173] 
and 
ON LAPLACES METHOD FOR THE ATTRACTION OF ELLIPSOIDS. 
59 
n (dV dV 7F\ , , 7 „ oN / — 1 \ , 7 /2VX 
-{dl + dm + ^)= (a ' + b ~ + c) {L71V + (®f+*”/ + <« (“7^ + 
2P 
fdW dW dW\ . „ 7 „ -Jcl \ . j ~(-k k. 
( dl + dm + +6 + c ^ V222 V-B/ + ^ + bv + C ®\In/R + LR 
\ D ‘ XyX, 
AX 2 \ 
XX^xJ 
Ski 
+ ■ 
3/^X X 3 
X 3 XyX 
AX 3 
2D\JR 2DR XX ’ 
and the value of the left-hand side of the second equation is therefore, 
( a Z + hr) + cO ^2^2 + 
XX SP \ SI XX X 3 
2D 2DsJR 
+ 
3AX 
X/ 3 
2XyX ' 2X 2 VX 2X 2 XVX’ 
which is not equal to zero, or the second equation is not satisfied. 
I consider again V, j 
by means of -the equations 
I consider again V, F as denoting the definite integrals, and I eliminate jrr, 
CLiC CL/b 
7 dV dV dV 7 dV A 
l —jj + VYl j (- 71 j— k -jy- — 0, 
dl dm dn dk 
. dF dF dF 7 dF . 
I —jf + m -j—b n -j—b k -jj- — 0. 
dl dm dn dk 
The first equation thus becomes 
7 dV dV dV 
l-jT + m-n—h«y- 
dl dm dn 
2 
dF dF dF\ 
~dl +m dFn + n dh) 
+ V-F 
dV 7 dV 7 
ci j—b b -jj- + 0 
da db 
dV\ 
dc J 
+ 
dF 7 dF dF 
a -j—b b jj- + c j— 
da db dc 
0, 
and the second equation becomes 
, 7 0X 1 ( 7 dV dV 
(a 2 + b 2 + c 2 ) j c (l-jj+m—+n 
dV\_ 
dm ' ' v dn ) 
(a 2 + b 2 + c 2 ) 
7 dF dF dF\ 
l jj -b m -j—b n j 
di dm dn) 
a dV^_bd V + c eAF\ fa dF + b_ dF + c dF^ _ (dF ^ dF ^ dF 
+ 2 t t: + 
2 \l da m db n dc 
l da m db n dc J 
dl dm + dn) 
) = 0; 
and it will be remembered that in these equations 
F= 
dV_ dV_ dV 
Ct da db C dc 
The first equation (it is easy to perceive) shows merely that V is made up of terms 
separately homogeneous in a, b, c, and in l, m, n, and such that the degrees in the 
two sets respectively being k, A, then A — (k — 2). In fact V being a function of 
the form in question, if we attend only to the term the degrees of which are /c,y>A, 
8—2