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

77. Substitute € ($ - r*) for Y, and e' (3-) for Y, in the equation of the last Proposition but one, and we have, after dividing by }-r?, $ (a)

d

a? de m a'd(a) P (a'&') da

da a За 5a

da

5 da' 6 a

a

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E

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

dae coefficient of

da"

i

{1

de 6pa de

pas | 6+

0. da? $(a) da

$ (a)s a This may be put into another form. Multiply by • (a), then d de d

6
+
(a)
da

da
{ } $

da

;

dp

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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;

Si pe came'
) da' = { a$(a) (e-)

m.

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=

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Now $la) ->s pa'da = 3Qt-con ga + sin qa).

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.. a?

d'af'a'a'da

Multiply by a' and differentiate;

dac
+ 2ax
3ax +

a' x'da' = 0.
da
Divide by a and differentiate, and then divide by a;

d'x

+ gex=0.

da2 The solution of this is

X + Cqo sin (qa + B) = 0, C and B being independent of a;

a

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a'x'da' Cqa cos (ga +B) + Csin (qa+B);
1

20
i$ (a) == Ca sin B a cos (qa + B)

9
20

С
sin (qa + B) + a cos (qa +B)

sin (qa +
2
3

3
sin (qa+B) +

cos (qa+B qa

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

of a.

tan qa qa

3

tan qa +

qa

Hence, if we substitute for (a)

3

3

tan qa+
Cq
q*a

qa
3Q
And e being the ellipticity of the surface,

3
tan
qa - qa q*a*
3

3
1

qa 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. 5

d a?

d '

é da

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tan qa qa

tan qat

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{ag'a– 15–

d'a' – 159a}

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

(tan qa - qa) sin qa
9
3

3
tan qa+
d'a

qa 3Q Also • (a)=

sin

qa qa

1 qo

tan qa

1

m Hence

26

q*a* — 3q*a+

3qoa

+11

1

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qa

tan qa) (gʻa–3+

5 (1

tan qa/

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

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

79. 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 ε = A + Ba" +..... Then

E 1

Fa" + ...=1 - Ha" + ..., H positive. φ (α)

n+ 3 D

pa®

n

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

B (m? + 5m) am-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 n=m and B (m* +5m) = 6 AH. Hence B must be positive. And therefore near the centre ε increases towards the surface.

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

da

{

de decreases. At this point = 0; and the equation of Art. 77, already used, gives de

pa® 6€ da? $ (a) a

a positive quantity. This corresponds to a minimum. Hence ε 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 pa'. This appears :: $ (a) =3

dp'

da'
do'
and is negative by hypothesis.

da

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

3Q 4πρ: 4ma'da'

cos qa +

sin 9

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.:. tan qa=

3nqa

3n-q*a?" If we take the mean density double of the density at the surface (see Art. 58), then

3qa

3 – 2q*a?? which is satisfied by qa = 2.4576 = 140° 45'. Then

tan qa=-0.812, 2=4:0266, gʻa? =6•0398, qa? - 2=1.5.

tan qa

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