718 
PROFESSOR CAYLEY ON PREPOTENTIALS. 
y—f sin 0, and multiplying by 2 in order that the integral, instead of being taken from 
0 to 2sr, may be taken from 0 to w, becomes 
_o n, C (sin Q) s ~ l dQ 
J J(/ 2 -2x/cose + x 2 ) is+9 ' 
Write cos then sin \ 0 =*Jl—x, sin 0=2#T1 — x) 1 ; d 6 ——x~\ 1 —x) *dx\ 
cos 0= — 1 -j- 2# ; 0=0 gives x=l, 0=w gives #=0, and the integral is 
= 2-i fs f 1 aP~\l -af-'dx 
Jo {(f+x) 2 — 4x/a?} is+2 ’ 
_ 2 s-1 /* a#* -1 (\—oof s ~'dac 
~(f+*) s+ 21 Jo (1 -ux) is+q ’ 
4xf 
if for shortness u ~ (obviously u< 1). 
The integral in x is here an integral belonging to the general form 
II(a,0, 7, w)=f XT * 1 (l—xy~ l (1 — ux)' y dx, 
viz. we have 
2-5—1 fs 
Ring-integral =— ~ ' U(^s, \s+q, u). 
yj+x) 
71 . The general function II(a, 0 , y, u) is 
n (a, 0, 7, «)=r ^|yF(a, 7, a+0, u), 
or, what is the same thing, 
F(a, 0 , y, = IT(a, 7~a> ft w )> 
and consequently transformable by means of various theorems for the transformation of 
the hypergeometric series ; in particular the theorems 
F(a, 0 , 7, m)=F( 0 , «, 7, u), 
F(a, 0 , 7s «)=(!— “) y-B_#, F(y— a, 7—0, 7, u) ; 
and if v= / 1 — ^1— Q w h a t j s the same thing, u—- 4 x/v - , then 
Vl+V'l-M/ (l + '/v) 2 ’ 
F(a, 0 , 20 , u)=( l+\/ v) 2 * F(«, a— 0 +^, v) ; 
in verification observe that if «t=l then also v=l, and that with these values, calcu- 
lating each side by means of the formula 
ir 7 r(y-«-/3) 
T(«,0, 7, 1)— r (y_a) r (y-/3) 
rr/ O n r«TQ8- y ) 
n(«, 0, 7, i)— r ( a+/ g_ y p 
the resulting equation, F(a, 0 , 20 , l)= 2 2 a F(a, a— 0 +|-, 0 +?r, 1 ), becomes 
rg0r(<3-«) r(0+i)r(2/3-2«) 
r(20-«)r0~ z r(2j8-«)r(0-a+i)’ 
