60 
on laplace’s method for the attraction of ellipsoids. [173 
— (a 4- /3 4- 7 + 30) © = 2( a + /^ + 7 + 30) V0, 
then, by the properties of homogeneous functions, F= — V, and the first equation is 
satisfied if only 
X 2Xk -(■ 1 4- k 4" ^k — k 2 = 0, 
i.e. if X (2k 4-1) = K 2 — § k — 1 = \ (k — 2) (2k 4-1) or X = \ (k — 2). Or, what is the same thing, 
we may say that the first equation shows that V is made up of terms the degrees 
of which in a, b, c and in l, m, n are — 2i — 1, and — i— | respectively. Attending 
henceforth only to the second equation, I write l =—— . , ^ ^ - • so 
J ^ k a + 6 k /3+0 k y + 6 
that a 4- 0, /3 4-6, 74-# denote the squared semiaxes of the ellipsoid. We have 
d _ (a 4- 6) 2 d „ 
Jl~ 1c da’ C ' 5 
and the equation becomes 
-(“ S + ii + C ’)( ^ + ^ Ta + <* + *> %+ (y + V f) 
+ (a‘ + & + <?)( (« + 0 ~ + (0 + 0) d -l+(y + e) 4T) 
+ (a(« + e) d £+b(0+e) f + c( 7+ «)f) 
+ ( (a+e),d £ + (/3+t>), % + (,y+e), %) =o - 
Put for shortness ® = (a 4- 6) (/3 4- 0) (7 4- 0), and write 
F=v\/®> 
(V® is to a constant factor pres the mass of the ellipsoid) then v, f are connected 
by the equation 
. _ dv , dv dv 
J U da db C dc’ 
and observing that 
(01 + i) “Sr + ( ^ + i) “A9“ + ( ' y + < ’ ) “3V =3e 27® = iV ®’ 
d/3 
dy 
(a+ 6) 
,dV® 
da 
+ (/3 4- 6) 
2 0V© 
¿/3 
+ (7 + <?) 
2 0V© 
dy