734 
PROFESSOR CAYLEY ON PREPOTENTIALS. 
In the last-mentioned equation, let e gradually diminish and become =0 ; then for an 
exterior point, viz. if 
($“ C 2 G/ 2 C 2 
ja--- +^ 2 > 1 , the equation • • • +^jTfl = 1 
has (as is at once seen) a single positive root, and 0 becomes equal to the positive root 
a 2 c 2 
of this equation ; but for an interior point, ot j- 2 . . . +^<1, the equation just written 
down has no positive root, and 0 becomes =0, that is the positive root of the original 
equation continually diminishes with e, and for e=0 becomes ultimately =0; its value 
• • £ 2 / £ 2 \ 
for e small is in fact given by -q— ^1 —p . . . — ^ J . Also a ... c, e or any of them inde- 
finitely large, 0 is indefinitely large, =a 2 . . . -\-c 2 -\-e 2 . 
95. We have an interesting geometrical illustration in the case s+l=2; 0 is here 
determined by the equation 
_?!_ i J!_ _i_ e __!. 
# 2 ^2 
viz. 0 is the squared 2 -semiaxis of the ellipsoid, confocal with the conic p-{-p=l, which 
■ a? b 2 
passes through the point ( a , b, e). Taking e=0, the point in question, if p-\-p>l, is a 
point in the plane of xy, outside the ellipse, and we have through the point a proper 
a 2 b 2 
confocal ellipsoid, whose squared 2 -semiaxis does not vanish ; but if then the 
point is within the ellipse, and the only confocal ellipsoid through the point is the 
indefinitely thin ellipsoid, squared semiaxes (f 2 , g 2 , 0), which in fact coincides with the 
ellipse. 
96. The positive root 6 of the equation 
a 2 ^ 
has certain properties which connect themselves with the function 
0, . . 4+A 2 )-i. 
We have (the accents denoting differentiations in regard to 6) 
„ e 2 
r =0 
where 
V d l 2 a 
J da Q+f 2 U ’ daT~ J' 0+/ 2 ’ 
T/— C' | g 
(/ 2 + 0 ) 2 ‘ ' ’ ‘ ( A 2 + 0) 2 ‘ 
0 2 ’ 
and we have the like formulae for . . . — , 
ac ae 
We deduce 
