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C 469 3
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X X I. Experiments to determine the Density o f the Henry Cavendish, Esq. F .R .S . and A .S .
. By
Read June 21, 1798.
JVIany years ago, the late Rev. John M ichell, of this Society, contrived a method of determining the density of the earth, by rendering sensible the attraction of small quantities of m atter ; but, as he was engaged in other pursuits* he did not complete the apparatus till a short time before his death, and did not live to make any experiments with it. After his death, the apparatus came to the Rev. F rancis John. H yde W ollaston, Jacksonian Professor at Cambridge, who, not having conveni­ ences for making experiments with it, in,the manner he could wish, was so good as to give it to me.
T he apparatus is very sim ple; it consists of a wooden arm, 6 feet long, made so as to unite great strength with little weight. T h is arm is suspended in an horizontal position, by a slender wire 4,0 inches long,, and to each extremity is hung a leaden ball, about 2 inches in diam eter; and the whole is in­ closed in a narrow wooden case, to defend it from the wind.
As no more force is required to make this arm turn round on its centre, than what is necessary to twist the suspending wire, it is plain, that if the wire is sufficiently slender, the most minute force, such as the attraction of a leaden weight a few inches in diameter, will be sufficient to draw the arm sensibly aside. T h e weights which Mr. M ichell intended to use w ere
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47o
Mr. Cavendish's Experiments to determine
8 inches diameter. One of these was to be placed on one side the case, opposite to one of the balls, and as near it as could conve­ niently be done, and the other on the other side, opposite to the other ball, so that the attraction of both these weights would con­ spire in drawing the arm aside; and, when its position, as affected by these weights, was ascertained, the weights were to be re­ moved to the other side of the case, so as to draw the arm the contrary way, and the position of the arm was to be again de­ termined ; and, consequently, half the difference of these posi­ tions would shew how much the arm was drawn aside by the attraction of the weights.
In order to determine from hence the density of the earth, it is necessary to ascertain what force is required to draw the arm aside through a given space. This Mr. M ichell intended to do, by putting the arm in motion, and observing the time cf its vibrations, from which it may easily be computed.*
Mr. M ichell had prepared two wooden stands, on which the leaden weights were to be supported, and pushed forwards, till they came almost in contact with the case; but he seems to have intended to move them by hand.
As the force with which the balls are attracted by these
weights is excessively minute, not more than
of their
weight, it is plain, that a very minute disturbing force will be sufficient t o destroy the success of the experiment; and, from the following experiments it will appear, that the disturbing
• Mr. Coulomb has, in a variety of eases, used a contrivance of this kind for trying small attractions; but Mr. Mi chell informed me of his intention of making this experiment, and of the method he intended to use, before the publication of any of Mr. Coulomb’s experiments.
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471
force most difficult to guard against, is that arising from the variations of heat and cold; for, if one side of the case is warmer than the other, the air in contact with it will be rarefied, and, in consequence, will ascend, while that on the other side will descend* and produce a current which will draw the arm sen­ sibly aside.* * As I was convinced of the necessity of guarding against this source of error, I resolved to place the apparatus in a room which should remain constantly shut, and to observe the motion of the arm from without, by means of a telescope; and to sus­ pend the leaden weights in such manner, that I could move them without entering into the room. This difference in the m an­ ner of observing, rendered it necessary to make some alteration in Mr. M icheli/ s apparatus; and, as there were some parts of it which I thought not so convenient as could be wished, I chose to make the greatest part of it afresh.
Fig. 1. (Tab. X X III.) is a longitudinal vertical section through the instrument, and the building in which it is placed* ABCDDCBAEFFE, is the case; and x are the two balls, which are suspended by the wires b x from the arm g b m b , which is itself suspended by the slender wire g l. This arm consists of a slender"deal rod bm by strengthened by a silver
* M .C a s s i n i , in observing the variation compass placed by him in the Observatory, (which was constructed so as to make very minute changes o f position visible, and in which the needle was suspended'by a silk thread,) found that standing near the box, in order to observe, drew the needle sensibly aside * which I have no doubt was caused by this current of air. It must be observed, that his compass-box was of metal, which transmits heat faster than wood, and also was many inches deep j both which causes served to increase the current of air. T o diminish the effect of this current, it is by all means advisable to make the box, in which the needle plays, not much deeper than
k necessary to prevent the needle from striking against the top and bottom.
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472
jMt. Cavendish's Experiments to determine
wire hgh ; by which means it is made strong enough to sup­
port the balls, though very light. *
The case is supported, and set horizontal, by four screws, rest­
ing on posts fixed firmly into the ground: two of them are
represented in the figure, by S and S ; the two others are not
represented, to avoid confusion. GG and GG are the end walls
of the building. W and W are the leaden weights; which are
suspended by the copper rods R rP rR , and the wooden bar rr,
from the centre pin P
p .This pin passes
beam HH, perpendicularly over the centre of the instrument,
and turns round in it, being prevented from falling by the
plate
p .MM is a pulley, fastened to this pin; and Mm, a cord
Wound round the pulley, and passing through the end wall;
by which the observer may turn it round, and thereby move
the weights from one situation to the other.
Fig. 2. (Tab. XXIV.) is a plan of the instrument. AAAA is the
case. SSSS, the four screws for supporting it. , the arm and
balls. W and W , the weights. MM, the pulley for moving them.
When the weights are in this position, both conspire in drawing
the arm in the direction b W ; but, when they are removed to the
situation w and w, represented by the dotted lines, both conspire
in drawing the arm in the contrary direction bw. These weights
are prevented from striking the instrument, by pieces of wood,
which stop them as soon as they come within -i-of an inch of the
* Mr. M ic hell’s rod was entirely of wood,-and was much stronger and stifFer than this, though not much heavier; but, as it had warped when it came to me, I chose to make another, and preferred this form, partly as being easier to construct and meeting with less resistance from the air, and partly because, from its being of a less complicated form, I could more easily compute how much it was attracted by the weights.
the
"Densityofthe E arth. 473
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case. T h e pieces of wood are fastened to the wall of the build­
ing ; and I find, th at the weights may strike against them w ith
considerable force, without sensibly shaking the instrum ent.
In order to determine the situation of the arm , slips of ivory
are placed within the case, as near to each end of the arm as
can be done w ithout danger of touching it, and are divided to
aoths of an inch. A nother sm all slip of ivory is placed a t each
end of the arm, serving as a vernier, and subdividing these
divisions into 5 p a rts; so th a t the position of the arm m ay be
observed with ease to looths of an inch, and may be estimated
to less. These divisions are viewed, by m eans of the short tele­
scopes T and T , (fig. 1.) through slits cut in the end of the case,
and stopped with glass; they are enlightened by the lamps L
and L, with convex glasses, placed so as to throw the light on
the divisions; no other light being admitted into the room.
T h e divisions on the slips of ivory run in the direction W
(fig. 2.) so that, w hen the weights are placed in the positions
w and w 9represented by the dotted circles, th e arm is draw n
aside, in such direction as to make the index point to a higher
number on the slips of ivory; for which reason, I call this the
positive position of the weights.
F K , (fig. 1.) is a wooden rod, w hich, by m eans o f an endless
screw, turns round the support to which the wire g l is fastened,
and thereby enables the observer to turn round the wire, till the
arm settles in the middle of the case, without danger of touch­
ing either side. T h e wire g l is fastened to its support a t top,
and to the centre of the arm at bottom, by brass clips, in which
it is pinched by screws.
In these two figures, the different parts are drawn nearly in
mdcgxcviii.
3 P
474
iWr. Cavendish's 'Experiments to determine
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the proper proportion to each other, and on a scale of one to thirteen.
Before I proceed to the account of the experiments, it will be proper to say something of the manner of observing. Suppose the arm to be at rest, and its position to be observed, let the weights be then moved, the arm will not only be drawn aside thereby, but it will be made to vibrate, and its vibrations will continue a great while; so that, in order to determine how much the arm is drawn aside, it is necessary to observe the extreme points of the vibrations, and from thence to determine the point which it would rest at if its motion was destroyed, or the point of rest, as I shall call it. To do this, I observe three successive extreme points of a vibration, and take the mean between the first and third of these points, as the extreme point of vibration in one direction, and then assume the mean between this and the se­ cond extreme, as the point of re st; for, as the vibrations are continually diminishing, it is evident, that the mean between two extreme points will not give the true point of rest.
It may be thought more exact, to observe many extreme points of vibration, so as to find the point of rest by different sets of three extremes, and to take the mean result; but it must be observed, that notwithstanding the pains taken to prevent any disturbing force, the arm will seldom remain perfectly at rest for an hour together; for which reason, it is best to deter­ mine the point of rest, from observations made as soon after the motion of the weights as possible.
The next thing to be determined is the time of vibration, which I find in this m anner: I observe the two extreme points of a vibration, and also the times at which the arm arrives at
the 'Density o f the Earth.
475
two given divisions between these extremes, taking care, as well as I can guess, that these divisions shall be on different sides of the middle point, and not very far from it. I then compute the middle point of the vibration, and, by proportion, find the time at which the arm comes to this middle point. I then, after a number of vibrations, repeat this operation, and divide the interval of time, between the coming of the arm to these two middle points, by the number of vibrations, which gives the time of one vibration. The following example will explain what is here said more clearly.
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Extreme Points.
Division.
Time.
Point o f Time o f middle rest. of vibration.
2 7 ,2 2 2 ,1
t
25 I O 2 3 4*1
24
57 J
t
IO S 3 2 3
-
- -
2 4 ,6
2 7
-
-
2 2 ,6
-
-
2 6 ,8
-
-
9 4 .7
2 4 .7 5 2 4 ,8
23
-
2 6,6
-
warn
mm
-
2 4 ,8 5 2 4 ,9
25 11 5 2 2 1
24
648}
23>4
11 5 22
The first column contains the extreme points of the vibra­ tions. The second, the intermediate divisions. The third, the time at which the arm came to these divisions; and the fourth, the point of rest, which is thus found: the mean between the first and third extreme points is 2.7,1, and the mean between this and the second extreme point is 24,6, which is the point of rest, as found by the three first extremes. In like manner, the
3 P 2
4,76 Mr. Cavendish’s Experiments to determine
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point of rest found by the second, third, and 4th extremes, is
24,7, and so on. The fifth column is the time at which the
arm came to the middle point of the vibration, which is thus
found: the mean between 27,2 and 22,1 is 24,65, and is the
middle point of the first vibration; and, as the arm came to 25
at io h 23' 4", and to 24 at io h 23' 57", we find, by proportion,
that it came to 24,65 at ic h 23' 23". In like manner, the arm
came to the middle of the seventh vibration at 1i h 5' 22"; and,
therefore, six vibrations were performed in 4 i' 59", or one vibra­
tion in 7' o".
To judge of the propriety of this method, we must consider
in what manner the vibration is affected by the resistance of
the air, and by the motion of the point of rest.
Let the arm, during the first vibration, move from D to B, (Tab.
XXIV. fig- 3.) and, during the second, from B to
d ;Bdbeingles
than DB, on account of the resistance. Bisect DB in M, and B in
m, and bisect in
n,and let x be any point in
then, if the resistance is proportional to the square of the velocity,
the whole time of a vibration is very little altered; but, if T is
taken to the time of one vibration, as the diameter of a circle
to its semicircumference, the time of moving from B to ex-
ceeds a vibration, by — near l y; and the time of moving
from B to
maflls short of ~ a vibration, by as much; and
time of moving from B to
x,in the seco
that of moving from x to B, in the first, by
supposing D<i to be bisected in so that, if a mean is taken, between the time of the first arrival of the arm at x and its re­ turning back to the same point, this mean will be earlier than
the true time of its coming to B, by 8b?tV -B* x *5*
the Density o f the
.
477
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T he effect of motion in the point of rest is, that when the arm is moving in the same direction as the point of rest, the time of moving from one extreme point of vibration to the other is increased, and it is diminished when they are moving in contrary directions; but, if the point of rest moves uniformly, the time of moving from one extreme to the middle point of the vibration, will be equal to that of moving from the middle point to the other extreme, and moreover, the time of two suc­ cessive vibrations will be very little altered; and, therefore, the time of moving from the middle point of one vibration to the middle point of the next, will also be very little altered.
It appears, therefore, that on account of the resistance of the air, the time at which the arm comes to the middle point of the vibration, is not exactly the mean between the times of its coming to the extreme points, which causes some inaccuracy in my method of finding the time of a vibration. It must be observed, however, that as the time of coming to the middle point is before the middle of the vibration, both in the first and last vibration, and in general is nearly equally so, the error produced from this cause must be inconsiderable ; and, on the whole, I see no method of finding the time of a vibration which is liable to less objection.
T he time of a vibration may be determined, either by pre­ vious trials, or it may be done at each experiment, by ascer­ taining the time of the vibrations which the arm is actually put into by the motion of the w eights; but there is one advantage in the latter method, namely, that if ther^e should be any acci­ dental attraction, such as electricity, in the glass plates through which the motion of the arm is seen, which should increase the force necessary to draw the arm aside, it would also dimi-
478
Mr. Cavendish’s Experiments to determine
nish the time of vibration; and, consequently, the error in the result would be much less, when the force required to draw the arm aside was deduced from experiments made at the time, than when it was taken from previous experiments.
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Account of the Experiments.
In my first experiments, the wire by which the arm was sus­ pended was 39^ inches long, and was of copper silvered, one foot of which weighed grains : its stiffness was such, as to make the arm perform a vibration in about 15 minutes. I imme­ diately found, indeed, that it was not stiff enough, as the at­ traction of the weights drew the balls so much aside, as to make them touch the sides of the case; I, however, chose to make some experiments with it, before I changed it.
In this trial, the rods by which the leaden weights were sus­ pended were of iron; for, as I had taken care that there should be nothing magnetical in the arm, it seemed of no signification whether the rods were magnetical or n o t; but, for greater se­ curity, I took off the leaden weights, and tried what effect the rods would have by themselves. Now I find, by computation, that the attraction of gravity of these rods on the balls, is to that of the weights, nearly as 17 to 2500; so that, as the attrac­ tion of the weights appeared, by the foregoing trial, to be suffi­ cient to draw the arm aside by about 15 divisions, the attraction of the rods alone should draw it aside about y% of a division; and, therefore, the motion of the rods from one near position to the other, should move it about | of a division.
The result of the experiment was, that for the first 15 minutes after the rods were removed from one near position to the other, very little motion was produced in the arm, and
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.
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hardly more than ought to be produced by the action of gra­ vity; but the motion then increased, so that, in about a quarter or half an hour more, it was found to have moved or i-§division, in the same direction that it ought to have done by the action of gravity. On returning the irons back to their former position, the arm moved backward, in the same m an­ ner that it before moved forward.
It must be observed, that the motion of the arm, in these experiments, was hardly more than would sometimes take place without any apparent cause; but yet, as in three experiments which were made with these rods, the motion was constantly of the same kind, though differing in quantity from | to i j division, there seems great reason to think that it was pro­ duced by the rods.
As this effect seemed to me to be owing to magnetism, though it was not such as I should have expected from that cause, I changed the iron rods for copper, and tried them as before; the result was, that there still seemed to be some effect of the same kind, but more irregular, so that I attributed it to some accidental cause, and therefore hung on the leaden weights, and proceeded with the experiments.
It must be observed, that the effect which seemed to be pro­ duced by moving the iron rods from one near position to the other, was, at a medium, not more than one division; whereas the effect produced by moving the weight from the midway to the near position, was about 15 divisions; so that, if I had con­ tinued to use the iron rods, the error in the result caused there­ by, could hardly have exceeded ~ of the whole.
480
Mr. Cavendish's Experiments to determine
EXPERIMENT I. Aug. 5.
Weights in midway position.
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Extreme points.
Divisions.
Time.
Point of rest.
Time of mid. of vibration.
Difference.
h- ,
H,4 9 42 O
ii,5
55 O
n ,5 10 5 O n ,5
t »/ *
At ioh 5', weights moved to positive position.
Z 3 >4s
s 7,6
24.7
-
s 7»3
-
25,1 1 -
M mm 25,82 - - 26,07 - — 26,1
- -
At n h 6 \ weights returned back to midway position.
5, 11 12
18,2 12 11
6,6 11 12
16,3 12 11
7,7
0 0 481 130/
16 29-1 17 &o /
30 24 \ 31 11/
45 58 \ 47 4 J
12
11,92
11,72 -
0 1 13
14 56 16 9
14 3° 45
*5 *3 45 58
Motion on moving from midway to pos. = 14,32
pos. to midway - - = 1 4 , 1
Time of one vibration
s s 14
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the Density o f the Rartb.
481
It must be observed, that in this experiment, the attraction of the weights drew the arm from 11,5 to 25,8, so that, if no contrivance had been used to prevent it, the momentum ac­ quired thereby would have carried it to near 40, and would, therefore, have made the balls to strike against the case. To prevent thisr, after the arm had moved near 15 divisions, I re­ turned the weights to the midway position, and let them remain there, till the arm came nearly to the extent of its vibration, and then again moved them to the positive position, whereby the vibrations were so much diminished, that the balls did not touch the sides; and it was this which prevented my obser­ ving the first extremity of the vibration. A like method was used, when the weights were returned to the midway position, and in the two following experiments. '
The vibrations, in moving the weights from the midway to the positive position, were so small, that it was thought not worth while to observe the time of the vibration. W hen the weights were returned to the midway position, I determined the time of the arm's coming to the middle point of each vibra­ tion, in order to see how nearly the times of the different vi­ brations agreed together. In great part of the following expe­ riments, I contented myself with observing the time of its coming to the middle point of only the first and last vibration.
MDCCXCVIII.
s Q
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48a
Mr. C a v e n d i s h ’s Experiments to determine
. EXPERIMENT II. Aug. 6 . Weights in midway position.
Extreme points.
Divisions.
Time.
Point of Time of mid. rest. of vibration.
Difference
/ //
11
IO 4 O
11
11 O
11
17 O
11
25 O
h* | * # *g
Weights moved to positive position.
*9.3
24,1
3°
-
26,2
-
®9>7 -
26,9
-
28,7
-
27,1
28,4
-
-
-
-
-
-
-
* -
-
- .
-
26,87
27,57
28,02
28,12
28,05
27,85
27,82
Weights returned to midway position.
is
13
18,5
13
12
6,5 11
12
*5>2 *3
12
} 3 5°
4 34
18 2Q\
19 l8 J
33 4 8 ) 34 J
} 45 8
46 22
12,37 11,67
1 4, 1
14 a
18 S3
14 46
33 39 13 46
47
the Density o f the
.
48S
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M 1 ^00
Extreme points.
Divisions.
Time.
Point of rest.
Time o f mid. of vibration .
Difference.
7» i
11
12
1 3>®
h. , . 10 75
2 3 48-I
5 18/
-I1* / 0 f
n
15 25
2 2 50
Motion of arm on moving weights from
midway to pos. -
pos. to midway
-
Time of one vibration - - -
= 15,87 = 15,45 = 1 4 ' 42"
E x p e r i m e n t h i . Aug. 7.
The weights being in the positive position, and the arm a little in motion.
3
29
—
3i
%9> 1
—
-
30,12
30,02
Weights moved to midway position.
9
20,5 15 15 *4
14 15
10 3418-1 35 8 /
49 S1! 5° 2 7 /
ix 5 71 6 18 J
- 10 34 55
1 4 44
49 39
i 4>°7
14 38
11 4 17
sQ *
484
M r. Cavendish's Experiments to determine
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Extreme points.
Divisions.
Time.
*3 10,1
13 14 15&
h* ,
*
11 18 46I 19 5 8 /
S3 46 \ 35 26 J
Point of rest.
Time of mid. of vibration.
Difference.
_
13.52
h*
;
, r9
a
14 47
19 4
13.3
14 *7
-
33 3*
h'
Weights moved to positive position.
3s 28 27
23>7 -
25,8 27 28
0 2 48 -i 3 J
-
-
-
44 8 j 46 50 /
-
37,8 28,27 28,62
■-
o s 59 47 40
3b1
Motion of the arm on moving weights from
pos. to mid.
4
= 15,22
mid. to pos.
- = 14,5
Time of one vibration, when in mid. position = 14' 39"
pos. position = 14 54
These experiments are sufficient to shew, that the attraction ©f the weights on the balls is very sensible, and are also suffi­ ciently regular to determine the quantity of this attraction pretty nearly, as the extreme results do not differ from each other by more than part. But there is a circumstance in them, the reason of which does not readily appear, namely, that the effect of the attraction seems to increase, for half an
the Density o f the Earth.
485
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hour, or an hour, after the motion of the weights ; as it may be observed, that in all three experiments, the mean position kept increasing for that time, after moving the weights to the posi­ tive position; and kept decreasing, after moving them from the positive to the midway position.
The first cause which occurred to me was, that possibly there might be a want of elasticity, either in the suspending wire, or something it was fastened to, which might make it yield more to a given pressure, after a long continuance of that pressure, than it did at first.
To put this to the trial, I moved the index so much, that the arm, if not prevented by the sides of the case, would have stood at about 50 divisions, so that, as it could not move farther than to 35 divisions, it was kept in a position 15 divisions distant from that which it would naturally have assumed from the stiff­ ness of the w ire; or, in other words, the wire was twisted 15 divisions. After having remained two or three hours in this position, the index was moved back, so as to leave the arm at liberty to assume its natural position.
It must be observed, that if a wire is twisted only a little more than its elasticity admits of, then, instead of setting, as it is called, or acquiring a permanent twist all at once, it sets gradually, and, when it is left at liberty, it gradually loses part of that set which it acquired ; so that if, in this experiment, the wire, by having been kept twisted for two or three hours, had gradually yielded to this pressure, or had begun to set, it would gradually restore itself, when left at liberty, and the point of rest would gradually move backwards; but, though the ex­ periment was twice repeated, I could not perceive any such effect.
The arm was next suspended by a stiffer wire.
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486
Mr. Cavendish's Experiments to determine
EXPERIMENT IV . Aug. 12. Weights in midway position.
Extreme points.
Divisions.
Time.
Point of Time of mid. rest. ofvibration.
Difference.
2 1 ,6
k* 1 n
9 3° 0
t aI*
• ■>{ 2 1 , 5
52 0
21,5 10 1 3 0 2 1 ,5
Weights moved from midway to positive position.
27,2
22,1
24,6
27
24,67
22,6
24>75
26,8
24,8
23,0
24,85
2 6 ,6
s 4>9
23,4
Weights moved to negative position.
17 19 22.4 20
19 15*1
19 20 21.5 20
19 15,3
18
19 20,8
19 18
15,5
19 20
25 41
}
10 20 31
18,7s 26 45 1 27 22 /
7 0 27 31
18,52
6 57
35 48 }
34 28
18,35
7 23
40 41
23 l8
}
18,22
41 51 6 4,8
48 36 1 49 24 J
18,1
48 39 6 58
54 45 \ 55 45 1
55 37
the Density o f the
.
W eights moved to positive position.
• 487
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Extreme points.
Divisions.
T im e.
Point o f T im e o f mid. rest. o f vibration.
Difference.
8^3 25 23
h.
11 IO 25 \ 11 3 J
1
n
11 IO 40
24,02
7 3
22
17 6 1
23
2 J
17 43
8°>6 25 23
18,4, 23
2 5
2 9 ,9
2 5 23
24 33 \ 25 17 /
31 21 \ 3* . 9 /
38 39 1 39 3 1 J
24,17
7 *
ft j: , r3;4] 44
24,32
7 5
31 49
24,4
6 59
38 48
i 9>4 23 2 5
24,5
46
\
46 12 J
7 6 45 54
*9>3
Motion of arm on m oving weights from
midway to pos. - — 3,1
pos. to neg„ -
18
neg. to pos.
-
= 5,92
T im e o f one vibration in neg. position - = y f 1"
pos. position - = 7 3
4&8
Mr. Cavendish's Experiments to determine
EXPERIMENT V. Aug . 20.
The weights being in the positive position, the arm was made to vibrate, by moving the index.
Downloaded from https://royalsocietypublishing.org/ on 01 August 2024
Extreme points.
Divisions.
Time.
Point of rest.
Time of mid. of vibration.
Difference.
2 9 ,6 :
21,1
—
29,
-
2 1 ,6 .
k* , H
-' — ~ :
25,3 25,17
/
St
/
n
Weights moved to negative position.
32,6
16,3
21,9
1 6 ,5 2 1 ,5 l6 ,8
21,3
1 7 ,1 2 0 ,8
17,4
2 0 ,6
17,5
20
*9
-
-
-
20
19
— .
10 e s 4 7 1
23 30 J
- -
■ -
—
-f
-
-
-
- -
-
mm
mm
11 32 l6 1
33 58
-— —
IO 2 3 11
19>s 7
i 9 »i5
1 9 ,1
19,07 19,07
1 9 ,0 7
19,05
1 9 ,0 2
1 9 ,0 5 1 9 ,0 2
mm n 33 53
1 8 ,9 7
19
20
20,3
41 16 V
43 0 y
j 41 6
the Density o f the Earth.
489
Weights moved to positive position.
Downloaded from https://royalsocietypublishing.org/ on 01 August 2024
Extreme points.
Divisions.
Time.
Point of rest.
Time o f midJ o f vibration.
Difference.
SO,2
24
2 6
29>4
-
2 6
25
20,8 -
2 8 ,7
-
2 1 ,3
-
2 8 ,1
-
2 1 ,5 ,
-
2 7 ,6
-
22
■ -
24
25
2 7 ,2
-
2 5
24
2 2 ,4
h. ; ,
h* , 0 $
0J 4
1 1 4 9 1 0 -I
5° *9
imm
n 49 37
-
-
3 4 >95 - . -
77
56 1 \ -
47 J \
-
— 2 4 ,9 2
5^ 44
-
- ' 2 4 ,8 7
-S
-
-
-
-
2 4 ,8 5
2 4 ,7 5 2 4 ,6 7
-
-
2 4 ,6 7
-
-
2 4 ,7
45 0
4 8 iJe%
\f
4P 43 J
-• -
2 4 ,7
0 T4
621
—— 7 *
53 I* )
mm
54 9 J
53 2 2
Motion of arm on moving weights from
pos. to neg. -
-
== 5,9
neg. to pos.
= 5>98
Time of one vibration, when weights are in
neg. position - - = 7' 5"
pos. position - -
=75
In the fourth experiment, the effect of the weights seemed to
increase on standing, in all three motions of the weights, con­
formably to what was observed with the former w ire; but, in
mdccxcviii.
3 R
490
Mr. Cavendish's Experiments to determine
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the last experiment, the case was different; for though, on moving the weights from positive to negative, the effect seemed to increase on standing, yet, on moving them from negative to positive, it diminished.
My next trials were, to see whether this effect was owing to magnetism. Now, as it happened, the case in which the arm was inclosed, was placed nearly parallel to the magnetic east and west, and therefore, if there was any thing magnetic in the balls and weights, the balls would acquire polarity from the earth; and the weights also, after leaving remained some time, either in the positive or negative position, would acquire po­ larity in the same direction, and would attract the balls; but, when the weights were moved to the contrary position, that pole which before pointed to the north, would point to the south, and would repel the ball it was approached to ; but yet, as re­ pelling one ball towards the south has the same effect on the arm as attracting the other towards the north, this would have no effect on the position of the arm. After some time, how­ ever, the poles of the weight would be reversed, and would begin to attract the balls, and would therefore produce the same kind of effect as was actually observed.
To try whether this was the case, I detached the weights from the upper part of the copper rods by which they were suspended, but still retained the lower joint, namely, that which passed through them ; I then fixed them in their posi­ tive position, in such manner, that they could turn round on this joint, as a vertical axis. I also made an apparatus, by which I could turn them half way round, on these vertical axes, with­ out opening the door of the room.
Having suffered the apparatus to remain in this manner for
the Density o f the Earth.
491
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a day, I next m orning observed the arm, and, having found it to be stationary, turned the weights half way round on their axes, but could not perceive any motion in the arm. Having suffered the weights to remain in this position for about an hour, I turned them back into their former position, but without its having any effect on the arm. This experiment was re­ peated on two other days, with the same result.
W e may be sure, therefore, that the effect in question could not be produced by magnetism in the w eights; for, if it was, turning them half round on their axes, would immediately have changed their magnetic attraction into repulsion, and have pro­ duced a motion in the arm.
As a further proof of this, I took off*the leaden weights, and in their room placed two 10-inch m agnets; the apparatus for turning them round being left as it was, and the magnets be­ ing placed horizontal, and pointing to the balls, and with their north poles turned to the n o rth ; but I could not find that any alteration was produced in the place of the arm, by turning them half round; which not only confirms the deduction drawn from the former experiment, but also seems to shew, that in the experiments with the iron rods, the effect produced could not be owing to magnetism.
The next thing which suggested itself to me was, that pos­ sibly the effect m ight be owing to a difference of temperature between the weights and the case; for it is evident, that if the weights were much warmer than the case, they would warm that side which was next to them, and produce a current of air, which would make the balls approach nearer to the weights. Though I thought it not likely that there should be sufficient difference, between the heat of the weights and case, to have
3 R2
492
Mr. Ca v e n d i s h ’s Experiments to determine
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any sensible effect, and though it seemed improbable that, in all the foregoing experiments, the weights should happen to be warmer than the case, I resolved to examine into it, and for this purpose removed the apparatus used in the last experi­ ments, and supported the weights by the copper rods, as before; and, having placed them in the midway position, I put a lamp under each, and placed a thermometer with its ball close to the outside of the case, near that part which one of the weights approached to in its positive position, and in such manner that I could distinguish the divisions by the telescope. Having done this, I shut the door, and some time after moved the weights to the positive position. At first, the arm was drawn aside only in its usual manner; but, in half an hour, the effect was so much increased, that the arm was drawn 14 divisions aside, instead of about three, as it would otherwise have been, and the thermometer was raised near i°-£; namely, from 6i° to 6s°4-. Gn opening the door, the weights were found to be no more heated, than just to prevent their feeling cool to my fingers.
As the effect of a difference of temperature appeared to be so great, I bored a small hole in one of the weights, about threequarters of an inch deep, and inserted the ball of a small thermometer, and then covered up the opening with cement. Another small thermometer was placed with its ball close to the case, and as near to that part to which the weight was ap­ proached as could be done with safety; the thermometers being so placed, that when the weights were in the negative position, both could be seen through one of the telescopes, by means of light reflected from a concave mirror.
the Density
ofthe
.
493
E x p e r i m e n t vi. Sept. W eights in midway position.
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Extreme points.
Divisions.
Time.
Point of Therm<jmeter rest. in Air, inWeight.
h '
18,9 9 4 3 18,85 1 0 3
18,85 :
55>5
W eights moved to negative position.
is.1
-
18,4, -
1 3 .4
missed.
! 3.6
-
17,6
-
! 3.8
-
i 7>4
-
14,0 1 ■-
17,3 , -
IO 12 j l8
23
1 5 ,8 3
39 46
53 11 0
7 14
-
1 5 ,6 5 1 5 ,6 5 1 5 ,6 5 1 5 ,6 5
-
55>5 55.8 55>5 55.8 55>5
W eights moved to positive position.
35,8
-
17,5
—
25,4 -
l8 ,l
-
25,0
missed.
24,7 -
*9>
24,4
—
23 3° 21,55
37 31,6
44 21,65 51
0 5 12 21,77
*9
Motion of arm on moving weights from midway to .— = 3,03 — to -f- = 5>9
4*94*
Mr. Cavendish's Experiments to determine
E x p e r i m e n t v i i . Sept. 18. Weights in midway position.
Downloaded from https://royalsocietypublishing.org/ on 01 August 2024
Extreme points.
Divisions.
Time.
Point of Therm*ometer rest. in Air. inWeight*
h 9
’ 9>4 8 3 0
-
5 6 .7
’ 9>4 9 3 V
56,6
Weights moved to negative position.
13.6 1 i8,8 13,8 -
40 -
-
47 16,25
54
Eight extreme points missed.
16,9 - IO 58
14.5 - 11 5 15,62
16',6 -
12
W eights moved to positive position.
26,4
-
17,2 -
26,1 -
20 - 56,5 28 21,72 35
Four extreme points missed.
1 9>3 • 0 10
25,1 -
17 22,3
i 9>7 -
*54
Motion of arm on moving weights from midway to — ■— to +
the Density o f the
.
495
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E x p e r i m e n t v i i i . Sept. 23.
W eights in midway position.
Extreme points.
Divisions.
Tim e.
Point of rest.
Therm'ometer in A ir. :inW eight.
h '
19>3 9 46 i 9>2 10 45 19>s
53*1 53’1
W eights moved to negative position.
13.5
-
56 -
- 53.6
18,6 - 11 3 16,07
*3>6
-
10
Four extreme points missed.
17.4
-
14.1 -
17,2
-
44 5 i 15.7 58 -
- S3>6
W eights moved to positive position.
15.7 26,7 16,6
-
0 1
-
8 21,42
—
15 — 53>!5
Tw o extreme points missed.
S5»9
-
18,1
-
25*5
-
36 43 21,9 50
Motion of arm on moving weights from midway to «— = 3,13 — to + = 5,72
4 g6
Mr. Ca v e n d is h 's Experiments to determine
Downloaded from https://royalsocietypublishing.org/ on 01 August 2024
In these three experiments, the effect of the weight appeared to increase from two to five tenths of a division, on standing an hour; and the thermometers shewed, that the weights were three or five tenths of a degree warmer than the air close to the case. In the two last experiments, I put a lamp into the room, over night, in hopes of making the air warmer than the weights, but without effect, as the heat of the weights exceeded that of the air more in these two experiments than in the former.
On the evening of October 17, the weights being placed in the midway position, lamps were put under them, in order to warm them ; the door was then shut, and the lamps suffered to burn out. The next morning it was found, on moving the weights to the negative position, that they were y°-~ warmer than the air near the case. After they had continued an hour in that position, they were found to have cooled i°-§-, so as to be only 6° warmer than the air. They were then moved to the positive position; and in both positions the arm was drawn aside about four divisions more, after the weights had remained an hour in that position, than it was at first.
May 22, 1798. The experiment was repeated in the same manner, except that the lamps were made so as to burn only a short time, and only two hours were suffered to elapse before the weights were moved. The weights were now found to be scarcely 20warmer than the case; and the arm was drawn aside about two divisions more, after the weights had remained an hour in the position they were moved to, than it was at first.
On May 23, the experiment was tried in the same manner, except that the weights were cooled by laying ice on them ; the ice being confined in its place by tin plates, which, on
Downloaded from https://royalsocietypublishing.org/ on 01 August 2024
the Density
ofthe Ea4r9th7.
moving the weights, fell to the ground, so as not to be in the way. On moving the weights to the negative position, they were found to be about 8° colder than the air, and their effect on the arm seemed now to diminish on standing, instead of in­ creasing, as it did before; as tne arm was drawn aside about 2^- divisions less, at the end of an hour after the motion of the weights, than it was at first.
It seems sufficiently proved, therefore, that the effect in ques­ tion is produced, as above explained, by the difference of tem ­ perature between the weights and case; for, in the 6th, 8th, and §th experiments, in which the weights were not much warmer than the case, their effect increased but little on stand­ ing; whereas, it increased much, when they were much warmer than the case, and decreased much, when they were much cooler.
It must be observed, that in this apparatus, the box in which the balls play is pretty deep, and the balls hang near the bottom of it, which makes the effect of the current of air more sensible than it would otherwise be, and is a defect which I intend to rectify in some future experiments.
E x p e r i m e n t i x . April 2 W eights in positive position.
Extreme points.
Divisions.
34>7
35
-
34
MDCCXCVIII.
T im e.
hi
*
Point of Tim e o f middle
rest.
o f vibration.
m
h• a
34.84
3S
498
Mr. C a v e n d i s h 's Experiments to determine
Weights moved to negative position.
Downloaded from https://royalsocietypublishing.org/ on 01 August 2024
Extreme points.
Divisions.
Time.
Point of Time of mid. rest. of vibration.
Difference.
eg,8 28 29
33>2 29 28
23>9 3s 24,15
/ H
h. ,
$
$
11 l8 29! 58 J
11 18 43
28,52
25 271 571 28,25
25 40
2 8 ,0 1 v
27,82
31
27^3
s>7>55
3°>4 28 27
27>47
o 7 4-1
53 J
0 , 7 26
24>7 Motion of arm
= 6,32
Time of vibration - = 6'58"
34>5 33>5 34>4 22,3
33>3
2 2 ,6
E x p e r im e n t x . May 5 . W eights in positive position.
A r 33>97
Weights moved to negative position.
28 29
10
43 44
42 6
}
10 43 36
27,82
7 0
28 2 7
50 331 5 1 °J
, 50 36
27,72
the Density o f the Earth.
499
Downloaded from https://royalsocietypublishing.org/ on 01 August 2024
Extreme points.
Divisions.
Time.
Point of rest.
Time of mid. of vibration.
Difference.
3 2>5
23,2
-
3 1>45 -
23>5 27
28
30,7
-
23>95 27^ 28
3°>25
/ />
-
.«• a7>7
—
27.58
— ) — 27.4
— — 27,28
11 25 201 -
58 j
-
27,21
3 2 ' Ol 32 40 J
-
-
27,21
39 19} 40 2J
-
h. , ,
11 25 24
-
32 27
- V
39 23
/ a
7 s
6 56
OC
0 0
rH
00 C* G*
Motion of arm - = 15 Time of vibration - = 6' 59"
3 4 .9
34.1
3 4 .8 3 4 .2 5
2 3 .3
3 3 .3
n | E xperiment xi. May
* > >
Weights in positive position.
-, -
-
-
3 4 .4 7
-
3 4 .4 9
Weights moved to negative position.
’ |
28
9 59 59 V
29
10 0 2 7 j
2 8 ,4 2
7 5
29
6 52
27
7 51 '
; Y? i f
3 8> 2
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5°°
Mr. Cavendish's Experiments to determine
Extreme points.
Divisions.
Time.
Point of Time of mid. rest. of vibration.
2 3 .8
-
32.5
24*4
missed.
2 4 ,8
3b S
-
2 9
28
* 5>3
-
28
Q9 3 °>9
h. t ,
-
- 2 8 ,3 5
2 8 ,3
-
- 2 8 ,1 7
10 4 8 371 -
49 «»J
-
- 2 8,2
56 8i 56 5 6 /
1
10 49 8 5613
Motion of arm - = 6,07 Tim e of vibration - = 7' 1"
In the three foregoing experiments, the index was purposely moved so that, before the beginning of the experiment, the balls rested as near the sides of the case as they could, without danger of touching i t ; for it must be observed, that when the arm is at 35, they begin to touch. In the two following expe­ riments, the index was in its usual position.
E xperiment x ii. May 9.
W eights in negative position.
17.4, 9 45 °
17.4
58 0
17.4 10 8 0
j 17.4
10 0 17,4 ,
the Density
ofthe
.
501
W eights moved to positive position.
Downloaded from https://royalsocietypublishing.org/ on 01 August 2024
Extreme points.
Divisions.
Time.
Point of Time of mid. rest. of vibration.
*8,85
h. , j.
18,4 a 8,3
19.3
37,8 so
2 4, 22
— — — -
IO 20 50 1
21 46 J
-
-
-
-
-
-
—
-
-
— IO 3 0 59
23,49 23.57 23,67 23,73 23,8
27 >4
-
24,
23
*0, 5 5
-
' -
-
11 3 131
54 J
-
—
23,83
— 11
23,87
3 14
23 24
9 451
_
10 28 /
10 18
*7
M otion of arm - - - =
Tim e of vibration -
- = ‘7 '3 "
e x pe r im e n t x in . M ay 25.
W eights in negative position.
j 6
18,3
-
16,2
- i 7 >2
W eights moved to positive position.
39,6 ’ 7.4 38,9 18,4
25 24
—
23 24
24 23
IO 22 22 1 O 45 J
- 2 9 30 2 3 /
36 581 37 24 J
3V3 ,3 3 23,4 23.52
22 56
/
3° 3
37 7
502
M r. Cavendish's Experiments to determine
Downloaded from https://royalsocietypublishing.org/ on 01 August 2024
Extreme points.
Divisions.'
Time.
Point of Time of mid. rest. of vibration.
23 24 38,4
19.3 27,8
24
23
19.9 23 24
37>3
/
It
1° 44 3 \
31 J
11 5 26 6 if
12 12 i 50 J
>
,
„
IO 4 4 1 4
23,62
23.7 23.7
11 5 31 23,72
12 35
W eights moved to negative position.
i 3>5 21,8
i 3>9 21,1 i 4>4 20,5 x4»7 20
15
19>5
18
17
-
17
18
-
-
— — — 18
17
-
17 18
-
-
17.75
37
38
ftl
-
-
44 2 6t
45 4 J
-
-
17,67 17,62
-
-
--
-—
-
—
0 19 57 1 20 52 y
17,6 17.52 17.47 17,42
-
-
- ■17.37
27 ! 5 \
28 15 J
37 39 44 45
o 20 24
27 30
Motion of the arm on moving weights from — to + + to —
Time of vibration at
- - -
+
6,12
5)97
7' 6"
7 7
the Density o f the
.
60S
xiv. e x p e r i m e n t May 26. Weights in negative position.
Downloaded from https://royalsocietypublishing.org/ on 01 August 2024
Extreme points.
'jDivisions.
Time.
•
h. , m
1^,1 9 18 O
l6 ,l
24 0
l6 ,l
46 0
l6,l
49 0
Point of rrime of mid.
rest. ofvibration.
h* /
l6,l
Weights moved to positive position.
«7.7 i 7»3 87,2 18,3 26,8
20 26,2
23 22
-
22 23
-
23 22
-
-
23 22
—
22 23
10 0 46 ] 116 J
-
lO 1 1
-
22,37
7 581
8 27 j
- - 22,5
08 a
15 2 v
32 j
- - 22,65
-
22,75
- 22,85
- .
22,97
15 9
43 40 1 44 22 J
-
-
23>*5
49 53 ) -
5° 37 J
43 3* 5° 41
Weights moved to negative position.
18,4 1 6
81,5 || 17— 17 16
11 7 53 \ 8 27 J
-
xi 8 25
- - 17,02
15 3° 1 16 3 J
15 27
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5 ° 4<
Mr. Cavendish's Experiments to determine
Extreme points.
Divisions.
T im e.
Point of Time of mid. rest. of vibration.
12,7
—
h- , —
-
h. , .
so,7 -
- - 16,85
13.3 -
20
-
-
- 16,82
-
- 16,72
* 3>6
-
16
17
t-9>5 -
-
- 16,67
11 50 33 \ - 11 50 58
19 J
-
- 16,65
17
16
57 53 \ 58 4 4 j
-
58 6
Motion of arm by moving weights from — to + = 6,27 + to — = 6, 13
Tim e of vibration at -{- == 7' 6"
—= 76
In the next experiment, the balls, before the motion of the weights, were made to rest as near as possible to the sides of the case, but on the contrary side from what they did in the 9th, 10th, and 1ith experiments.
e x pe r im e n t xv. M ay 27.
W eights in negative position.
3>9
3>35 -
-
3»85 — -
3>4
3.6 1 3.61
the Density o f the Earth.
505
Weights moved to positive position.
Downloaded from https://royalsocietypublishing.org/ on 01 August 2024
Extreme points.
Divisions.
Time.
Point of Time of mid. rest. of vibration.
*5>4
IO
9 4>8 -
9
10
14,8 10
9
5»9
—
14.35 «*
6,8
-
13.9 -
11
10
7.5
-
10
11
13.5
Motion of the arm
Time of vibration.
h- , «
h. , „
10 5 59 6 27 J
-
lO 5 56
-
- 9>95
i s 43 ) 13 11 J
-
13 5
10,07
20 24, "i
56 ■»
-
-
10,23
20 13
-
- i°.35
-
10,46
—
-
10,52
48 30 i
49 11
-
-
55 26 i
56 10 J
10,6
-
48 42
55 48
- -
-
-
-
< 7'7"
The two following experiments were made by Mr. Gilpin, who was so good as to assist me on the occasion.
experiment xvi. May 28.
Weights in negative position.
22.55 8,4 21 9.2 mdccxcviii.
15.09 14.9
3T
506
Mr. Ca v e n d i s h 's Experiments to determine
Weights moved to positive position.
Downloaded from https://royalsocietypublishing.org/ on 01 August 2024
Extreme
points.
Divisions.
Point of Time of mid. rest. ofvibration.
,2 6 ,6 22
21
15.8
20
21
25.8
22
21
16.8
20
21
25.05
22
21
*7>57 21
22 24.6
22
21
18,3 Motion of the arm Time of vibration
h.
IO 2 2 5 3 23 20
30 7 36
37 23 1
5 J
44 45
29 4
1J
51 54 52 32
59 3 i
o 13
6 24
7 9
t «
10 23 15
21,05
3° 3°
21,11
37 45
21,11
45 1
52 20
2 1 ,2
59 34
21,28 11 6 49
= 6,1 = 7' 16"
e x p e r i m e n t xvii. May 30.
Weights in negative position.
1 7 , 2 10 19 0
17,1
17,07
25 0
29 0
1 7 .1 5
40 0
1 7 .4 5
49 0
1 7 .4 3 51 0
1 1 7 . 4 2 11 1 0
17,42
the Density o f the
. Earth
507
Weights moved to positive position.
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Epxotirnetms.e.#Divisions. Time.
Point of Time of mid. rest. ofvibration.
28,8
24 23 l 8,l 22
23 87,8 -
34
33 18,8 -
23 24
®7>38 24
23
19.7 23
24
«7
-
24
so,4 23-
23 24 36,5 -
24 23 30,8 -
23 24 36,35
» j»
h. , ,
11 11 23 V
11 11 37
- 49- ' 23.2
18 4 3 J} - - 23,12
2519 \ ~ 4 -9 J 23,2
32 4 1 1 ~3 3 1 -3 i 23,31
3938) 4° 3 ■* — — 23,44
46 33 ) 47 n J
-
— - 23>52
53 36 \
54 17 ;
- - ^3>57
0 0 34 i 1 18 J
-
-
2 3>55
7 34 1 8 21 J - - 23>59
H 3° ) 15 24 J
-
l8 42 25 40 32 43 39 44 46 46 53 48 0 0 55
7 50 14 58
3T 2
5dS
Mr. Cavendish’s to determine
Weights moved to negative position.
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Extreme points.
Divisions.
Time.
Point of Time of mid. rest. of vibration.
* 3>3 17
18
-
l8
17 * 3>7
17
18
2 1 ,6
-
l8
17 14
17
18
2 0 ,8
-
18
17
* 4>3
-
17
18
2 0 ,1
-
18
17
1 4 ,6
!*• / H
1 «
O 32 191 48/
-
-
39 4 6 1
40 1 9 /
-
-
46 2 6 1
47 ° /
53 431
54 20 /
-
-
i 0 391
1 20 /
- —
7 391
8 21 J
-
-
14 541
15 42 J
-
21 3 2 n
22 22 /
O 32 44
* 7>95
-
17^5
-
39 44
46 48
1 7 ,7 2
-
53 50
1 7 ,6
-
1 0 55
1 7.47 -
7 59
1 7 .3 7 -
1 7 ,2 7
15 4
22 5
Motion of the arm on moving weights from — to + = 5,78 + to — =3 5 ^ 4
Time of vibration at
+ = 7'2 ' —= 7 3
the Density of the Earth.
509
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On the Method of computing the Density of the Earth from these Experiments.
I shall first compute this, on the supposition that the arm and copper rods have no weight, and that the weights exert no sensible attraction, except on the nearest ball; and shall then examine what corrections are necessary, on account of the arm and rods, and some other small causes.
The first thing is, to find the force required to draw the arm aside, which, as was before said, is to be determined by the time of a vibration.
The distance of the centres of the two balls from each other is 73,3 inches, and therefore the distance of each from the centre of motion is 36,65, and the length of a pendulum vibrating se­ conds, in this climate, is 39,14; therefore, if the stiffness of the wire by which the arm is suspended is such, that the force which must be applied to each ball, in order to draw the arm aside by the angle A, is to the weight of that ball as the arch of A to the radius, the arm will vibrate in the same time as a pendulum
whose length is 36,65 inches, that is, in
seconds; and
therefore, if the stiffness of the wire is such as to make it vibrate in N seconds, the force which must be applied to each ball, in order to draw it aside by the angle A, is to the weight of the
ball as the arch of A x 4N* x &39,14
the radius. But the ivory J
scale at the end of the arm is 38,3 inches from the centre
of motion, and each division is ~ of an inch, and therefore
subtends an angle at the centre, whose arch is ; and there­
fore the force which must be applied to each ball, to draw the
5io
Mr. Cavendish's Experiments to determine
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arm aside by one division, is to the weight of the ball as
if 3^21 to 1, or as
to 1.
766 N a 39,14
818 N a
The next thing is, to find the proportion which the attraction
of the weight on the ball bears to that of the earth thereon,
supposing the ball to be placed in the middle of the case, that
is, to be not nearer to one side than the other. When the
weights are approached to the balls, their centres are 8,85
inches from the middle line of the case; but, through inadver­
tence, the distance, from each other, of the rods which support
these weights, was made equal to the distance of the centres of
the balls from each other, whereas it ought to have been some­
what greater. In consequence of this, the centres of the weights
are not exactly opposite to those of the balls, when they are ap­
proached together; and the effect of the weights, in drawing the
arm aside, is less, than it would otherwise have been, in the tri­
plicate ratio of ~ ~ to the chord of the angle whose sine is
or in the triplicate ratio of the cosine of •§• this angle to
the radius, or in the ratio of ,9779 to 1. Each of the weights weighs 2439000 grains, and therefore
is equal in weight to 10,64 spherical feet of water; and there­ fore its attraction on a particle placed at the centre of the ball, is to the attraction of a spherical foot of water on an equal par-
tide placed on its surface, as 10,64 x , * bTsJ to 1. The mean diameter of the earth is 41800000 feet;* and therefore, if the mean density of the earth is to that of water as D to one, the attraction of the leaden weight on the ball will be to that
• In strictness, we ought, instead of the mean diameter of the earth, to take the diameter of that sphere whose attraction is equal to the force pf gravity in this cli­ mate ; but the difference is not worth regarding. ]
the Density
ofthe Earth. $11
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of the earth thereon, as 10,64 x ,9779 x g-g~ to 41800000 D :: 1 to 8739000 D.
It is shewn, therefore, that the force which must be applied to each ball, in order to draw the arm one division out of its
natural position, is 8i8^ - of the weight of the ball; and, if the
mean density of the earth is to that of water as D to 1, the at­
traction of the weight on the ball is 8739J’oo'r>'
w^ight of
that ball; and therefore the attraction will be able to draw the
arm out of its natural position by
d or 10683 d &Visions.;
and therefore, if on moving the weights from the midway to a near position the arm is found to move B divisions, or if it moves 2 B divisions on moving the weights from one near po­ sition to the other, it follows that the density of the earth, or D,
N4
10683 B *
W e must now consider the corrections which must be ap­ plied to this result; first, for the effect which the resistance of the arm to motion has on the time of the vibration: 2d, for the attraction of the weights on the arm : 3d, for their attrac­ tion on the farther ball: 4th, for the attraction of the copper rods on the balls and arm : 5th, for the attraction of the case on the balls and arm : and 6th, for the alteration of the attrac­ tion of the weights on the balls, according to the position of the arm, and the effect which that has on the time of vibration. None of the^e corrections, indeed, except the last, are of much signification, but they ought not entirely to be neglected.
As to the first, it must be considered, that during the vibra­ tions of the arm and balls, part of the force is spent in accele­
512
Mr. Cavendish's Experiments to determine
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rating the arm; and therefore, in order to find the force re­ quired to draw them out of their natural position, we must find the proportion which the forces spent in accelerating the arm and balls bear to each other.
Let EDC e d c (fig. 4.) be the arm. B and b the balls. C s the suspending wire. The arm consists of 4 parts ; first, a deal rod D cd,73,3 inches long; 2d, the silver wire DC weigh­ ing 170 grains ; 3d, the end pieces D E and e dy to which the ivory vernier is fastened, each of which weighs 45 grains; and 4th, some brass work C c, at the centre. The deal rod, when dry, weighs 2320 grains, but when very damp, as it commonly was during the experiments, weighs 2400 ; the transverse sec­ tion is of the shape represented in fig. 5 ; the thickness BA, and the dimensions of the part DE , being the same in all parts; but the breadth B b diminishes gradually, from the middle to the ends. The area of this section is ,33 of a square inch at the middle, and ,146 at the en d ; and therefore, if any point x (fig.
4.) is taken in c d, and ~
is
called
x,
7
this rod
cd7
weighs
■2°4°°
73>3LX
per inch at the middle; *4°° ^ 23^ at the end, and
3320—1848x ^ . an(j therefore, as the weight of the wire is 7 3»3
170
73T3
per
inch,
the
deal rod
and wire
together may be considered
as a rod whose weight at
x = —l84— per inc
But the force required to accelerate any quantity of matter
placed at x f is proportional to
x %; that is, it
quired to accelerate the same quantity of matter placed at d as
x %Xo 1 ; and therefore, if
c d i$called /, and x i
flow, the fluxion of the force required to accelerate the deal rod
the Density of the Earth.
513
and wire is proportional to - — *-349°— the fluent of which,
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generated while x flows from
cot =
x~p —
so that the force required to accelerate each half of the deal rod
and wire, is the same as is required to accelerate 350 grains
placed at
d.
The resistance to motion of each of the pieces , is equal to
that of 48 grains placed at
d;as the distance of their centres of
gravity from C is 38 inches. The resistance of the brass work
at the centre may be disregarded; and therefore the whole force,
required to accelerate the arm, is the same as that required to
accelerate 398 grains placed at each of the points D and d.
Each of the balls weighs 11262 grains, and they are placed at
the same distance from the centre as D and d ; and therefore, the
force required to accelerate the balls and arm together, is the
same as if each ball weighed 11660, and the arm had no weight;
and therefore, supposing the time of a vibration to be given, the
force required to draw the arm aside, is greater than if the arm
had no weight, in the proportion of 11660 to 11262, or of
1>°3 5 3 to *• To find the attraction of the weights on the arm, through d
draw the vertical plane
d w bperpendicular to D d, and let w
be the centre of the weight, which, though not accurately in this
plane, may, without sensible error, be considered as placed
therein, and let b be the centre of the b&ll; then is horizontal
and = 8,85, and
d ibs vertical and = 3,5; let =
b, and let ~ , or 1 —
x =
%;ht en the attraction of the weig
on a particle of matter at
x,in the direction d , is to its attra
tion on the same particle placed at b :: b3: + / , or is pro-
MDCCXCVIII.
o0 T^T
514
'Mr. Cavendish’s Experiments to determine
rportional to a=%=+=*=4=3, and the force of that attraction to move
the arm, is proportional to -======: and the weight of the
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deal rod and wire at the point , was before said to be
3490—1848X 1642 + ■84-- per inch; and therefore, if
73*3
73*3
flows,
the fluxion of the power to move the arm = x 21~3**~3l848z
b3x 1— + z1
2x821
b3« x 821 + 1 0 3 2 :— 9 2 4 ^
a4+ Z4z4}'!
2x821+ 1035;-f 924
a4
924J3K X ^+2*
«
; which, as
a4+/4z4]s
1!>3ZX895-|-1032T z4)r
924 ft3 g; P^/a'+Pz7- ‘
The fluent of this
= ,08
895 b3z a1* / a4+ lx z 7
Tva1Q43+ F F +, 1Q3 t>3 -----91H0g* inr. ^ 4 ^ 7*--+---^- >„annda tthftee WIorcee
with which the attraction of the weight, on the nearest half of
the deal rod and wire, tends to move the arm, is proportional
fluent generated while
zflows from o to 1, that is, to
128 grains.
The force with which the attraction of the weight on the b3
end-piece de tends to move the arm, is proportional to 47 x -5,
or 29 grains; and therefore the whole power of the weight to
move the arm, by means of its attraction on the nearest part
thereof, is equal to its attraction on 157 grains placed at bT
which is
or ,0139 of its attraction on the ball.
It must be observed, that the effect of the attraction of the weight on the whole arm is rather less than this, as its attrac­ tion on the farther half draws it the contrary way; but, as the attraction on this is small, in comparison of its attraction on the nearer half, it may be disregarded.
the Density
of the Earth. 515
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The attraction of the weight on the furthest ball, in the di­
rection
bw,is to its attraction on the nearest ball : : w : w D 5
:: ,0017 : 1; and therefore the effect of the attraction of the
weight on both balls, is to that of its attraction on the nearest
ball :: ,9983 : 1.
To find the attraction of the copper rod on the nearest ball,
let b and w (fig. 6.) be the centres of the ball and weight, and
ea the perpendicular part of the copper rod, which consists of
two parts, ad and de. ad weighs 22000 grains, and is 1
inches long, and is nearly bisected
de weighs 41000,
and is 4
6niches long, wb is 8,85 inches, and is perpendicular
to ew. Now, the attraction of a line ew, of uniform thickness,
on b,in the direction
bw,is to that of the same quantity
matter placed at
w : : bw: eb; and therefore the attraction of
the part da equals that of ~2°°?*Zfc,Z>, or 16300, placed at w;
and the' attraction of
deequals that of 41000 x “ “41000
x x or 2500, placed at the same point; so that the at­
traction of the perpendicular part of the copper rod on is to
that of the weight thereon, as 18800 : 2439000, or as ,00771
to 1. As for the attraction of the inclined part of the rod and
wooden bar, marked P
rand rr in fig. 1, it may safely be
neglected, and so may the attraction of the whole rod on the
arm and farthest ball ; and therefore the attraction of the
weight and copper rod, on the arm and both balls together, ex­
ceeds the attraction of the weight on the nearest ball, in the
proportion of ,9983 + ,0139 >°°77 *° one>or of 1,0199 1* *
.The next thing to be considered, is the attraction of the ma­
hogany case. Now it is evident, that when the arm stands at
3U 2
316*
Mr. Cavendish’s Experiments to determine
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the middle division, the attractions of the opposite sides of the case balance each other, and have no power to draw the arm either way. When the arm is removed from this division, it is attracted a little towards the nearest side, so that the force re­ quired to draw the arm aside is rather less than it would other­ wise be; but yet, if this force is proportional to the distance of the arm from the middle division, it makes no error in the re­ sult ; for, though the attraction will draw the arm aside more than it would otherwise do, yet, as the accelerating force by which the arm is made to vibrate is diminished in the same proportion, the square of the time of a vibration will be in­ creased in the same proportion as the space by which the arm is drawn aside, and therefore the result will be the same as if the case exerted no attraction; but, if the attraction of the case is not proportional to the distance of the arm from the middle point, the ratio in which the accelerating force is diminished is different in different parts of the vibration, and the square of the time of a vibration will not be increased in the same pro­ portion as the quantity by which the arm is drawn aside, and therefore the result will be altered thereby.
On computation, I find that the force by which the attrac­ tion draws the arm from the centre is far from being propor­ tional to the distance, but the whole force is so small as not to be worth regarding; for, in no position of the arm does the at­ traction of the case on the balls exceed that of ^th of a spheric inch of water, placed at the distance of i inch from the centre of the balls; and the attraction of the leaden weight equals that of 10,6 spheric feet of water placed at 8,85 inches, or of 234 spheric inches placed at 1 inch distance; so that the attraction of the case on the balls can in no position of the arm exceed
the Density of the Earth.
517
yyyq of that of the weight. The computation is given in the Appendix.
It has been shewn, therefore, that the force required to draw the arm aside one division, is greater than it would be if the arm had no weight, in the ratio of 1,0353 to 1, and therefore
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= b°38s^T- of the weight of the b all; and moreover, the attrac­ tion of the weight and copper rod on the arm and both balls together,exceeds the attraction of the weight on the nearest ball,
in the ratio of 1,0199 to 1, and therefore =
of the
weight of the b all; consequently D
is really equal to
818 N1 1*0353
x
or 7^ p r > instead o f a s by the former
computation. It remains to be considered how much this is
affected by the pos lion of the arm.
Suppose the wc.yjhts to be approached to the balls; let W ,
(fig. 7.} be th venire of one of the weights; let M be the centre
of the nearest ball at its mean position, as when the arm is at
20 divisions; let B be the point which it actually rests a t ; and
let A be the point which it would rest at, if the weight was re­
moved; consequently, AB is the space by which it is drawn
aside by means of the attraction; and let M be the space by
which it would be drawn aside, if the attraction on it was the
same as when it is at M. But the attraction at B is greater
than at M, in the proportion of W M h W Bl ; and therefore,
AB = M (3 x ~ = M @x 1 +
very nearly.
Let now the weights be moved to the contrary near position, and let w be now the centre of the nearest weight, and b the point of rest of the centre of the ball; then A6 = M /3 x i + 2MMWb>and B b M/3 x 2 + ^ + M?MWB sM/3 X 1 + MW
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ji8
Mr. Cavendish's Experiments to determine
that the whole motion B
bsi greater than it would
attraction on the ball was the same in all places as it is at M,
in the ratio of 1 -|- to one; and, therefore, does not depend
sensibly on the place of the arm, in either position of the
weights, but only on the quantity of its motion, by moving
them.
This variation in the attraction of the weight, affects also the
time of vibration; for, suppose the weights to be approached to the balls, let W be the centre of the nearest weight; let B
and A represent the same things as before; and let x be the
centre of the ball, at any point of its vibration; let AB repre­
sent the force with which the ball, when placed at B, is drawn
towards A by the stiffness of the w ire; then, as B is the point
of rest, the attraction of the weight thereon will also equal AB;
and, when the ball is at
x, the force with which
towards A, by the stiffness of the wire, = A x , and that with
which it is drawn in the contrary direction, by the attraction,
= AB x ; so that the actual force by which it is drawn to-
wards A = A.r —
=A B + B x - A B x i + -^f =
B.r — —’^x Ah, very nearly. So that the actual force with
which the ball is drawn towards the middle point of the vibra­ tion, is less than it would be if the weights were removed, in
the ratio of 1 — ^ 5 . to one, and the square of the time of a
vibration is increased in the ratio of 1 to 1 — which differs
very little from that of 1 + to 1, which is the ratio in which the motion of the arm, by moving the weights from one near position to the other, is increased.
theDensity of the Earth.
519
The motion of the ball answering to one division of the arm
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= 20x3s"3’ anc^ ^ ^ t^le moti°n of the hall answering to
d divisions on the arm,
and there-
fore, the time of vibration, and motion of the arm, must be cor­
rected as follows : If the time of vibration is determined by an experiment in
which the weights are m the near position, and the motion of the arm, by moving the weights from the near to the midway
position, is d divisions, the observed time must be diminished
in the subduplicate ratio of 1 —— to 1, that is, in the ratio
of 1 — ~ to 1; but, when it is determined by an experiment
in which the weights are in the midway position, no correc­
tion must be applied.
To correct the motion of the arm caused by moving the
weights from a near to the midway position, or the reverse,
observe how much the position of the arm differs from 20 di­
visions, when the weights are in the near position: let this be
n divisions, then, if the arm at that time is on the same side of the division of 20 as the weight, the observed motion must be
diminished by the ~ part of the whole; but, otherwise, it must
be as much increased.
If the weights are moved from one near position to the other,
and the motion of the arm is 2
ddivisions, the observed m
tion must be diminished by the ^2d part of the whole.
If the weights are moved from one near position to the other, and the time of vibration is determined while the weights are in one of those positions, there is no need of correcting either the motion of the arm, or the time of vibration.
52°
Mr. Cavendish’s Experiments to determine
CONCLUSION. Thefollowing Table contains the Result of the Experiments.
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Exper. Mot. weight Mot. arm Do. corr. Time vib Do. corr. Density.
m . to
l\
to m .
„ J m . to +
-f- t o m .
-j- to m .
1 4 .3 2 1 4 .1 15.87
15.45 15,22
m . to +
c m . to + 4-< + to —
c — t o ~|r I - f to — 5 1 — t o -f*
m . to —
1 4 .5 3 .1 6 ,1 8 5 .92
5 ,9 5 .9 8 3>°3
1 — to + m . to —
— t o —j~
8 $ m . to — i — to +
9
~j“ t o
lO -j- to —
5.9 3 .1 5 6 ,1
3.1 3 5.7 2 6,3 2 6 ,1 5
n
-j- to — 6 ,0 7
12 1Q ( J3 <
1A 14 <
— to + ——t o —J— -f* t o -— —• t o —j~ + t o ----
1 5 •— t o + 1 6 —— t o —|—
6 ,0 9 6 ,1 2
5 .9 7 6 ,2 7 6 ,1 3 6 ,3 4 6,1
— to -j-
17 {
t o ——
5 ,7 8 5>6 4
13.42 1 3 .1 7 h .6'9 14.14
/ " 1 4 .5 5
-
1 4 ,4 2
!3>56 14^39 13.28 1 4 .5 4
2 .9 5
-
7 .i
-
7 .3
-
7 .5
-
7 .5
2 ,9
-
5 .7 i 3 .0 3 5 .9 3>°° 5 .5 4
-
7.4 y by
mean.
J
-
6 ,5 8
-
6 ,5 9
-
7.1
-
7 .3
-
7 .6
-
7.7
-
7 .6
-
7 .6
-
7.7
-
7 ,1 6
-
7 .2
-
7 .3
-
-
-
-
6 .5 4 -
-
-
6 ,5 7 -
-
-
-
-
-
5 ,5 5 ,6 1 4,8 8
5 ,0 7 5 ,2 6
5 ,5 5 5 ,3 6 5 ,2 9 5 ,5 8 5 .6 5 5 ,5 7 5 ,5 3 5 ,6 2
5 ,2 9 5 ,4 4 5 .3 4 5 .7 9 5 ,1 5 ,2 7 5 ,3 9 5 .4 2 5 ,4 7 5 .6 3 5 ,3 4 5 ,4 6 5 ,3 5,7 5 5 .68 5 .8 5
the "Density o f the
.
52 X
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From this table it appears, that though the experiments agree
pretty well together, yet the difference between them , both in
the quantity of motion of the arm and in the tim e of vibration,
is greater than can proceed m erely from the error of observa­
tion. As to the difference in the motion of the arm , it may
very well be accounted for, from the current of air produced
by the difference of tem perature; but, w hether this can account
for the difference in the tim e of vibration, is doubtful. If the
current of air was regular, and of the same swiftness in all parts
of the vibration of the ball, I think it could n o t; but, as there
will most likely be much irregularity in the current, it may very
likely be sufficient to account for the difference.
By a mean of the experiments made with the wire first used,
the density of the earth comes out 5^4,8 times greater th an th a t
of w ater; and by a mean of those made with the latter wire, it
comes out thejsame; and the extreme difference of the results of
the 23 observations made w ith this wire, is only ,75; so th a t
the extreme results do not differ from the mean by more than
,38, or of the whole, and therefore the density should seem
to be determined hereby, to great exactness. It, indeed, may
be objected, that as the result appears to be influenced by the
current of air, or some other cause, the laws of which we are
not well acquainted with, this cause may perhaps act always,
or commonly, in the same direction, and thereby make a c o n ­
siderable error in the result. But yet, as the experiments were
tried in various weathers, and with considerable variety in the
difference of temperature of the weights and air, and w ith the
arm resting at different distances from the sides of the case,
it seems very unlikely that this cause should act so uniformly
in the same way, as to make the error of the mean result nearly
MDccxcvnr.
3 X
522
Mr.C a v e n d is h 's Experiments to determine
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equal to the difference between this and the extreme; and, therefore, it seems very unlikely that the density of the earth should differ from 5,48 by so much as T*T of the whole.
Another objection,, perhaps, may be made to these experi­ ments, namely, that it is uncertain whether, in these small distances, the force of gravity follows exactly the same law as in greater distances. There is no reason, however, to think that any irregularity of this kind takes place, until the bodies come within the action of what is called the attraction of co­ hesion, and which seems to extend only to very minute dis­ tances. With a view to see whether the result could be affected by this attraction, I made the 9th, 10th, 11th, and 15th expe­ riments, in which the balls were made to rest as close to the sides of the case as they could ; but there is no difference to be depended on, between the results under that circumstance, and when the balls are placed in any other part of the case.
According to the experiments made by Dr. M askelyne, on the attraction of the hill Schehallien, the density of the earth is \ \ times that of w ater; which differs rather more from the preceding determination than I should have ex­ pected. But . I forbear entering into any consideration of which determination is most to be depended on, till I have exa­ mined more carefully how much the preceding determination is affected by irregularities whose quantity I cannot measure.
the Density of the Earth.
5* 3
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APPENDIX.
On the Attraction of the Mahogany Case on the Balls.
T he first thing is, to find the attraction of the rectangulaf
plane
c k 13(bfig. 8.) on the point a, placed in the line a c per­
pendicular to this plane.
Let a c =■ a, c k = b,
7
7
*= vx, then the attraction of the line
h Q
a b, =
7FxT@* anc*therefore, if
6c =
a* +
9
-f
on , in the direction
x,and le c fblows, the fluxio
traction of the plane on the point , in the direction c bf
^
b X
v'aa + x * x ,S a l +b*-\-xl
X— b W
v'cf+ x*
__ J
_______ ij;
the variable part of the fluent of which = — log. + y / i + v \
and therefore the whole attraction = log.
x W+a& ’ SQ
that the attraction of the plane, in the direction c , is found readily by logarithms, but 1know no way of finding its attract tion in. the direction ac, except by an infinite series.
The two most convenient series I know, are the following:
First series. Let - ^ =
tf,and let A = arc whose tang, is *r,
B = A — 7r, C == B + —, D = C — &c. then the attrac-
tion in the direction &c.
3X 2
a c= \ / i — w%x A +
1 2 1 2 * 4 1 2. 4 «»
Mr. Cavendish's Experiments to determine
For the second series, let A = arc whose tan g .= ~ ,B = A—^ ,
C = B -J*—3 —, D = vC3 -—
5 5
&c^hten the attract
y i -f-
v*x A Bva + 3Cr/4 32•.54D. 6p*•>
It must be observed, that the first series fails when tt is
greater than unity, and the second, when it is less; but, if b is
taken equal to the least of the two lines ck and cb, there is no
case in which one or the other of them may not be used con­
veniently.
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By the help of these series, I computed the following table.
> 1 9 6 2 ’ .3714 , 5145! ,6 2 4 8 , 7 0 7 1 ’ ,7 8 0 8 >8575 ,9 2 8 5 > 9 8 1 5
,1062 ,00001
>3714 ,0 0 0 3 9 O O I48 >5*45 ,0 0 0 7 4 0 0 2 7 7 0 0 5 2 1
,6 2 4 8 OOllO 004 0 6 ’OO778 0 1 1 8 3
>7° 7i OOI4O 0 0 3 2 2 0 1 0 0 8 0 1 5 2 5 0 2 0 0 2
, 7 8 0 8 0 0 1 7 1 0C6’3 7 0 1 2 4 5 0 1 8 9 6 0 2 4 0 5 0 3 2 4 7
.8575 OO2O7!0 0 7 7 2 0 1 5 2 2 0 2 3 3 9 0 3 1 1 6 0 3 9 6 4 0 5 0 5 7
, 9 ^ 5 OO244!0 0 9 1 0 O l S l O 0 2 8 0 7 0 3 7 7 8 0 4 8 6 7 0 6 3 1 9 0 8 l l 9
, 9 8 1 5 0 0 2 7 if O I O I 9 0 2 0 8 4 0 3 1 9 3 0 4 3 6 8 0 5 6 3 9 ' 07478 0 9931 1 2 8 4 9
1*
1 0 0 2 84I'O IO 54 0 2 1 3 5 | 0 3 3 4 7 1 0 4 5 6 0 ° 5975i0 7 9 7 8 1 0 7 8 9 1 4 6 3 2 1 9 6 1 2
Find in this table, with the argument ~ at top, and the argu­ ment in the left hand column, the corresponding logarithm ; then add together this logarithm, the logarithm of and the lo­ garithm of —; the sum is logarithm of the attraction.
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the Density of the
*
$25
To compute from hence the attraction of the case on the
ball, let the box DCBA, (fig. 1.) in which the ball plays, be divided into two parts, by a vertical section, perpendicular to
the length of the case, and passing through the centre of the
b all; and, in fig. 9, let the parallelopiped ABDE ahde be one
of these parts, ABDE being the abovementioned vertical sec­
tion ; let x be the centre of the ball, and draw the parallelogram
13n p m i x parallel to B
h Dd , and parallel to jGB and
bisect
f3 $in
c.Now, the dimensions of the box, on the inside,
are B 6 = 1,75; BD = 3,6; B/3 = i ,75; and/ 3A = 5 ; whence
I find, that if
x cand (3x are taken as in the two upper lines of
the following table, the attractions of the different parts are as
set down below.
xc
>75 >5 >25
(3x
1,05
1.55
Excess of attract,
of Dab.2o3ve74B
,0813
------------- --------- m drp above nbrp .2374 ,1614 ,0813
-----------------------mesp above nasp >37°5 ,2516 ,1371
Sum of these
Excess of attract, of B ---------------- ------
>8453 >5744 .2897 bab»5oove°7D .3271 ,1606 Aba>o4v6'e77 .3079 .1525
Whole attraction of the inside surface! of the half box - - - j
,1231
,0606 ,0234
It appears, therefore, that the attraction of the box on x in­ creases faster than in proportion to the distance xc.
The specific gravity of the wood used in this case is ,61, and its thickness is -J of an inch; and therefore, if the attraction of the outside surface of the box was the same as that of the in­ side, the whole attraction of the box on the ball, when = ,75*
526
Mr. Cavendish's , &c.
would be equal to 2 x ,1231 x ,61 x f cubic inches, or ,201 spheric inches of water, placed at the distance of one inch from the centre of the ball. In reality, it can never be so great as this, as the attraction of the outside surface is rather less than that of the inside; and, moreover, the distance of x from c can never be quite so great as ,75 of an inch, as the greatest motion of the arm is only !-§- inch.
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K
/
\
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o
.Philos. Trans. M D C C X C V H I . Tab. X X IE L p. 5 z6 .
i
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fhilos.Tmns.MD
CC X C V T I T .
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