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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)
a? de m a'd(a) P (a'&') da
da a За 5a
5 da' 6 a
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
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
Cor. 1. By putting a =a in the first equation of this last Article, we have the following equation, which we shall find
Si pe came'
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=
Now $la) ->s pa'da = 3Qt-con ga + sin qa).
Multiply by a' and differentiate;
a' x'da' = 0.
da2 The solution of this is
X + Cqo sin (qa + B) = 0, C and B being independent of a;
a'x'da' Cqa cos (ga +B) + Csin (qa+B);
sin (qa +
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
tan qa — qa
tan qa +
Hence, if we substitute for • (a)
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
tan qa — qa
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
qa 3Q Also • (a)=
q*a* — 3q*a+
tan qa) (gʻa–3+
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
Fa" + ...=1 - Ha" + ..., H positive. φ (α)
n+ 3 D
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
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
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 +
.:. tan qa=
3n-q*a?" If we take the mean density double of the density at the surface (see Art. 58), then
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.