MR. HOPKINS’S RESEARCHES IN PHYSICAL GEOLOGY. 
203 
Putting - % — r cos 6, 
y = r sin $sin <p, 
x = r sin 6 cos <p, 
and arranging - the results in terms which severally satisfy the equation of Laplace’s 
coefficients, we obtain 
c = v + |V + (j + ^- a2 (1 _ f ), 
+ if tb (1 — cos 2 A) a 2 (1 — y 2 ) cos 2 <p + - 3 - — 3 sin 2 A . a 2 y \/l — y 2 cos <p, 
where y = cos 6, and a is a mean value of r for any surface of equal density. Or 
transposing the constant term, and putting a 2 F (6 <p) for the sum of the other small 
terms, 
Const. = V -j- a 2 F (0 <p). 
Putting r = a (1 + a y), where a is a small coefficient, and y here denotes a function 
of 6 and <p, we have 
V = — (1 - ay) f a s ' f a' 2 + a/7 (~ Y/ + .. . + t 4— n Y/ + &c.) } da' 
a v J’J 0 s I 1 da' \ 3 a 1 n T (2 i + 1) a* * 1 / J 
+ 4 *f a 1 f' { + a da' ( 3 Y i' + • • ‘ + 
fi — 2 A i 
/ + &c.) d , 
(2 z + 1 ) a' 
a being so taken that Y 0 = 0, and = value of a at the earth’s external surface. 
Putting y — Y L + Y 2 +, &c., and substituting the resulting value of V in the above 
equation, we shall have = 0, except i — 2 ; and we shall have for the determina- 
tion of Y 2 the equation 
° = — K ./„ ? dZ\I? Y t) da - — aY J a e a2da 
+ 4 «■ “X”' s' X (t Y ' 2 ) " + « 2 F 0> ?)• 
Y' 2 will differ from Y 2 only by small quantities, and therefore in determining the 
influence of the smaller terms in F (d <p) (those which involve ^ on the value of Y, 
we may consider Y 2 = Y 2 , and we then obtain 
“ ^2 — 4 7T fa 
a 2 F (0 <f>) 
Hence 
— a ( 1 + a Y 2 ) = a< 
I w / >a , , 2 7 . 4 7r r a , /4 , 
— / p' a 1 da' 0- / e e 4 « a' 
a J 0 cr «/ 0 
a 2 F (S <p) 
1 + 
4 7T Pa 
a 
have 
-J‘ a g' a n da' — g' a ' 4 d a' 
Ifg' denote the force of gravity at the distance a from the centre of the earth, we 
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