720 
PROFESSOR CAYLEY ON PREPOTEN TIALS. 
where, instead of (l+\/ v) 2a (l—vy p ~ 2a , it is proper to write (1+^/ vf° J (1 \J v) 2/3 ~ 2 “; and 
then to each form applying the transformation 
we have 
F(a, /3, y, w) — p a r(y— a) 7 w), 
r«r(2j3-«) ^ — & u ) 
Yj 3 T/3 n(/3, (3, a, u ) 
=(l w) 3 a r(2/S-«)r« — a ’ a? 
= (1 - uf- a y r- r ^_ a) n(a, 2/3 — a, 2(3- a, u) 
— (i+\/^) 2a rar(/3— «+^) n(«, /3 a +!> a /^-H2> w ) 
=(l-}~\/' y ) 2 “ r(2j3 — a) ^( a 2/3 cc,k,v) 
=(l- s r\/v 2p 2 “ p^_ a+ ijp a n(/3— a+i) 2/3 — a, ?;) 
= (1 +\/ ^ (1 — y ) 2 ^ 2a r( 2 / 3 -«)r («-/3 + i) n(2/3 — a, a— ^+-2 5 /3 — a+^, y )’ 
I select on the left-hand the second form, and equating it successively to the four 
right-hand forms, attending to the relation ^^-^-=2 1-23 T|, we find 
n(3,0,<vO=(i+\AO 2a 2 1-23 r«r|?!+i) n(«,j9-«+*,«H3+*,«) 
— (!+>/ -y) 2a 2 l ~ v r(«-^+f)r(2/3-«) n ^~ — a > a > ^ 
= (l + \/ y ) 2,3 (l — \Z v ) 2fi 2 “2 1 2P p^_f + \)r a HO — a+-|, a, 20— a, v) 
= (1 + \A0 213 (1 — \/ V Y ? 2a 2 1 ^r( 2/3 — a) r(a— /3 + i) ^(2/3— a, a— /3 — a + tt). 
Putting herein a.=\s-{-q, the formulae become 
n(is,|s,|s+!?!»)=(l+\/ vy + ' Q 2 l_i Tiis+q-fT^-q) n(l s +?>i — 2>i+?>*) • • • CO 
=(iW'v)^2'~ ra+ r ^_ g) n(i+g,^-g,i»+g^) • • ( IL ) 
=(l+yi)'(l-^)- 2 *2'-' r5 ^J 7T?y n(|- 2) is+f.i*-?.»)- • -( ra -) 
=(l+v^)-(l -^)-» 2- r( is -;)ri +g) n (^-g' i+g- * -g. »)' • -(IV.) 
