PROFESSOR CAYLEY ON PREPOTENTIALS. 
74; 
factor 
l 1 X+/ 2 "' a + A 2 a)’ 1S- °- 
2°. Suppose then as e diminishes to zero, A tends to become =0, but ^ 
x 2 z 2 e 2 
is finite and = 1 — — . . — — , whence is indefinitely large; and since - 
z 2 
(a + A 2 ) 5 
p -- w * ( A +/ 8 ) 2 
becomes =^... + 74 , which is finite, the denominator may be reduced to and the term 
therefore is 
x 2 
— 2 1 
■-STT. -ni.-y 
which, the other factor being finite, vanishes in virtue of the evanescent factor 
l _ & 
A +/ 2 A + A 2 A / 
Hence the second term always vanishes, and we have ( e being =0) 
de 
109. Considering first the case ^...+^>1, then as <? diminishes to zero, A does not 
become = 0 ; the integral contains no infinite element, and it consequently vanishes in 
virtue of the factor e 2q+2 . 
yz _ z z 
f 2 '" l 'h 2 
But if then introducing instead of t the new variable f, =fi, that is 
t=%, and writing for shortness, 
£ £ 
R=l- 
/2+ £ 
A 2 + 
the term becomes 
=Jrf|. 2 m(R-f)—g’(/*+| ...4*+' 
where, as regards the limits corresponding to t=cc , we have £= 0 , and corresponding to 
i—'k we have | the positive root of R— £=0. But e is indefinitely small ; except for 
indefinitely small values of |, we have 
£j, and (>+f. ■**+£)'=(/•• • *)- ; 
and if g be indefinitely small, then whether we take the accurate or the reduced 
