INVERSE METHOD OF DEFINITE INTEGRALS. 319 



If it be desired that Q„, R^ should be functions of the same 

 nature, differing only in the order expressed by m and n, that is 



self-reciprocal, put \ = W = \/{-~\, and having found any kind of 



self-reciprocal functions in which the limits are and 1, as for ex- 

 ample, the functions denoted by P,„, P„ in the preceding Memoirs, we 

 then obtain 



3. If a function V can he determined so that the quantity 



d°f(ttyV} dt^ 

 dt" ■ d0 



may he of n dimensions in t, (where t' = 1 — t as in the former Memoirs), 

 this quantity will he a self-reciprocal function when the integrations are 

 performed relative to (p. 



Denote this quantity by Q„, and supposing m to be an integer 

 less than n, it is necessary to show that f,pQmQn — 0, or that 



d''{{ttyn 



^'^- di" -^' 



the limits of t being and 1. 



Now Q„ being of m dimensions in t, let its general term be re- 

 presented by Oj.f, where it is evident that p cannot exceed n — 1, 

 since m<n; the part of the preceding integral dependant on this term is 



""'^'^ dv' — • 



The latter integral may by partial integration be put in the form, 



the last term being 



and therefore the index of differentiation never becomes negative. 



