518 ON THE ATTRACTION OF ELLIPSOIDS (jACOBl’S METHOD). [89 
Multiplying this by X 1 
aX + aï Ÿ + a"Z, the terms containing XY, XZ vanish after the 
X 2 . 
integration, and we need only consider the term ( ^- X 2 , or what is the same — 
CLCti QjOL\ yj i 'll) 
Hence 
A = 
(u — q) (u — r)) a 2 a 
{u + q)(u- r)j aa x (/+ u)J X 
X 2 dlS 
+ 
Y 2 
+ 
Z 2 
f + u g Au h A u 
The value of the corresponding function A (that is, the attraction of the exterior 
ellipsoid upon P) is 
^_ a f X 2 dS 
~/AuJ X 2 + F 2 ~ I 2 ’ 
f A u g A u h A u 
the limits being the same, whence 
or we have 
A -r-A = 
(u — q)(u — r) 
(u — q)(u — r) 
fAu aa-L _ V(u Ag) Y( u + h) 
/A ü aa x \J(u + g) \J(u A h) ’ 
a __VO + ff) Y(uAh)j = V<M Ah) *J(u +/) = V(u +/) V(a A g) f1 
\/( u A g) Y( u A h) ’ hj(u + h) *J(u +ÿ*) ’ V('^+/) V( M Y g) ’ 
formulae which constitute in fact Ivory’s theorem. 
Let K, K denote the attractions in the directions of the normals at P, P, we have 
K=^ixM K=fxM 
ffhj J 
K = i AK; 
and it is important to remark that this is true not only for the entire ellipsoids; 
but if denote the attractions of the cones standing on the portions M, N of 
the surface of the interior ellipsoid, and JW, J2, the attractions of the portions of the 
exterior ellipsoid bounded by the tangent plane at P, and the portions M, N of the 
surface of the exterior ellipsoid, then 
where obviously 
gui 
B = ^B, 
gcti 
K=3-M K = 3Ai№] 
this theorem is so far as I am aware new.