28 
ON GAUSS METHOD FOR THE ATTRACTION OF ELLIPSOIDS. 
[164 
and thence, effecting the integration, 
( A 2 + 6$ (B 2 t y\j - t iVj ¿iw j o tv)- yjo- t 0) 2 (C 2 + 6J 1 
where X t refers to the ellipsoid whose semi-axes are (A 2 +0 / )*, (B 2 + 0)^, (C 2 + 0j*. 
In the case where 0 t = oo, we have 
X, 
(A*+0 / )ì(B 2 + 0,)ì(C* + 0,)ì 
and consequently 
which is the expression for the attraction, in the direction opposite to that of the axis 
of x, of the ellipsoid (A, B, C) upon an interior point, the coordinates of which are 
(a, b, c). 
In the case of an exterior point, let A n B n C / be the semi-axes of the confocal 
ellipsoid passing through the attracted point; so that putting A / =\f(A 2 +rj), B / = \f(B 2 +rj), 
C / = \J(C 2 + rj), we have 
a 2 b 2 c 2 
A 2 + rj B 2 + 7] C 2 + 7] 
the attraction is equal to x attraction of the ellipsoid which passes through the 
point, i.e. 
or, putting 0 — 7] instead of 0, 
which is the expression for the attraction upon an exterior point. The formulae coincide, 
as they ought to do, in the case of a point upon the surface. 
2, Stone Buildings, 9th April, 1855.