118 Mr MURPHY'S SECOND MEMOIR ON THE 



not vanish ; and since in general 



/■p.. _/_ix. ^-(^-1) ....{x-n + 1) 

 ■'' " " ^ '^^ •{x+l){x + 2)....{x + n + l)' 



it is evident that in this case ftPj 



2n + l ' 



4. To illustrate the observation in Art. 1, with respect to the 

 generality of this method, let it now be required, to find a rational function 

 of t, as f{t), of the lowest possible dimensions, to satisfy the equation 

 fif{t).t' = 0, when x is any number of the series 



p, p + 1, p + 2, .p + n-l. 



Putting as before /(^) = \ + A,t + A-J'- + + Aj\ we have 



/• f(f\ ft _ 1 , ^' , -^2 , -^- 



■"•^^^' x + 1 x-^2 x + S x + n + 1' 



the sum of all which fractions must by the reasoning of Art. 2, be 



c . jx—p) {x—p—l)....{x — p — ti + 1) _ 



{x + l){x + 2){x + 3)....{x+p + x) ' 



and determining c, Ai, Ai, &c. in the same manner as in the Article 

 referred to, we have 



1.2.3....W 



c = (-l)». 



(j9 + l).(jO + 2)....(jO + W)' 



. _ n n+p+1 

 "^'~~1- p^\ ' 



. _ n.{n-\) (»+/? + l).(«+jP + 2) 

 '~ 1.2 • (^ + l).(;j + 2) ' 



&c &c. 



and therefore 



'^ ' 1 ^ + 1 1.2 (jo + l).(jo + 2) 



t^ d^ j /t _ ^ , n.{n-l) ^ „ 1 



~ip + l).{p + 2)....(p + n)-dt''-y -V 1-^+ 1.2 .^-&c.|; 



