MR. HOPKINS’S RESEARCHES IN PHYSICAL GEOLOGY. 
49 
r = a, — x. 
Suppose 
r 
= °i 0 “ 0 ’ 
oc • 
where — is small compared with unity. Then we have approximately 
«i 
i l . r x ( x\ 
+aj’ 
(N 0 ) “ V ^ V’ 
£1 
7T : 
N 2 r 2 7 r 2 — 9 ^ 
(Nj ' «? = 1 3 % 
7T 3 - 9 
COS 2 4. 
and therefore 
( ^\ 2 Qg f ^ 
— j and the products of — and g l5 except in the last term, i 
-) 
ay 
X 
which we may substitute for — its approximate value, G. We have thus (putting 
a x — r for x), 
G =>— g) «.««•<! 
7T 2 — 9 
whence 
i - 
X = (1-6)1 1 + 
G Sj cos 2 (3 
1 - G 
and 
r = (l -G)«, [l + {l + (l - ^ 3 -^) G] si cos 2 «] 
the aproximate equation to the isothermal surface. 
Hence it appears that the ellipticities of the isothermal surfaces within the earth 
X 1 
are greater than that of the surface. Thus if G = — = we have 
ellipticity = (1 + '07) ^ nearly. 
It will also be observed that it increases with G, i. e. with the depth. A further ap- 
proximation gives a somewhat slower rate of increase, but the inference from the 
above formula is sufficient for our purpose. 
§. Ellipticity of any Surface of equal density within the Earth. 
6. If we assume the density of the earth (§) at any distance (a) from its centre to 
be such that 
= A 
sin q' a 
a 
MDCCCXLII. 
H 
