75] 
ON THE ATTRACTION OF AN ELLIPSOID. 
433 
For 03 = 0, the cone coincides with the circumscribed cone; as to increases, the 
aperture of the cone gradually diminishes, until for a certain value, co = il, the cone 
reduces itself to a straight line (the normal of the confocal ellipsoid through the 
attracted point). It is easily seen that il 2 is the positive root of the equation 
Pa? T)?lr v?c 2 
W+ m + D? + m8 + DJ-\- n8 = ’ 
. . JcS 
a different form of which may be obtained by writing fl 2 = ^, f: being then determined 
by means of the equation 
la 2 mb 2 nc 2 
k + U; k + + k+ n% ’ 
that is, 
k 
m 
+ 1 • 
are the semiaxes of the confocal ellipsoid through the attracted point. 
In the case where co remains indeterminate, it is obvious that the cone intersects 
the ellipsoid in the curve in which the ellipsoid is intersected by a certain hyperboloid 
of revolution of two sheets, having the attracted point for a focus, and the plane of 
contact of the ellipsoid with the circumscribed cone (that is the polar plane of the 
attracted point) for the corresponding directrix plane: also the excentricity of the hyperboloid 
is — \J(J?a* + w?b 2 + n 2 c 2 ), which suffices for its complete determination. For = 0, the 
hyperboloid reduces itself to the plane of contact of the ellipsoid with the circum 
scribed cone, and for co = il, the hyperboloid and the ellipsoid have a double contact, 
viz. at the points where the ellipsoid is intersected by the normal to the confocal 
ellipsoid through the attracted point. 
If oo remains constant while k is supposed to vary, that is, if the ellipsoid vary in 
magnitude (the position and proportion of its axes remaining unaltered), the locus of 
the intersection of the cone and the ellipsoid is a surface of the fourth order defined 
by the equation 
('!x 2 + my 2 + nz~ — lax — mby — ncz'f = t» 2 (¿tr + y 2 + z 2 ), 
and consisting of an exterior and an interior sheet meeting at the attracted point, 
which is a conical point on the surface, viz. a point where the tangent plane is 
replaced by a tangent cone. The general form of this surface is easily seen from the 
figure, in which the ellipsoid has been replaced by a sphere, and the surface in question 
c. 
55