zotero/storage/2WNYUPK5/.zotero-ft-cache

Journal
of the
Optical Society of America

Devoted to all Branches of Optics

Vol. V

JULY, 1921

Number 4

THE PROPAGATION OF LIGHT IN ROTATING SYSTEMS*
BY
L. SILBERSTEIN
The purpose of the present papert is to investigate some questions concerning light propagation in a uniformly rotating rigid system, such as the Earth, on both the aether theory and the relativity theory.
On both theories we shall have to understand by "rotation" a rotation of our rigid system' S with uniform angular velocity relatively to the fixed stars, or to any other inertial system, which will be shortly referred to as the reference system S*. That such a specification of rotation is still necessary even in the relativity theory, in spite of appearances to the contrary, will become clear in the sequel where we shall also have the opportunity to point out some outstanding difficulties of the relativistic gravitation theory with respect to the concept of rotation.
With regard to the rotating system S itself, it will perhaps be well to have in mind our own Earth. The more so as the most interesting experiment in connection with our subject will be a purely terrestrial one.
1. To begin with the aether-theory treatment, let be the (scalar) angular velocity of the Earth (S) relatively to the fixed
* Communication No. 123 from the Research Laboratory of the Eastman Kodak Company.
t Paper read December 29, 1920, at the Chicago Meeting of the Optical and the Physical Societies.
1 It is well known that among the possible motions of a relativistically rigid body (as defined by Born and Herglotz) there is uniform rotation, such as is familiar to us from the kinematics of ordinary classical rigid bodies.
291

292

L. SILBERSTEIN

[J.O.S.A., V

stars (S*) and let K-1 be the rotatory dragging coefficient at and near the surface of the Earth, in other words, let KC5be the relative angular velocity of the aether and the Earth. If the unit vector k is taken along the positive axis of rotation, the vector velocity of the aether stream past a point P of S will be
u=KCoVrk,
r being the vector drawn from the center of the Earth (or from any point fixed on the axis) to the point P; in Cartesians, with
r=xi+yj+zk,
Ux=KCy, Uy=-KWX, Uz=O,
and the resultant velocity, = KCorw, here is distance from the axis. As to the factor K, it is not our intention to prejudice its value, which may be any fraction from zero to unity, corresponding to a full drag and to no drag, respectively.
The propagation of light in S* being isotropic, of constant velocity c, the velocity of light in S, always in vacuo, along the wavenormal n (unit vector) will be

V=c+un=c I+-~ co (,.(la)

C

.. . . . . .. . . . .

Notice that, in the case of the Earth, c = 27r/86164, the reciprocal

length Co/camounts only to about 2.43X 105cm.-', so that even if r

be of the order of the Earth's radius, the factor - is a very small
C
fraction. Such being the case it will be enough to retain in all our formulae KC.r/citself, rejecting its square and higher powers. Now, rigorously speaking, formula (1) is valid for the wavenormal and not for the ray or the tangent to the "light path." But n appears in (la) only in the term multiplied by KC.r/c, and since n differs from the light ray only by small terms, we have up to higher order terms, simply

-= 1+- (up)=1+cos 7 ...........................

(1)

C

C

C

where p is a unit vector along the optical ray, and y the angle

between p and u. Such being the expression for the velocity of light along the
ray, we can at once find the shape of the ray or light path in S

July, 1921]

OPTICS OF ROTATING SYSTEMS

293

by means of Fermat's law, which it will be enough to work out
in detail for the case of a light ray contained in a plane parallel to the equatorial plane.2 In fact let do- be a line-element of the

ray; then Fermat's principle is

f

do-

=

o,

the

limits

of

the integral

V

being fixed. Now, introducing in the said plane polar co-ordinates

r, 0, the latter measured positively in the sense of the rotation of S,

we have

cos e=-r do/do=r 0' siny,

so that, by (1),

-=V1I +-rr2-WCnKdOi s' iny,

C

C

and Fermat's principle becomes, after easy reductions, and

considering as corresponding points those having the same r,

f{A?.{4[v] }8'

Thus, d =o, and the required equation of the light path dr

becomes

1l 0a'[1v~sin]-y

Now c vsiny

1+rO

. Thus after simple reductions

2V-KIji

22o

and putting y =X+h,
2

r(I -2 sinc,) A

sinn+

= -, where co=-, A=const.

(1-rsinn) I r

c

Rejecting second order terms, the left hand member of this equa-

tion can be written sinq+owr.

Ultimately, therefore, the equation of the light path in S be-

comes

r( sinq+-)=A ............

(2)

2 If the end-points (any two points) of a light path be in such a plane, the whole light path is contained in that plane.

2941

L. SILBERSTEIN

[i.O.S.A., V

where atis the angle under which the light path cuts the radius vector (Fig. 1).
Fig.

Notice that in absence of rotation (2) reduces, as it should, to

r sin -q= const, this being the equation of a straight line. There is

thus, due to rotation, a first-order deviation of the light rays from

straight lines.

In order to introduce into the ray equation the polar co-

ordinates r, instead of r, q, it is enough to remember that

sing=r

ddo-:

I '\Al+r

2

i'do -

dr ~dr

Thus (2) becomes, with p, B =1+2A , co=KC,
-do- rdr + Adp
-VBr2-A2 VB-A2p2
whence, integrating, putting A /VB = r,, and counting 0- 0 from the radius vector r = ro (in the sense of rotation of S),

cos [OaOoOwXr2r-

rO

Here B=1+2wo A and A=ro (1+row), so that rejecting second order terms and with the same right also replacing /r2 - r.2 by r, tanO, we have ultimately

ro= CosF 0 O_

r

L0

r,

J1=/rCosLFO0O

±-tan oCa

(o-oo) (3)
Il I

July, 1921]

OPTICS OF ROTATING SYSTEMS

295

the upper or the lower sign to be taken for light travelling in the sense of rotation of S or in the opposite sense. Or, to put the sign
Fig. 2

a ,O~~~~~~=

rule in a more convenient form, the light path is convex (or bulges out) always to the left of a person3 walking in the direction of propagation (Fig. 2).
This equation of the light ray can also be obtained more directly by transforming the light rays of S* to our system S by means of the substitution r = r', 0' = 0+Kct (and t' = t). In fact, the equa-
Fig. 3
I*
tion of a straight line, and such is in S every optical ray, can be written
r' cos'=ro "-const.,
and this becomes, through the said transformation,
= COS(+Kc~t). r 3 For whom the rotation of S is clockwise.

296

L. SILBERSTEIN

[J.O.S.A., V

In other words the light path is the path of a particle moving with
uniform velocity c along a straight line which is itself spinning
uniformly around 0 (Fig. 3). If t =o for r = r. we have ct = ct'=

V/r2- rI2, which gives again

ro

F _

- COS O_o +

1
r2 ro2 X

as under (3). Developing the cosine, up to the second order, we can write

r Cos(0-)F. [I -tcan(0-o) I . (3a)

The constants 00,rocan be determined if either the initial direction

or any two points, A, B of the light path, say the sending and the

receiving stations, are given through two pairs of r, 0. It will be

noted that the light path or ray BA (i.e., with B as sending and

A as receiving station) does not coincide with AB. The two

optical rays AB and BA enclose between them a certain area,

having the shape of a biconvex lens. In other words light propaga-

tion in S is irreversible. Under appropriate circumstances A

will see B without being seen by B.

The last ray-equation can still be simplified. Putting the x-

axis along the radius vector r(O = 0o),so that

x=r cos (0-0o), y=r sin (0-0o),

we have x(1 + ey2/x2) = ro, whence

x =rO~l ey2

roK.-

,

x~o1x-2),

1E=....c.....(4)

In the small term we can write, with the same approximation

x = r0, so that the ray equation becomes

X=ro± y2.......... .

(4a)

C

Thus, up to the second order, the optical ray (which more rigor-

ously is a complicated spiral) becomes simply a parabola, with

apex in shortest distance (ro)from 0. If the propagation is in the

sense of rotation of S, the parabolic ray turns its convex side

towards the axis of rotation. In the opposite case it will turn

its concave side towards 0. Various problems of what may be

called optical trigonometry of the rotating system, i.e., problems

concerning triangles or polygons built up of optical rays, can

July, 19211

OPTICS OF ROTATING SYSTEMS

297

now be easily dealt with, namely by constructing the said parabolic arcs as light paths and keeping well in mind the

sense of propagation.

By way of illustration consider an optical triangle traversed,

say, in the sense of rotation, and having its corners A, B, C on the

circle r=a. Given the angles 2=BOC,

2 2 =COA,

2 3=AOB

(f1 +c 2 +03 = r), find the sum of the angles a, I3, y of the optical triangle ABCA, the order of the letters giving the sense of propa-

Fig. 4

A

gation. Denote by z the angle between the optical ray CA and the radius vector at A, and let fl2, 3 have analogous meanings for B and C, as in Fig. 4. Then, since the angle OCA is also equal nq,and similarly for the remaining angles, we shall have
a=771+772, 0=12+773, -y=773+ni, and a+,#+.y=2(7+72+73) ........... (5) Now, by (2) and remembering that A = r0 (I+ rw), cO= KCC,
r(sin-q+rw) =ro(1+row).
Apply this to the ray CA at the corner A. Then

sin,71

=a

(I+wro)-a.................

(6)

and it remains only to find r0. Now, the radius vector r. bisects

the angle COA=u 2; thus, applying (3a), with OA- Oo= 2, we have

ro - =Cos 0'2-

I-- awoSin2a2

a

COSa2

Substitute this in (6) and reject second order terms; then the

result will be

SiH11

O 2- 2 acosin2a2.

298

L. SILBERSTEIN

[J.O.S.A., V

Since l +of2 differs but little from a right angle, put here nq=

2- (2 +61); then the last equation will give for the small angle
2

a=2aw sinu2.
Similarly the defects of fl2+a3 and 73+u1 will be 62=2 aco sin 0T3, 33 =2acw sing1. Thus by (5), and since oi+ 02+ f3=7r,

a+,0+7=7r-2(61+32+83),
i.e., the defect of the optical triangle ABCA will be

K&

4a- (sinaI+sinr C

2 +sinas) ......

(8)

For the optical triangle (or triangular circuit) ACBA we shall

have an equal excess of the angle sum.

Thus, for instance, for an equilateral triangle, al- =02 = 3= 60°,

the defect, or the excess, will be 4aK. 3 V =6\/3a . For the

c

2

c

Earth (and a triangle parallel to the equatorial plane) even if

K= 1 (no drag) this would amount to 0. "00052 per kilometer of a,

and the difference between the angle sum of ACBA and ABCA

would be the double of this. Thus, even for a= 10 or 20 km. the

difference would certainly be too small to be measured directly.

The experimental possibilities with regard to the optical effects

of the rotation of the Earth lie in another direction, to wit in the

phase retardation in an optical circuit (i.e., closed light path) as

in the well-known laboratory experiment of Sagnac4 with a small

spinning interferometer as our system S. The corresponding

formula used by Sagnac and before him by Michelson (Phil. Mag.

vol. 2, 1904, p. 716-719) who actually proposed but never carried

out a terrestrial experiment' of the kind here aimed at, can be

IG. Sagnac, "L'ether lumineux dmontr6 par 'effet du vent relatif d'ether dans
un int6rferometre en rotation uniforme." C. R. Paris, 157 (1913), p. 708. Ibidem, p. 1410, "Sur la preuve de la ralit6 de ether lumineux par l'experience de l'interf6rographe tournant." The titles seemed interesting enough to be quoted in full. But Sagnac's experiments (even apart from the question of the reliability of his measurements) by no means decide for the aether as against relativity.
5Such a terrestrial experiment was already hinted at by Oliver Lodge, in 1897, Phil. Trans. Roy. Soc. A, vol. 189, p. 151, where also the experiment carried out by Sagnac twenty-five years later is suggested, only with "telescope and observer" instead of a photo camera mounted on a rotating "turn-table."

July, 1921]

OPTICS OF ROTATING SYSTEMS

299

most simply deduced in the following way. Consider any optical circuit s, traversed by light in the, sense of rotation or "positive" circuit, say.6 By formula (1), the time taken to traverse an ele-
ment ds of the circuit is, with u, written for up,

ds.(c+us)-=l I1_u 4

c

c

giving a retardation - 2 us ds per line-element, and thus for the

whole circuit the phase retardation AT-= --Ifuds,

as already noticed by Lorentz (Wiss. Abhandlungen, vol. I). Now, by Stokes' theorem, this can be written, if a-be any surface laid through s, and n its normal,
AT=--fn curul .do-.

This holds for any circuit, plane or not, and for any distribution of velocity. Since-curl u is the double angular velocity, 2KJ.k, we have, writing c,.= oknfor the normal component of the spin,

AT=

Co&- ....

.......

(9)

This, the required formula, is valid for any, not necessarily

constant value of KU throughout the surface of integration. In

our case, a, and therefore co,,for a plane circuit, are manifestly

constant, but the drag of the aether, if any, may vary from point

to point, thus giving rise to a variable coefficient K.

If the optical circuit is plane and small compared with the dimensions of the Earth, we have simply Ar =2KOn/C 2, or if T

be the period of oscillation and X= cT,

AT

2
Xa.

.

(10)

T c,

where a is the total area embraced by the circuit. This gives the

retardation, in parts of the period, for a positive circuit. As was

already mentioned, the same path cannot, rigorously, serve for

light propagation in the opposite sense [or, which is the same

6 As we already know, the same light path cannot be described in the opposite sense.

300

L. SILBERSTEIN

[J.O.S.A., V

thing, with inverted normal n in (10) ]. Thus, for instance, in the case of a triangle (Fig. 5) we have to take for the positive circuit the area al- of the inner, and for the negative circuit the area 2 of the outer, convex triangle, so that rigorously we would

Fig. 5

M

have for the phase difference of the two beams (say, separated at A by a semi-transparent plate and reflected at B, C) in parts of the period, or for the corresponding shift in fringe widths,

2KJJ
e= - (a-I+02).........................

(11)

But the difference of u- and 2 is itself small of the first order. Moreover, up to higher terms, the paths AB and BA, etc., are symmetrical with respect to the corresponding (dotted) straights, so that even up to terms of an order higher than the second we can replace al+f2 by trice the area (a) of the rectilinear (dotted) triangle, and similarly in the case of any polygons. Thus :7

.(12)
cX

' Prof. Michelson's paper of 1904, (l.c.) has by a manifest slip the factor 2 instead of our 4.

July, 1921]

OPTICS OF ROTATING SYSTEMS

301

Four horizontally placed circuits we have Co =C sin s, if o be the geographic latitude at which the experiment is performed. For
a latitude 5p=450 and the wave-length X=5000A. U., formula
(12) gives

e=1.38 a .' ..........................

(12a)

i.e., about 1. 4xfringe widths per each square kilometer embraced by the circuit. If the drag is complete (K = 0) there should be no shift; if there is no drag, we should have the total amount of 1 .4 per kM2, and intermediate values if the Earth in its spinning motion drags the aether, even at its surface, only partially.
In Sagnac's experiment the spinning motion of the (disc-shaped) table bearing the interferometer, the light source, as well as the photographic camera, was reversed, and thus the double of (12) was observed 8 as the shift of the system of interference fringes.
No such reversal, of course, is possible in the case of the Earth as the rotating system. But as Prof. Michelson has already pointed out in 1904 (loc. cit.) there is an easy way out of this difficulty, to wit by silvering heavily one, e.g., the upper, half of the dividing glass plate (such as A in the case of Fig. 5 corresponding to three stations; B, C, being mirrors) and leaving the ldwer part clear or but lightly silvered. A beam of parallel rays from the collimator M impinging upon the plate A is here divided into ACBA and ABCAby reflection and transmission respectively. Now, according to Michelson's suggestion, cover the lower half and observe first by reflection from the upper half only (ACBA) when simply the image of the slit is seen. Place the cross-hair of the eyepiece in the center of this image. Next, covering the upper half and leaving the lower half of the plate clear, observe the interference fringes. Then, if there is an effect , the midpoint of the central fringe will be displaced from the crosshair by e fringe widths. The effect sought for will be easily discernible from undesired accessorial shifts by being proportional to the area . Instead of comparing the position of the interference

8 In Sagnac's case K=1, since such small masses as was his table certainly do not drag the aether, as follows from the widely known older experiments of Sir Oliver Lodge.

302

L. SILBERSTEIN

[J.O.S.A., V

fringes with that of the slit image Prof. Michelson contemplates

also the comparison of the two fringe systems given by two con-

siderably differing values of the area .

It is estimated that the terrestrial experiment could be carried

out with the required precision on comparatively small areas,

such as 1/10 of a square kilometer. The beams can, of course, be

sent around twice or more times making as many times larger;

as many times, however, would the light path be increased which

may not be convenient. A more radical way is to leave the cir-

cuits simple but to extend the linear dimensions of the circuits;

for then the value of e will be increased in the squared ratio.

Again the more stations (arranged in a regular polygon) the greater

o-for the same light paths. Yet, to avoid too many reflections

and other inconvenients, the triangular arrangement as suggested

in Fig. or a quadrangular one may turn out to be preferable

But details of a technical kind need not detain us in the present

paper.

2. Let us now try to find out what aspect the same problem

assumes from the standpoint of the theory of relativity, the special

and the generalized one. It goes without saying that with

neither of these theories can there be any question of an aether

and its being dragged by the Earth in its daily rotation around its

axis, or in its annual motion around the sun.

First of all, then, the special relativity theory is wholly incom-

petent to deal with the problem of light propagation in a rotating

system rigorously, simply because it has nothing to do with any

reference systems other than the galilean or inertial ones, i.e.

the fixed-stars system S* and those moving relatively to it with

uniform translational velocity. The only thing the special theory

can do is to treat our problem approximately up to the second

order, i.e., rejecting the square (and higher powers) of the ratio

j3 of the velocity of motion9 to the light velocity. Now up to

2
p3

the

relativistic

addition

theorem of velocities does not

differ

at all from that of ordinary, Newtonian kinematics. This amounts

simply to a neglect of the Lorentz-Fitgerald contraction. Under

I Of any point of S relatively to S say.

July, 1921

OPTICS OF ROTATING SYSTEMS

303

these circumstances the propagation of light in S becomes simply

a superposition of that in S*and of the reversed spinning motion of

S relatively to S in much the same way as on the aether theory.

The only difference is that our previous Kcohas now to be replaced

by the full angular velocity co of S relatively to S (fixed stars).

Thus formula (1) for the light velocity v will be replaced by

-v=1+-1us=1+-cosy,

C

C

C

and the shift formula (12) by

X

. ................

(12r)

In fact v. Laue, the chief exponent of Einstein's older theory in

Germany, gives in his well-known book, in a section on Sagnac's

experiment, a formula identical with (12r) at which he arrives

by a rather roundabout way, instead of using the simple relation (9), valid for a circuit of any shape. 10

Similarly all other formulae given above will continue to hold

on the special relativity theory with unity written for K.

From the standpoint of the general relativity theory all ques-

tions concerning light propagation (in vacuo) relatively to any

system whatever and therefore also to our terrestrial system S will be answered if the four-dimensional line-element ds2 =

2gK dxtdx, belonging to that system be known, to wit, by putting

ds=o ................

(13)

Thus the problem is reduced to building up ds in terms of terres-

trial or S-co-ordinates, say xi, X 2, x3, x4 -r, , z, c respectively, keeping in mind that in the fixed-star system S* (disregarding

in our present connection the extremely minute terms due to

the Earth's gravitation) the line-element is

ds' = c2dtt-dre- r2dO"2-dz' 2 ...............

(14*)

To be faithful to its own leading principles, the relativity theory

ought to deduce the terrestrial ds or its coefficients gCK as a gravi-

10M.v. Laue, Relativitgtstheorie, vol. 1, 3rd ed. Braunschweig, 1919, p. 125-
127. Laue seems, by the way, to be under the misapprehension that the light rays relative to the rotating table are straight lines, which they are not. As we saw before, their departure from straight lines is a first order effect and does, therefore, by no means disappear though "the Lorentz contraction" be neglected. The influence of the curvature of the rays on (12) or (12r) remains, of course, negligible.

304

L. SILBERSTEIN

[J.O.S.A., V

tational effect of the stars and all the remaining matter of the universe rotating around the Earth. This, however, it proved unable to do. As a matter of fact Einstein himself never entered into the details of this important problem of rotation. Thirring" tried to solve it by considering a huge massive spherical shell in uniform rotation and evaluating by means of Einstein's approximate integrals of his field-equations the gK thus produced at comparatively small distances from the centre of that gigantic sphere. But the result, though mathematically interesting, was a complete failure,12 although the treatment of the same problem on Einstein's newest cosmological views (elliptic space and so on) seems more promising. At any rate, the relativity theory is unable to construct the required line-element on its own great principles, and is content to transcribe it from the galilean line-element (14*) by putting simply r', z', t = r, z, t, and
'=O+at,

where a is an appropriate constant. In fact, de Sitter, one of the chief exponents of Einstein's theory, and even Weyl in his interesting book" write down without much discussion this simple transformation in order to pass from one to the other system. This gives for the latter,

ds2= I-

c'dt-.-2r--dOcdt-dr 2-rkOl2..................

14)

/

C

The constant a is to be chosen so as to describe correctly the terrestrial laws of physical phenomena, approximately, at least, in our present connection up to the second order only.
Now, if but such an approximation is required, the fundamental principles of general relativity offer us a good test of whether a chosen value of the constant a is or is not the appro-

1'H. Thirring, Ueber die Wirkung rotierender ferner Massen in der Einsteinschen
Gravitationstheorie. Phys. Zeitschrift, Vol. 19, 1918, p. 33-39. 12Not only that Thirring's "centrifugal force" had also a component along the axis
of rotation, but the coefficients of the centrifugal and the Coriolis force, apart from being very unsatisfactory in their structure, bore a wrong numerical ratio to one another.
13H. Weyl, Zeit-Raum-Materie, 3rd ed., Berlin, J. Springer, 1920.

July, 1921]

OPTICS OF ROTATING SYSTEMS

305

priate one. In fact, according to one of these principles ds=0 represents light propagation, and according to the other the geodesics of the world, determined by the same line-element, i.e.,

fds=0 ........................

(15)

give the equations of motion of a free particle. And since the

terrestrial laws of motion are certainly known to that degree of

approximation, we can derive from (15) the correct value to be

attributed to the constant a.

Now, using (14) in (15) the terrestrial equations of motion of

a free particle (always apart from gravitation proper) follow at

once. Of these equations it will be enough to write down only

that which most interests us here, i.e., the equation corresponding

to the variation r. This is

d-2r =aOIdrt2I-I +2ard-o dt IIdrt-\U

ds' \ds/

ds ds ds

and since, up to the second order, ds2 = c2 dt2, we can write

d2r / d-=rt

do 2 da+-

The right-hand member will represe-it the correct value of the centrifugal acceleration (or "centrifugal force" per unit mass), in size and direction, provided that a stands for the full angular velocity of the Earth relatively to the fixed stars, i.e.

a=w.

This then is the value to be substituted into (14) to suit our dynamical experience, and at the same time, as already explained, into the light equation ds = 0. In short, on the relativity theory the same ds and therefore the same cois required for light as for mechanics, while on the aether theory we may have any fraction KC) of X for light, though the full angular velocity X is required for mechanics.
Thus, on the relativity theory the terrestrial velocity of light for any direction of the ray will be determined by

(1--c2)cdt2-2C-rdcdt-dr'-r'd. .(16)

whence also the form of the optical rays and the circuitous shift

306

L. SILBERSTEIN

[J.O.S.A., V

formula can be deduced.14 Or, equivalently and much more simply, we may deduce all these terrestrial optical laws from those of S* (uniform propagation with velocity c) by means of the transformation
0'=O+at=0+6t.
Consequently, all our previous formulae for the light velocity, for the shape of rays and the phase retardation in a terrestrial optical circuit, will reappear, with the only difference that the unknown fraction K will be replaced by the special and definite value 1, i.e., our KO will be replaced by a. Thus also the shift formula, which from the experimental point of view is the only important thing, will again be

cxn

as on the special relativity theory. The planned terrestrial experiments with the optical circuit
might thus enable us to decide between relativity and aether theory. That is to say, if the result of such experiments will be a full effect (a shift of 1. 38 fringe widths under the stated conditions, per square kilometer), there will be no discrimination between the two theories. But should there be either no shift at all or only a fraction K of the full effect, sensibly different from unity, the relativity theory, special or general, will be irremediably disproved, while on the aether theory we would have only to assume a complete or partial rotational dragging of the aether.
In fine, the optical circuit experiment may easily become crucial and fatal for Einstein's theory especially if it gives a nil-effect. Should it, on the other hand, give a full effect, it will certainly cease to be decisive either way but will even then (as every new positive experiment) be certainly a valuable contribution

14The only effect of terrestrial gravitation on the optical-circuit experiment would according to relativistic gravitation theory, be represented by the term
8 CosS -R to be subtracted from the factor 2 of the second term in (I 6), V being the
geographic latitude, M the mass and R the radius of the Earth. But since M/c'= 0.45 cm., the correction term due to gravitation amounts, even at the equator, only to 1.2X10-9 which is entirely negligible in presence of 2.

July, 1921]

OPTICS OF ROTATING SYSTEMS

307

to our experimental knowledge of the Earth as a physical reference system.
Speculations on a possible rotational drag of the aether by the Earth were made, in connection with the diurnal aberration, by Christian Doppler (1845) and after him by v. Oppolzer.'5 But owing to an almost complete absence of direct observational data on daily aberration, a state of things prevailing even at the present time, Doppler's formula, generalized by Oppolzer, could neither be proved nor disproved. More recently the question of a rotational drag was taken up by Lorentz 6 but it had again to be left unsettled in absence of daily aberration data. If there is anything on the experimental side to make the rotational drag unlikely it is the absence of shift of a star near occultation by Jupiter, as mentioned by Lorentz (loc. cit. p. 413). On the theoretical side the question of the influence of a spinning planet on the adjacent aether could not be answered definitely without some special and more or less artificial hypotheses about the properties of that medium in addition to those given to it already by Stokes-Planck. In this respect we can say only that even if there is an almost complete translational (annual) drag of the aether due to its condensation around the planet, there may yet be no appreciable rotational (daily) drag. Again, from Lodge's experiments (loc. cit) one can judge only that there is no such drag by comparatively small spinning masses, such as can be used in a laboratory, but not by such massive bodies as the Earth or other planets. In fine, the only sound way of settling the question would be to carry out the terrestrial experiment with optical circuits embracing as large an area as is technically
possible. ROCHESTER, N. Y.
March, 24, 1921.

15 E. v. Oppolzer, Erdbewegung and Aether, Sitzber. Akad. Wien, vol. CXI, Ha, Febr. 1902, pp. 244-254.
161H. A. Lorentz, Abhandlungen, Vol. I.