742 
PROFESSOR CAYLEY ON PREPOTENTIALS. 
viz. the prepotential equation is satisfied by the value 
V=C dt l m T, 
where m -f- 1 is positive. 
107. We have consequently a value of g corresponding to the foregoing value of V 
and this value is 
____ TQs + g) ( c ^dW\ 
S 27rKT(2 + l)\ de ) e=0 ’ 
where, writing X for the positive root of 
vve have 
1—JL- , 
A+/ 2 
--= 0 , 
x + A 2 x 
W=jJW 
we thence obtain 
eL=fV-?~( 
1 
de J* t \ 
v, £ p/ 2 ^ P 
-l\ 
\ *+/ 2 
X + /i 2 X/ ' 
multiplying by e 2?+1 , and substituting for ^ 
we have 
I Im7T'” + 0^T 2+ ^/ 
^ «,*) I 1 ' ' X + h2 '>' i ’ 
PA 2 ) 2+ X 2 j 
2g2?+2 
X^+2" 
( X <1 
W+pf"- + ^+h^) 
where the second term, although containing the evanescent factor 
1- 
X -) -f~ X -(- Id’ X 
is for the present retained. 
108. I attend to the second term. 
x i £ 2 
1°, Suppose j 2 ...-}-^>l, then as e diminishes and becomes =0, X does not become 
zero, but it becomes the positive root of the equation 
