136 Mr MURPHY'S SECOND MEMOIR ON THE 



and tlie numerator will be a function of ft — \ dimensions, represented 

 by v„, so that 



v„ 



<p{x) 



{x+ l)(ar + 2) (a; + M)' 



when X is any integer from to (^^ - 1) inclusive; and if we multiply 

 by a;+ 1 and put x= — 1, and again by a; + 2 and put a-= — 2, &c. as in 

 the preceding Section, we get 



'~ 1.2 ■ 1.2.3....(« - J)' 



&c.= &c. 



Now the equation 



^{x) . {x + 1) (a; + 2) {x + n)- v„ = 0, 



is of m + n dimensions, and is by hypothesis satisfied, when 



^• = 0, 1, 2, («-l); 



therefore if u^ represent some function oi x of m dimensions, we must 

 have the identity 



(p{x) .{x + V) (ar + 2) (ar + «)-», = M,.ar . (^-l)(ar- 2) (x-n + \), 



hence if we divide 



<f>{x){x + \){x^2) (x + w) by x{x-'\){x-^) {x-n^l\ 



and retain only the part of the quotient which is an entire function of x, 

 u, will be completely determined. 



Put now —1, —%,...— n successively for x in the preceding identity, 

 and we get 



t;., = (-l)»+M .2.3....».«_,, 



