PROP.

To obtain an equation for calculating the ellipticity of the strata.

2

77. Substitute (-) for Y, and e' (μ3) for Y' in the equation of the last Proposition but one, and we have, after dividing by - μ3,

Divide both sides by a2, and differentiate with respect to a; then multiply by ao, and differentiate again, and divide by the

coefficient of

d2e

da2;

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

This may

d

da

COR. 1. By putting a = a in the first equation of this last Article, we have the following equation, which we shall find of use;

PROP. To find an expression for the ellipticity of the strata, with the law of density deduced in the last Proposition but one.

78. In the equation of last Article put p

a

α

Now $(a) =3[® p ́a”da' = 3Q {— cos qa + sin qa).

{

1

de = {2 cos ga - sin ga } = - Zi 4 (a),

dp

Also

da

la

qa

a2

6

3a2

+ g2. $ (a) ɛ = — a $ (a) . ɛ.

a

α'

a' a'x'da'2;

To integrate this put 4 (a) e = — [ ̃ ̈à' [“ a'œ'da" ;

a

a

a

a

0

0

a'x'da'2

-

1

0

"a'x' da" + = ["a'x'da' ;

2

α

2

0

a

α

0

0

Multiply by a2 and differentiate;

Divide by a and differentiate, and then divide by a;

x+Cq2 sin (qa + B) = 0,

C and B being independent of a;

In our case B = 0, otherwise the ellipticity at the centre would be infinite, as is easily seen by expanding & in powers of a.

Hence, if we substitute for ø (a)

This gives the law of decrease in the ellipticity of the strata in passing down from the surface to the centre.

By Art. 77, Cor.

1

a

d

a

{ a°p(a) (e — '{ m) = [ "p' ~; (a'"e') da' = Q f* sin qa' d

da'

α

da

(a'') da'

a'"'e' (sin qa' - qa' cos qa') da'} by parts.

Substituting for e' from the expression already found for ɛ, integrating and reducing, the integral in this expression

€ (tan ga-ga) sin qa{6qa2

3

=

4

3

tan qa+
qa

tanga) (6g'a2-15 — qa3 — 15qa)

tan qa

When this is calculated for the surface, we shall be able to find the ellipticity of any stratum we like by the ratio of ɛ to e already found above.

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

Ea" +

79. We assume that the density of the Earth increases from the surface to the centre. Let then where E is positive: and ɛ = A + BaTM+.... Then

p = D

Put these in the differential equation in ɛ of Art. 77; it gives

Neither m nor B can equal zero, because then the second term of 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 & increases towards the surface.

In thus increasing, suppose it attains a maximum, and then

dε

decreases. At this point = 0; and the equation of Art. 77,

already used, gives

d2ε

da2

da

This corresponds to a minimum. Hence & 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

PROP. To calculate the numerical value of the ellipticity of the surface in the case of the Earth.

80. In order to do this it is necessary to find the values of qa and tan qa at the surface. Let n be the ratio of the density of the surface to the mean density of the Earth. Now the mean density

If we take the mean density double of the density at the surface (see Art. 58), then

which is satisfied by qa= 2.4576 = 140° 45'. Then

tan qa=-0.812, z=4'0266, q'a2=6'0398, q'a2÷z=1·5.