106 THE REV. F. H. JACKSON ON 
2. 
If we invert the base element », we see that [7]! is transformed into p-"’-»?[7]! and 
that (2), becomes transformed into p~”’~”"(2),. These transformations hold whether r 
be integral or not.. Inverting the base p in the series J,,,(A), we obtain 
os (- lyon nN n-+2r 
eco.) ie 
which we denote 
n> Xr 
p au(s) . é i) 
LomMEL has shown that 
r=0 m+n+ 2r tee 
THATn() = Dy(- 1)" ae (5) as) 
a T(m+r+1)0(n+r4+1) > 
The function Yj, was formed while seeking to extend the above theorem. The 
extension was found to be 
= 4 aS (-1)'T,»([m + 2+ 27 + 1)) Dd \mtner 
: Fenm(A)Sem() = Tem A) den) ar 2 TO [m tnt+rt+ 1])D,o([m 474 1)D,2([n +r + 1))P,([r rs ipa) (4) 
The relation between J and Y was surmised from the following simple but Senior 
theorem. 
E,(A) = cb see ce ee 
* 2 — 1-+p)r2 
1(A) = eet SNe 
BOVE) = 1+ yt Sy bee 
which suggested that g,,, might be derived from J,,, by inverting the base p. As | 
have given the proof of (4) as an example of the use of generalised Gamma-functions in 
a paper communicated to the Royal Society, London, it will be sufticient to say here, 
that the theorem may be proved by showing, that the coefficients of the powers of A on 
both sides of the equation are identical, being cases of the extension of VANDERMONDE’S 
theorem (Proc. Lond. Math. Soc., series 2, vol. i. p. 68). 
In the notation of Art. (1) we may write the theorem 
(= 1)'{2m + 2n + 47}! 
Jima) = = {Qi + In + Ww} Im + Aw} !{2Qn + 27} any : : (5) 
3. 
Consider the series 
J o(A) Fp(A) + iy o(rA)an(A)+ .. 0... + gD pA) SpA)t .-- 2 ees . (6) 
