AND THE FIGURE OF THE EARTH. 
93 
2nd. Let n = 2, and we get 
T 2 ~ t) r2 f- 1 d P' (f*' 2 ~y) \f* + (V a - <p a) ( - a>' 2 + p' 4 ( y - A' ) ) 
+ {a<p' a — a p' a) ^y- + o /^ yrffl' + (F«-Fa)( - s' ^' 2 ) n «' . d .d d 
which gives 
4 C“ 2 - f) 4 0 (- T+k (i - A )) + a ?' a .l £2 +L0 4 rfa ' ~ I f 
a . s 
+ f 5 f o d a' | — 4 77 g>(y 2 — y) r 2 (same function of a, s l and A x ). 
3rd. Let n — 3, and we have 
1225 
64 
(f * 4 - f^ + l) 2 ’ r f r 4 X. d f i '(f‘' 4 -yf‘'* + ^){X''¥' ia ' + (?- v) 
(- (£ + A')) + ( £ - £) 4 +/* W ' + 4 - ?) <- £ V 2 > 
+ / a ^' rfa '}> 
or 
* / 6 2 3 \ 4 J 1 ( ^ ■ A \ , $' a g2 e Fa 1 ,] 
4' - + 35/ 4 TT^j- a2 9 l 2 + A / + X 18 ~ -9 “flT + “9 ./o 4" 3 (la \ 
— (y 4 — y ^ 4 Tr^r 4 (same function of a, s 1 and Aj). 
19. To the sum of these must be added V for the inner spheroid, for which we shall 
have to find the value of V generally, when the attracted point is without the body. 
The general term of the series for V is 
j d ft' P» _ 2 £ d n . (p r 1 . d r 1 + t d 2 J^ r' n F r' . d r' -j- r' n . II r' . d r' , 
and 
f o r' n <pr' dr { = d n . <p d . d d -j- a n + 1 <p a ( — s p? + pfi (s 2 — A) -J- y g2 ) 
-f- a n + 2 <p' a s -~- 
and similar equations hold for the other functions. 
1st. Let n — 2, then we have 
2 
^f-\dpJ^f Q d 2 <p d dd + a 3 <p a( — s pJ 2 + pJ* (2 s 2 _ A)) -f cfi<p' a ^ 
+ J‘“d 2 F d d d + a 3 F a ( - s pJ 2 ) +f* d 2 n d d d | , 
which is equal to 
{X d 2 <pd dd + a? p a (- ~ + y (2 s 2 - A)) + a*<p' a ^ + y^ d 2 F d . dd 
— y a 3 F a . s + \ Sa a ' 2 n d d d j> . 
