PROFESSOR CAYLEY ON PREPOTENTIALS. 
719 
that is 
T2/3 02a Y(2{3-2ct) 
r/3r(/3+i) — 1 r(/B- a )r(s -«+i)’ 
which is true, in virtue of the relation 
Y2xY\ o 2 j — i 
ra?r(«“+-^) 
72. The foregoing formulae, and in particular the formula which I have written 
F(a, (3, 2j3, u)=(l-j-\/v) 2a F(a, a— /3-j--^-, /3+-|, v), are taken from Rummer’s Memoir, 
“ Ueber die hypergeometrische Reihe,” Crelle , t. xv. (1836), viz. the formula in question 
is under a slightly different form, his formula (41) p. 76 ; the formula (43), p. 77, is 
intended to be equivalent thereto ; but there is an error of transcription, 2a — 2/3 + 1, in 
place of /3+^, which makes the formula (43) erroneous. 
It may be remarked as to the formulae generally, that although very probably 
II(a, (3, y, u ) may denote a proper function of u, whatever be the values of the indices 
(a, (3, y), and the various transformation- theorems hold good accordingly (the T-function 
of a negative argument being interpreted in the usual manner by means of the equation 
r.r=^r(l+#), = X ^J + ^ r(2+tf) &c.), yet that the function U(a,(3, y, u ), used as de- 
noting the definite integral 1 oc a ~ l (1— (1 —ux)~ y dx, has no meaning except in the 
Jo 
case where a and (3 are each of them positive. 
In what follows we obtain for the ring-integral and the disk-integral various expres- 
sions in terms of H-functions, which are afterwards transformed into ^-integrals with a 
superior limit go and inferior limit 0, or ; but for values of the variable index, q 
lying beyond certain limits, the indices a and (3 , or one of them, of the Il-function will 
become negative, viz. the integral represented by the Il-function, or, what is the same 
thing, the ^-integral, will cease to have a determinate value, and at the same time, or 
usually so, the argument or arguments of one or more of the T-functions will become 
negative. It is quite possible that in such cases the results are not without meaning, 
and that an interpretation for them might be found ; but they have not any obvious 
interpretation, and we must in the first instance consider them as inapplicable. 
73. We require further properties of the II-functions. Starting with the foregoing 
equation, 
F(a, (3, 2/3, w) =(!+>/ «) 2a F(a,a— /3+i, /3 + i v ), 
each side may be expressed in a fourfold form : — 
F(a, 0, 2/3, u) 
=F(/3,a, 2/3, u) 
=(1— m) p_ “ F(2/3— a, /3, 2/3, u) 
— (1 _ uf-o- Y(cc, 2/3 - a, 2/3, u) 
(l+xA;) 2 “F(a, a -/3+i,/3+A v) 
=(1 +\/ v ) 2a F(a— /3+-|, a, /3-f^, v) 
= (1+xA;) 2 “ (1 -«)*"«» Ffl3-a+i, 2/3 — a, (3+± v) 
= (l+ N /^(l-^- 2 “F(2/3-a,/3-a+i,/3+i,i;), 
