and, as before, 3N is the whole decrease which would thus be produced in gravity from the equator to the poles by this change of distribution of the mass. We shall calculate N for the second, third, and fourth shells. = 1.20 25 1.23 25 1 243-32 578 3125 1 32 1 578 3125 0.0005354, 0.0001876, 0.0000322. The ratios these, multiplied by 3, bear to the actual increase of gravity are 0.310, 0.167, 0.019. And therefore the number of beats gained at Spitzbergen upon the pendulum at the equator would be 214 multiplied by these fractions, or 66, 23, and 4. The first and second of these might be detected, though not the third. This calculation shows that a comparatively small change in the form of the strata would have a very perceptible influence upon the pendulum. The effect is greater than is produced in the former case, by a mere change in densities without altering the form of the strata: and this tells very strongly in favour of the fluidarrangement, and indeed of the fluid-theory itself. We might PENDULUM EXPERIMENTS. 95 perhaps conceive the external surface of an irregular mass revolving round a fixed axis assuming, after an enormous period, a generally spheroidal form, because the perpetual weathering of the surface would set free parts of the solid materials, which with the fluids would arrange themselves according to fluid principles. But the interior parts could not thus arrange themselves, as these calculations seem to show they have done, unless they had at one time been fluid or semi-fluid, so as to partake of that bulging form about the equator of each stratum which the motion of rotation tends to produce. 98. Third re-arrangement. The following is perhaps a still better hypothetical arrangement of the earth's mass with a view to testing the fluid theory of its origin. Suppose that the earth is a solid mass which, as described above, has acquired its external spheroidal form by the action of time; and imagine its mass to be made up of a homogeneous spheroid of the earth's present form, but of the density only of the surface, with the remainder of the mass distributed anyhow in spherical shells around the centre. The density of the surface is half the mean density of the earth; hence the mass of the homogeneous spheroid will be half the whole mass; and the consequent increase of gravity between the equator and the poles will or, when compared with the actual increase of gravity, This is nearly half the actual increase. Hence if this were the actual distribution, the gain of the pendulum over its rate at the equator would everywhere be only about half what experiment makes it to be. Experiment shows no sudden changes, nor any marked deviation from a regular increase, varying as the change in the square of the latitude, in the rates of the pendulum in passing from the equator towards the poles. Hence the excess of matter above the homogeneous spheroid cannot be distributed irregularly. We have supposed it to be distributed in spherical shells, and the change on the pendulum would be, as we have shown, very great, and would be very perceptible indeed. Any departure from the spherical form, not towards the oblate spheroids required by the fluid-theory, but in the opposite direction, would produce a result still more discordant with experiment; whereas every approach in the distribution to those spheroids will bring the calculation into nearer accordance with fact. No stronger testimony can well be borne to the truth of the fluid-arrangement and fluid-theory. 99. COR. We may find the effect of a large departure from regularity in the mass in the following manner. Suppose there is a preponderance of matter the effect of which may be represented by a spherical mass m, the distance of the centre of which from the centre is c, = x.a suppose. Then the difference of gravity in consequence of this at the two points nearest and furthest off from this mass Ifb be the number of beats lost or gained by a seconds' pendulum between the two places in 24 hours PENDULUM EXPERIMENTS. 97 where e is the radius of a sphere of which the density is the mean density of the earth and the mass = m. a mass of only radius 96 miles and only fourteen millionths and a half of the earth's mass, and as far down as 1000 miles from the surface, will have a perceptible influence upon the pendulum. Ex. 2. If the depth below the surface be 500 miles the radius of the disturbing mass will be only 62 miles, and the mass three millionths and three-quarters of the earth's mass. Ex. 3. If x=1, e = 236 miles and m is 1-5000th part of the earth's mass, and 3000 miles below the surface; and yet in each of these cases the effect is the same as before: Accurate pendulum experiments all over the world must bring to light such masses, if they exist. None have as yet been detected. Second Test. PERTURBATION OF THE MOON'S MOTION IN LATITUDE. 100. Laplace first pointed out that the ellipticity of the Earth would have an effect upon the Moon's motion. This we shall use as our second general test of the truth of the fluid-theory. PROP. To find the effect of the Earth's mass, arranged according to the fluid law, upon the Moon's motion in latitude. 101. By the Planetary Theory (see Mechanical Philosophy, second edition, p. 329; or Cheyne's Planetary Theory, p. 35), di na dR (E+M) i di' where n is the mean motion of the moon about the earth, a the mean distance, E and M the masses, i the inclination of the moon's orbit to the ecliptic (the square of which is neg P. A. 7 lected), the longitude of its node, R the disturbing function such that its differential coefficient with respect to any line drawn from the moon is the disturbing force acting on the moon in that direction, reckoned positive if acting on the side of the origin of the co-ordinates. If V be the potential of the earth with reference to the moon condensed into its centre, then MV÷E will be the potential of the moon with reference to the earth; and in calculating the motion of the moon about the earth, we must imagine the earth reduced to rest by the moon's attraction being applied in an opposite direction to both the earth and moon. Hence the disturbing function R, which refers to the difference of attraction of the earth's mass as condensed into its centre and as arranged according to the fluid-law, r being the distance of the moon from the earth's centre. Let λ and be the latitude and longitude of the moon, e' the epoch, the longitude of the perigee, I the obliquity of the ecliptic. Then, as λ and i are both small, tan λ = tan i sin (nt + e' — N), or λ = i sin (nt + e' — N). ́ = = sin I cos λ sin + cos I sin λ sin I sin (nt + e') + i cos I sin (nt + e' — ↓) -e sin I sin + e sin I sin (2nt + 2e' — ☎). Substituting this in R, and preserving only the terms which are periodical and also independent of nt+e', since these last go through their changes so rapidly as to neutralize their effects very quickly, we have |