[173 
63 
value 
3rences 
>o, the 
l\ is 
inde- 
is, and 
is, are 
ipsoids 
h = 0, 
173] on laplace’s method for the attraction of ellipsoids. 
where Ui is of the degree i in a, /3, y, and of the degree 2i in a, b, c. We have 
“ T + «» W + = + * + <*** (“ f + W § + 07 f) 
- (4i + 1 )(a 2 + b 2 + c 2 )“ 2l “t (a 2 a + b 2 /3 + c 2 y) Qi 
cl XTi cL Ui - cl U$ i / r\ \ tj- 
“ * +/3 ^ + ^ w +4(a + /3+7)£7i 
= (a* + ¥ + c 2 ) -2i— i (*§ + ^ + 7 s + i (« + 8 + 7) ft- 
Hence, putting in like manner 
0' 4- 1) (2i + 5) Qi+i = - (2i + f ) |(a 2 + 6 2 + c 2 ) (aa + &£ fr + c 7 
— (4% + 1) (a 2 a + 6 2 /3 + c 2, y) Q;j 
-(2i+l)( (t 2 + 6 2 + c 2 )(a 2 ^ i + / 3 2 ^ + 7 »^ i + l( £ < + ) 3 + 7)e i ), 
(* +1) (2» + 5) ft +1 = - (a* + 6 s + (?) |(2i + |) (aa ^ ^ ^ + 07 ^j-') 
+ (2i +1) («" tj + /3* ^ + 7* ^7 + è (« + fi + 7) «<)} + (» + f) (« +1) (<*» + W + 0=7) Qi, 
K:. 
d(3 1 dy 
from which the functions Qi may be calculated successively. 
We may, it is clear, write 
n.-^L I 
1 3 ’ 2 < 1.2.3..»5.7..2t + 3' 
and we shall then have 
(<** + V + o’) + < U + 8 ) (“In + b P W + in) 
+ <« + *)(*7§‘+*f ^f) 
where 
I proceed to show that 
+ (2i + 1) (a 4- /3 + 7) Ki = 0, 
1 
K 0 = 
V(a 2 + b 2 + c 2 ) ’ 
T _ ( d? Q d? d? y 
Ki= { a ds +/3 ® +7 ® 
V(a 2 + ò 2 + c 2 ) '