INVERSE METHOD OF DEFINITE INTEGRALS. 333 



Put # = in all these equations successively, thence we have 



{m + l)./'{0) =-n.{n + 2m + l).f{0), 



(m + 2) ./" (0) =- in~l){n + 2m + 2) ,/' (0), 



m + 3 ./'" (0) = - (ra - 2) . (« + 2w + 3) ./(O), 



&e. 



it follows from this by Maclaurin's Theorem, that the preceding equa- 

 tion will be satisfied, as a particular solution, by taking 



^•/^v ^/«v(i ^ n + 2m+l .n.{n-\) (w + 2w^ + 1) (w + 2>» + 2) ., , . 

 ./(0=/(0){l-i.-i-^^-.^ + ^^. (i + ^).(2 + «.) ^ ^^-&c-}» 



and ,/(0) being arbitrary if we put it equal to 



(1 + m) (2 + w<) (3 + ?») (n + m) 



i '. 2 ; 3 TTTT^ n ' 



this value oi f{t) will become the same as the value found for Q„ in 

 the preceding article ; hence, replacing \ — thy its equal t', we get 



(«') ^ + (« + 1) (1-2^) . ^ + « . (w + 2m + 1) . Q„ = 0. 



But if in the value of /{() we change the sign of m, putting 



... _ (l-m)(2-?w) {n-m) 



'^^"^~ 1.2 n ' 



then y*(#) becomes equivalent to q„ {tt')" ; and if we put this for y (#) 

 in the first supposed equation, and divide the result by {tf)"', we get 



{ti')^ + im + l){l-2t).^+{n+l){n-2m).q„ = 0. 



(2) Particular inferences resulting from the preceding theory. 



15. Denoting as before by <f> the" indefinite integral fi(tt')'", and 

 putting 



xxS 



