173] on laplace’s method for the attraction of ellipsoids. 57 
Now considering V, F, A, B, G as standing for the definite integrals, we may 
replace A, B, G by the differential coefficients of V, and retaining for shortness F to 
stand for 
dV .dV dV 
Cl da db C dc ’ 
the first equation becomes 
j fdV dF\ Tr 
k ^~dk~dk) + V ' 
, dF 
dl 
dF dF 
+ a 
(il. 
\da 
and the second equation becomes 
a 
+ -Ï 
— m -j n 
am an 
j dV" 
dF 
2 da , 
) + b{ 
M + 
dF 
dF 
dm 
dn 
dF 
, dV\ 
b i 
sla 
2 da) 
8 
m 1 
, (dF . dV\ (dF ,dV\ n 
db J ^ \dc 
■ + l-îr +- 
dV 
dc 
= 0. 
T / t> 
if we put as before V = , the preceding values of the differential coefficients 
2I\JR kl 
of V give ^ = , or as we may write it F= — 2V + W, where W = 
-L h y L\JR 
I put then for the moment 
y = FJR yy_ kl 
L* ’ L*JR’ 
F=-2V+ W. 
It should be remarked that there is nothing in what has preceded which tends 
to show that these values must satisfy the differential equations. The definite integrals 
must, of course, satisfy as before the equations, but it does not follow that the equations 
are satisfied by the elements separately. And in fact only the first equation is so 
satisfied ; the second equation is not satisfied. To verify this I form the differential 
coefficients 
dW _ 
lt\ 
( k 
da 
\LbJR 
dW 
I 
dk 
L»JR 
dW 
( k 
dl 
[ljr 
kP \ , j ( \ 
LRFJr) + {WRJ ’ 
kl 
2 R*JR’ 
kP \ (Ski kP 
LR\/R) + s \2LWR 2RR^R) 2R\/R ' 
c. III. 
8