422 On the Summation of a certain Factorial Expression, 

 whence 



n(«-3 ) n(-7-i) 

 n(a-i)n(-7-3) 



n(«-2) .n(-7-i) 



+ ^P + '^)U{u-^l) n(-y-2)- 

 The second and third terms are 



- (»-4_2) (y+i)(r+ii)-(/a+a)^fa+i). 



which are 



= - (»-iKL8) '"^+^'-^^+v-a). 



For the reduction of the first term we have 



n(^«_7+^+l)=[/3 + l-a-7]^+2n(-a-y-l), 

 n(^-y-l) :=l3-y-iyU{-j-l) ; 



and we thus find 



iGS + l)(^ + 2)(«-l)(«-2)S = 



I^i^i:f^r^'^(7 + l)(«/3 + 2«-2^ + 7-2); 

 [/9-7-l]^ 



where, as before, 



b = l+ q 1 Q ^ r- + &c., 



0.7 0.4.7. 7 — I 



which is the formula in hypergeometric series requii'ed for the 

 present purpose, and which is certainly true when the series is 

 finite. 



Write now 



« = f, /3 = r+l, y=r—^; 



then the first term is [l]'"^^-r-[2^]'^"'"V which vanishes on account 

 of the numerator, and the second term is —^r {?•■+-), and we 

 have consequently 



-ir{r + 2){r + S)l.l8=-ir{r + ^), 

 which gives 



4r(/- + ^) 

 (r + 2)(»- + 3)' 



S being here the series in r, the sum of which was required, and 

 the particular case of Mr. Kirkman'a formula is thus verified. It 



1 



J 



