89] 
ON THE ATTRACTION OF ELLIPSOIDS (jACOBl’S METHOD). 
515 
X X _ 1 / a a \ 
V(f + u) *j\(f + u) p w (f+ u) \/(f + u)) ’ 
Y Y _ 1 / b _ b \ 
V(/ + u) >J{g + u)~ p w(g + U) V(<7 + u)j ’ 
Z Z 1 / c c \ 
\/(h + u) y/(h + u) p\\/(h+u) \/(Ji + u)) ^ 
which are in fact the equations which express that Q and Q are corresponding points. 
It is proper to remark that supposing, as we are at liberty to do, that P, P are 
situate in corresponding octants of the two ellipsoids, then if the curve of contact of 
the circumscribed cone having P for its vertex divide the surface of the interior 
ellipsoid into two parts M, N, of which the former lies contiguous to P: also if the 
curve of intersection of the tangent plane at P divide the surface of the exterior 
ellipsoid into two parts M, N, of which M lies contiguous to the point P; then the 
different points of M, M correspond to each other, as do also the different points of 
N, N. 
we see that 
and similarly 
We may now pass to the integral calculus problem. The Attraction parallel to 
the axis of x is 
^ _ f xdx dy dz 
J (x 2 + y 2 + Z 2 )- 
the limits of the integration being given by 
or putting 
o +_«): + (y + &) 2 + o + c) 2 =1 
f +u g + u h + il 
x = rX, y = rY, z = rZ, 
where X, Y, Z have the same signification, as before, we have 
dxdydz — PdrdS, 
and then A = jXdrdS = JpXdS, 
where p has the same signification as before : it will be convenient to leave the formula 
in this form, rather than to take at once the difference of the two values of p, but 
of course the integration is as in the ordinary methods to be performed so as to 
extend to the whole volume of the ellipsoid. The expression dS denotes the differen- 
65—2