MR. HOPKINS’S RESEARCHES IN PHYSICAL GEOLOGY. 
47 
a series likewise of Laplace’s coefficients, independent of r. Then shall we have* 
U = R Z z~ mt , 
n n n ? 
v ' m 
where R re — M B r n + cos a m cos co) sin 2K + 1 a .da, M re being an arbitrary con- 
stant. Hence if we denote the value of the definite integral by N tt , we shall have 
w' = {M 0 N 0 Z 0 + . . . . + M re N yz n + . . . 
When r = a l3 %£ must = £ ; and therefore if the corresponding value of N re be de- 
noted by (NJ, we shall have 
M 0 (N 0 ) Z 0 -f . . . . + M b (NJ a n 1 Z n -\- ... . = Z 0 + .... + Z n + ... ., 
which gives for the determination of M n , 
= 
<(NJ 
whence 
.1 f Np y , . r y , 1 — m t 
1 ~ 1 (N 0 ) Z o + * * • * + (NJ « * Z n + * * « « j £ • 
According to our assumptions for the value of £ (Art. 4.), we must have £ equal to 
the temperature (wj in a sphere, whose radius = r', at a distance r' — a x from its 
surface. Therefore^ 
if i 
£ = C.^|l +b(r'- ai ) ja ^ 
or omitting 
£ = C 
-p— £ r ' 2 , 
C being the temperature at the centre. Or since 
r' = a x (1 -j- s 1 cos 2 6), 
a? 5T 2 
y = Cs 1 cos 2 6 
omitting smaller terms. 
It is here supposed that the temperature of the surface of the earth is constant and 
equal to zero. If we take into account the variation of external temperature in pass- 
ing from the pole to the equator, we have only to consider s 1 as the ellipticity of that 
surface of equal temperature which touches the earth’s surface at the equator, the 
temperature there being also assumed as zero. Then z x will be rather greater than 
the ellipticity of the earth. 
Putting the expression for ^ under the form of Laplace’s coefficients, we have 
= { If" 1 + c s i ( cos2 6 ~ j) } 2 
* Theorie de la Chaleur, Art. 173. 
f Ibid. Art. 171. 
