1 1, Prof. %vs: Joseph Larmor concludes that we must treat the aether as a plenum. Under the influence of electrical theory, he even speaks as if the aether were not molecular. In dis- — cussing the transparency of the celestial spaces, to which much attention was given by Cheseux and Olbers, W. Herschei — and W. Struve (cf. Etudes d'Astron. stelL, St. Pdtersbourg, 1847) — Larmor first recalls the well known transparency of space shown by astronomical research, and then adds »If the aether were itself Constituted of discrete mole, cules, on the model of material bodies^ such transparency We would not be conceivable. must be content to treat the aether as a plenum, which places it in a class by itself; and we thus recognize that it may behave very differently from — matter, though in some manner consistent with itself, a remark which is fundamental in the modern theory:* The first part of this reasoning apparently implies that the aether is not molecular, at least »on the model of ma.terial bodies«. This may be correct in part, because no one would suppose the aether to be made up of complex mole- cules, underlaid by a finer medium, such as the aether is to the more complex masses of common matter. On the other hand there is not the smallest objection to an aethereal medium made up of spherical perfectly elastic monatomic elements, so called aetherons, having a diameter of I:4oo5'^ of a hydrogen molecule, and a mass of 15.56 millionths pf a millionth of such a molecule, such as we show do really exist. As no finer medium would underly such a monatomic aether, it coiild not dissipate the energy of wave motion, »on the model of material bodies*, and thus it would fulfill Larmor's condition of a plenum. This would give such an excessively fine monatomic molecular structure that the me- dium would penetrate all material bodies, but waves in such an aether would be very noticeably retarded in solid or liquid bodies, and much less so in gases, in accordance with physical experience. That the aether must necessarily be molecular Tollows at once from our every day experience with such granular bodies as fine gravel, grains of corn, sand, shot or mustard seed. If we fill a glass vessel with such coarse granular masses, and insert the fingers or any solid body, such as a rod, into the granules, we perceive that they are thrust aside to make way for the hand or solid rod. If we fill the vessel with water, oil, alcohol, ether, or any similar liquid, our experience in such displacement is the same. The liquid is visibly thrust aside and this holds even when the molecular structure is relatively so fine that a drop of water might be magnified to the dimensions of the earth without exhibiting — the molecules of larger size than footballs, as shown by Lord Kelvin in his well kiiown researches on the size of atoms. But it will be said that the aether penetrates all bodies, and thus we cannot sensibly displace it, as we can water, We oil, alcohol or ether. reply that it is -perfectly true that the aether penetrates freely all bodies, even the dense and' highly elastic or rigid masses of the earth, sun and stars, almost as if their molecular structure were absent: yet we learn from the phenomena of refraction and diffraction in our laboratories, that light waves in the aether are very perceptibly retarded in their motions through transparent bodies; and in our investigation of celestial phenomena, we find from the investigation of the motion of the moon that the sun's gravitational 'waves, though of such length as to pass through the earth, are yet sensibly Refracted; and perhaps dispersed or partially absorbed at the time of total eclipses — of the moon, whence arises the fluctuations of the moon's mean motion established by Newcomb in 1909, and explained by the present writer in 19 16, (cf. Electrod. Wave-Theory of Phys. Forces, vol. i). From these considerations it appears that we have both terrestrial and celestial evidence that the aether is molecular, but of such excessively fine grained structure that no finer "rnedium whatever underlies it: thiis it penetrates all bodies freely, under an elastic power, or expansive tendency, 68932 1600000, tirries greater than our atmosphere exhibits in proportion to its density, as more fully shown in the first paper, sect. 4. 10. The Kinetic Theory of the Aether accords with the Views o{ Newton, 1721, and oi Maxwell, 1877. In order to further illuminate the above discussion we may recall the earlier though little known views of Newton and Maxwell, on the physical constitution of the aether. a) Views of Sir Zr««(:iVi'2<'2'^z, Treatise on Optics, 3''''ed., 172 I, p. 325 et seq. ^) »Qu. 20. Doth not this Aethereal Medium in passing out of Water, Crystal, and other compact and dense Bodies, into empty Spaces, grow .denser and denser by degrees,, and by that means refract the Rays of Light not in a point, but by bending them gradually in curve, lines? And doth not the gradual condensation of this Medium extend to some distance from the Bodies, and thereby cause the Inflexions of the Rays of Light, which pass by the edges of dense Bodies, at some distance from the Bodies-?« »Qu. 21. Is not this Medium much rarer within the dense Bodies of the Sun, Stars, Planets and. Comets, than in the empty celestial Spaces between them? And in passing from them to great distances, doth it not grow denser and denser ') In the Optics, 1 721, pp. 342-3, Netvion discusses the very problem here treated of in the fbllowing manner: "The resistance of water arises principally and almost entirely from the vis inertiae of its matter; and by consequence, if the heavens were as dense as water, they would not have much less, resistance than water; if as dense as quick-silver, they would not have much less resistance than quick-silver if absolutely dense, or full of matter without any vacuum, let the matter be never so subtile and fluid, they would have a greater resistance A than quick-silver. solid globe in such a medium would lose above half its motion in moving three times the length of its diameter, and a globe not solid (such as are the planets) would be retarded sooner. And therefore to make way for the regular and lasting motions of the planets and comets, it's necessary to empty the heavens of all matter, except perhaps some very thin vapours, stea,ms or effluvia, arising from the atmospheres of the earth, planets and comets, and from such an exceedingly rare aethereal medium as we described above. A dense fluid can be of no use for explaining the phaenomena of nature, the motions of the planets and comets being better explain'd without it." '') Quoted at length, because (his edition is very inaccessible to the modern reader. « : «: 179 5048 i8o perpetually, and thereby cause the gravity of those great Bodies toward one another, and of their parts towards the Bodies; every Body endeavouring to go from the denser parts of the Medium towards the rarer ? For if this Medium be rarer within the Sun's Body than at its surface, and rarer there than at the hundredth part of an inch from its Body and rarer there than at the fiftieth part of an inch from its Body, and rarer there than at the Orb of Saturn; I see no reason why the increase of density should stop anywhere, and not rather be continued through all distances from the Sun to Saturn, and beyond. And though this Increase of density may at great distances be exceeding slow, yet if the elastick force of this medium be exceeding great, it may suffice to impel Bodies from the denser parts of the Medium towards the rarer, with all that power which we call Gravity. And that the elastick force of this Medium is exceeding great, may be gathered from the swiftness of its Vibrations. Sounds move about 1140 English feet in a Second Minute of Time, and in seven or eight Minutes of Time they move about one hundred English Miles. Light moves from the Sun to us in about seven or eight Minutes' of Time, which distance is ^bout 70000000 English Miles, supposing the horizontal Parallax of the Sun to be about 12". And the Vibrations or Pulses of this Medium that they may cause the alternate Fits of easy Transmission and easy Reflexion, must be swifter than Light, and by consequence above 700000 times swifter than Sounds. And therefore the elastick force of this Medium, in proportion to its density, must be above 700000 times 700000 (that is above 490000000000) times greater than the elastic force of the Air in proportion to its density. For the Velocities of the Pulses of elastic Mediums are in a sub-duplicate Ratio of the Elasticities and the Rarities of the Mediums taken together.* »As Attraction is stronger in small Magnets than in great ones in proportion to their bulk, and Gravity is greater in the surfaces of small Planets than in those of- great ones in proportion to their bulk, and small Bodies are agitated much more by electric attraction than great ones; so the smallness of the Rays of Light may contribute very much to the power of the Agent by which they are refracted. And so if any one should suppose that Aether (like our Air) may contain Particles which endeavour to recede from one another (for I do not know what this Aether is) and that its Particles are exceedingly smaller than those of Air, or even than those of Light: The exceeding smallness of its Particles may contribute to the greatness of the force by which those Particles may recede from one another, and thereby make that Medium exceedingly more rare and elastick than Air, and by consequence exceedingly less able to resist the motions of Projectiles, and exceedingly more abl6 to press upon gross Bodies, by endeavouring to expand itself. »Qu. 22. May not Planets and Comets, and all gross Bodies, perform their Motions more freely, and with less resistance in the Aethereal Medium than in any Fluid, which fills all Space adequately without leaving any Pores, and by consequence is much denser than Quick-silver or Gold ? For instance; If this Aether (for so I will call it) should be supposed 700000 times more elastic than our Air, and 700000 times more rare; its resistance would be above 600000000 times less than that of Water. And so small a resistance would scarce make any sensible alteration in the Motions of the Planets in ten thousand years.* In Newton' % views above quoted, Qu. 20, dating from 172 1, it will be noticed that he not only held the aether = to be a superfine gas, of enormous elasticity, but also cal- culated this elastic power to be ^7 490000000000 times greater than that of air in proportion to its density. By the = most careful calculations that can be made today, we find this relative elastic power to be «Y+(«^/cr ^ (17) It will be proved hereafter that there is a fundamental error in the wave-theory of light, as handed down by tradition from the days of Young and Fresnel; and that in a ray of common light the aether particle not only has trans- verse motion, but also a corresponding longitudinal motion, depending on the small ratio of the amplitude a to the wave length X. After polarization these natural free motions of the aether are restricted, by the resistance impressed upon natural light, in the surface action of reflection, or transmission through transparent bodies, crystals, etc., and by unsymmetrical transparency in different directions, as in tourmaline, which forces half the light into one plane and destroys the other half. Originally the general path of the aetheron was elliptical, and although now transformed into oscillations near one plane the vibrations in most cases still are narrow ellipses, because it is proved by the reflection of plane polarized light from « 241 5079 242 a silver surface that an almost circular polarization results, whereas that reflected from galena has very narrow ellipses. This could not well result unless the polarized light before reflection from these metals described narrow ellipses, which are not exactly straight lines. Now the elliptical paths established by equations (16), (17), (18), are similar to those" analysed by Herschel in Section 618 of his great article Light, 1849. Suppose we consider the part of these waves which in a polarized ray have only right-handed rotations. Then if such a selected beam traveling along the ^-axis be looked at flat on, from a point on the 2-axis, the paths of the aetherons would resemble the motions of the particles of water in Airy's figure given as fig. i, except that the aetherons may have paths more highly elliptical than are shown by At'ry. This is the simplest form of the oscillations in the new wave-theory of light, which will be developed in the fourth paper: and we shall now see if it is possible to find corresponding oscillations in the field of a magnet and of an electric current. In the year 1845 Faraday made a celebrated experiment in which he passed a beam of plane polarized light along the lines of force; and discovered that when the light travels in a material medium such as heavy lead glass, carbon- disulphide, etc., the plane of polarization is twisted by the action of the magnetic field. Not only is the plane of polarization rotated, but the rotation increases in direct pro- portion to the length of path traversed; and even when the light is reflected back and forth many times the twisting of the plane of polarization is always in the same direction like the helix of a circular winding stairs, as was long ago noted by Sir John Herschel. In the article Wave-Theory, Encyclopedia Britannica,