INVERSE METHOD OF DEFINITE INTEGRALS. 341 



2d, of a rational and entire function />„ which satisfies the equation, 



dP 



since the term 2 ( - 1)". -^ is the result which arises if the logarithmic 



term ( — 1 )" P„ . h. 1. -> be put for u in the actual equation. 



3d, of an appendage containing n + 1 arbitrary constants, which as 

 before remarked must be rejected altogether. 



Differentiating the equation for p„ above obtained, we get 



(«',^- + Mi-20.^-+>-i)(«+^)# + 2(-ir.^--o,. 



(«-l)(l-20.^" + 2(2«-l)^^" + 2(-l)»^^=0, 



• : ■ ^" df-^ + ^^^> ~dF~^' 



when these equations terminate, since j9„ is of « — 1 dimensions. 



Put ^ = 0, in all these equations beginning with the last, observing 

 that then 



^ = (-l)-.(« + l)(» + 2)...(2«), 

 ^^ = - ( - 1)" . « (« + 1) (« + 2)...(2« - 1), 



'^=^-^)"-^^i^^-^'*''^)^'' + ^) (2«-2),&c. 



Y Y2 



