d"v 

 dF 



INVERSE METHOD OF DEFINITE INTEGRALS. 337 



and the law of the successive formation of these equations being very 

 simple, we have generally 



(„.,^-(»+».-*-i)(.-.o^>.{(*+i)(»+»)-*i*±L>}.g=o. 



Put k = n, hence 



d"v 

 Transpose n(n + 2m + l)-j—, and multiply by {tt')-", hence 



from which it follows that M = (it')"" .d". ' satisfies the equation of 



Art. 16. 



dv' 

 Again put «j' = («')"'"*'"^^ or tt' -j- +(n + m + l)(l-2t)v=0, and by 



successive integrations we obtain 



tt' . v' + (n + m) (1 -2t) ftv' + 2 (n -\-m) ft'v =0, 

 tt' . ftv' + {n + m-l) (1 -2t) . ft' v -\-2(2n + 2m- 1) J^'v; = 0, 

 and generally * 



tt' ft*-'v' + {n + m-k + l){l-2t) ft'v' + 2h{n + m) - ^ '^^~^^ \ . ft"-' v==0. 



Put k = n, hence 



tt' fr' V + (m + 1) {1 -2t) . ft" v' + 91 (n + 2m + 1) . ft"*' v' = ; 



from which it appears that m = jJ" +*(»') is also a particular solution, and 

 therefore the complete solution of the general equation is 



