This may be put into another form. Multiply by (a), then

de d

6

dp

da { (a) da} + 2 (8pa°e) = { 1 $ (a) e + 3a°e da;

da

6

=

dp

or de (ø (a) e) = 1 1⁄2 4 (a) e + sae do.

da2

da'

81. COR. 1. By putting a = a in equation (3) of Art. 75, we have the following equation, which we shall find of use;

a d

5

[P' adƒ (a′′e') da' = 3 a2p (a) (e — m).

PROP. To prove that the ellipticity of the strata decreases from the surface towards the centre.

82. We assume that the density of the Earth increases from the surface to the centre. Let then p =D - Ea" + ...,

where E is positive: and e = A + Ba+.... Then

Put these in the differential equation in e of Art. 80; it gives

m-2

B (m3 + 5m) aTM-3-6AHa"... = 0.

Neither m nor B can equal zero, because then the second term of e only merges into the first. Nor can m=- 5, a negative quantity. Hence the first term will not vanish of itself. But we may make the first and second vanish together by putting nm and B (m2 + 5m) = 6AH. Hence B must be positive. And therefore near the centre e increases towards the surface.

In thus increasing, suppose it attains a maximum, and then decreases. At this point 0; and the equation of Art. 80,

de

da

already used, gives

={1

σε

d' = (1 - p) &e, a positive quantity.

da

a3

This corresponds to a minimum. Hence e does not attain a maximum, and therefore it continually increases from the centre to the surface. In the above we have assumed that (a) is greater than pa3. This appears

α

α

13

· · Þ (a) = 3 [ " p'a'”da = pa3 — ["a de da',

α

is negative by hypothesis.

83. The following Proposition, the converse of that of Art. 77, is of considerable importance, as it leads to the conclusion, that the form of the surface of the earth and the arrangement of the earth's mass are intimately related the one to the other. The opposite of this has sometimes been stated. For example, it has been said, that the law of gravity at the surface of the earth can be obtained theoretically without any reference to the arrangement of the mass. That this is erroneous will appear from what follows.

PROP. To To prove that if the form of the Earth's surface be a spheroid-of-equilibrium, the earth's mass must necessarily be arranged according to the fluid law, whether the mass is or has been fluid or not, in part or in whole.

84. The meaning of a "spheroid of equilibrium " has been explained in Art. 78, and also of a "surface of equilibrium in general. In consequence of the first property stated in that Article, a surface of equilibrium may be also thus defined. It is such that the resultant force at every point of the surface is at right angles to the surface at that point.

Now suppose some change were to be made in the arrangement of the earth's mass, without altering its external form. It is evident that, although the resultant attraction of the whole mass on the surface might possibly be unaltered by this change at particular points of the surface, it could not remain the same as before at every point of the surface.

SURFACE-FORM DEPENDS ON INTERNAL ARRANGEMENT. 81

Hence on this change being made in the internal arrangement, however slight it might be, the surface would cease to be one of equilibrium. In fact, if it were fluid it would at once assume another form, consequent on the internal change in the arrangement of the mass. Hence the form of the surface, if it be a surface-of-equilibrium, depends upon the arrangement of the mass. Suppose the arrangement of the mass throughout its solid and fluid parts follows that of the fluid law. That is, suppose that, not only the external surface is (what our hypothesis assumes it to be) a "spheroid of equilibrium," but that all the interior mass, whether solid or fluid, follows the fluid law of density. It is evident that in this case the surface would retain its form, even if the whole mass were to become fluid. Here is, then, one arrangement of the mass which we know accords with the form of the surface. And by what is said above it appears, that there can be but one such arrangement; as any departure from it will alter the attraction at the surface, and therefore deprive the surface of its character of being one of equilibrium. This, therefore, viz. the fluid law, is the only possible law of arrangement of the interior mass, if we know that the surface is a spheroid of equilibrium.

PROP. To find the potential of the earth for an external point, on the hypothesis of the arrangement of the mass being according to the fluid law.

85. By putting a = a in the formula for the potential of the earth given in Art. 75, it becomes, for an external point, bearing in mind Art. 76,

for an external point, E being the mass.

86. Professor Stokes has obtained this formula in a somewhat different manner, which we introduce below on account of its elegance. He also deduces the results of Art. 79 regarding the centre and axis of the earth. But his investigation, in neither case, is more general than Laplace's theory here developed. For he assumes, not only that the mass is arranged in concentric strata all nearly spherical (see the course of the following investigation); but that the surface is one of equilibrium and also spheroidal, assumptions which, it will be seen from what goes before (see Art. 78), involve the whole fluid hypothesis, and need proof such as we have given above. The following is taken from his demonstration in the Cambridge Philosophical Transactions for 1849.

87. Let V be the potential of the mass. Then because the surface is a surface of equilibrium, (see Art. 75),

By Art. 24 we have, for an external point,

Let V be expanded in a series of Laplace's Functions,

Then since the above equation is linear with respect to V, and a series of Laplace's Functions cannot equal zero unless the Functions are separately zero (see Art. 35), we have, by substituting the above series for V and remembering the condition given by Laplace's Equation,

r

d2.r V
dra

i (i + 1) V¿ = 0.

Multiply by and integrate;

·+iri (rV) = const. (2i+1) Z; suppose.

=

where W, and Z are independent of r. The complete value of V becomes

Now evidently vanishes, from its very definition, when r is infinite. Hence Z1 =0, Z1 = 0, Z,=0...

If the earth's strata were exactly spherical instead of being nearly so, as shown in Art. 74, this expression would be reduced to its first term. Hence in our case W1, W,... must be all small quantities of the first and higher orders,

Substitute this in the equation of equilibrium and put

Equate the sums of Laplace's Functions of the same order

to zero;

.. W const.-} w2a3, W1=

=

= 0,

W1 = (W1e - † w3a3) a2 (} − μ3), W ̧=0, W1=0...

2

a2

* Professor Stokes's words are simply "If the surface were spherical"; but this is not sufficient. The surface of a body may be spherical and yet its mass so arranged that its potential on an external point will not be the mass divided by the distance from the centre.