555 
figure of the earth . 
and therefore z will not exceed 4. And as even powers only 
of f J are found in R, there will be no terms as Z' in which 
i is odd, and therefore it is not necessary to consider any but 
the even values of i. Now for the interior spheroids, when 
i = o, 
R‘ +3 = R 3 = a 3 { 1 + 3 e.i-v.” + 3 e-.T=^l 2 -(3 A : 
and since by Laplace’s formula (liv. 3. n°. 16), Z' , when it 
does not depend on is a multiple of f/ z — 
2 . 2- 
1 1— -2 
2.21 I 
2.2 I . I 2 .2 3 
jj/ &c. we must resolve R 3 into mul- 
2.4.ZZ 1.22 3 
6 
pies of p' 1 — - f * /2 + -A- > of f / 2 — A- , and constants. Thus 
7 35 3 
we have R 3 = 
a »( 1+!e + f ._ii)_ 4 s( s{ + iL e 4^( (l '*-L) + ^ e . + 
whence B 0 = a 3 (1 + 2 e 4 e* — -j- A) , Z ( 1. 
Similarly when z = 2 , R 5 = 
a constant — a’(s e + ^ «“+ f A )(d~ 7) + a multiple of ((.'*— - ^ ) 
whence B z = — a’ (5 e + ^-e‘+ A A), Z <2) = j. 
And when z == 4 , R 7 = 
a constant + a multiple of y + a 7 <? a + 7 A), — - f/ 2 -f- : 
2 7 35 / 
whence B=a 7 ( — e 2 +7^), Z VT ' = u. 
(4) 
6 2 [ 3 
7 *"+ U- 
(9.) Collecting then the different terms of 
x n d B , :( i ) ( A ) 
— TT+T f p -r-f.fQ. Z' , which by (7) has no value 
(i + 3)r T aaJu'Jp 
except k = z, in which case^, Q 
(0 Z/ W__ 4£_ z (0 we 
• ^ 22 + I ^ , 
