[173 
173] on laplace’s method for the attraction of ellipsoids. 
65 
b 2 + c 2 ) 
1 
FTcO 
the third line of which is 
d 2 
d 2 
d 2 \ 
1 O' + 1) ( a 2 ^ + /3- + 7 2 — 4 ( a- ^ ¿p + 7 2 ^ ¿+i > 
dò 2 
dc 2 ) 
d 
d 
d 
da. 
d(3 ' dy 
by a foregoing equation, and the assumed equation of difference thus leads to 
d , _ d d^ 
[ i—f - bp u +C7 -T- 
da db dcj 
+ (« + 8)(^ + ^ + yg^ 
(a 2 + 6 2 + c 2 ) Z i+2 + (4 i +7)(aa‘*-+b(3^ + cy £) K i+l 
+ ( 2i+ 3) ( a + /3 +7 ) K{ + 1 — 0, 
which is the assumed equation, writing i+ 1 instead of i. The equation, if true for i, 
is therefore true for ¿ + 1, and it is easily seen to be true for ¿ = 0; hence it is true 
generally, or the value 
K; = 
da 2 
d 2 
d 2 \ 
^ db 2 + ^ deV J{a 2 + b 2 + c 2 ) 
satisfies the equation obtained by Laplace’s method, and gives, moreover, the proper 
value for K 0 . We have thus the value of ; and remembering that 
V — \/(a + 6) (/3 + 6) (7 + 6) v, 
and observing that the symbol A may be replaced by 
A= ^ a+ ^£ + ^ +e) m +(r)+0) 3?’ 
the value pf V is 
7 - T V( “ + 0Hf3 + B) (7 + 6) 8i « (2C.2...J.7...2,- + 3 A ‘VOMTO)) ; 
which is in fact the value which I have found by a much more simple method in 
the Cambridge Mathematical Journal, t. III. p. 69 [2]. 
C. III. 
9