[173- 
173] 
ON LAPLACES METHOD FOR THE ATTRACTION OF ELLIPSOIDS. 
55 
2L^R’ 
the 
= 0. 
shes,. 
the 
3gral 
I write this equation in the form 
+mb '- + nc ° + 
+ < m - z ){-Kf-D + s 
fdV 
\dfc dkj ' n V dc 2 dc ~) dn 
'dF x dV F 
m\db 2 db j 
dF) 
dm) 
= 0; 
and I remark that this equation may be broken up into two equations, each of which 
separately is satisfied; these equations are 
. (dV dF\ , dF 
- k [dk-Tk) +( - r - F >- l dT- 
dF dF 
m 7 — — n -7— 
dm dn 
fdF dV .\ 7 fdF dV n \ fdF . dV F\ A 
+ a[^--h — -A) + b(- ir -±-^-B) +0(^-1 — -C) = 0, 
v da 
and 
/0 79 dZ' 1 
da 
dF_dF_dF 
dl dm dn 
+ 
a fdF 
1 dV . . „ 
ix-4 +- 
k d& 
5 /dF 
dc, 
dc 
l \da 2 da "V ' m\db 2 db ) ' n\dc 2 dc 
FZ-b^+U^-F-I-c' 
0. 
It may be added that the functions under the integral signs, and consequently 
the integrals, are all of them homogeneous of the degree zero in l, m, n, k. The 
thing to be verified is that the foregoing two equations are satisfied by the functions 
under the integral signs, independently of the integrations, in fact by the values 
f A _ £ FR B _ v q _ 
v= 
p (af + by + c£) V-R 
F = 2 ' 
We find by differentiating these values, and after a few obvious reductions, 
F \ , / -1 \ 
dV _ it(*J R . 
da ? V F F */Rj 
-t- la 
¿*JR)' 
dV y (FR I 
~di =ai (t? + — 
1 3 
I a *( - 1 
F sjR) + V2X sJR) + * V 2Z 3 2ZV^V ’ 
I