INVERSE METHOD OF DEFINITE INTEGRALS. 1S8 



Put now h = r, the first factor under the sign of integration becomes 

 simply P„, and the second factor is then of m dimensions; and there- 

 fore, by the nature of P„, the integral vanishes; and therefore, when 

 n>m, ftIl„Bm' = 0: and the same reasoning applies when m>n, only sub- 

 stituting RJ instead of R^ throughout the process, hence R„ and R,„' are ' 

 reciprocal functions. 



When m = n, then in the general expression 



j;R«.' = (-irj;p.^(«-^); 



we need only take the term involving the highest power of t in 



dr K^ dr )' 

 namely, 



/ ,.... (« + !)•(« + 2)...(2w) d^ (..rd-.n 

 ^ ' 1.2... n dr \ dt' 1 



. ,, , (« + l) . (« + 2)...2« , . , ,^ , ,x , 



and observing that /JP„#" = ( — 1)" . -, .,, , — ''" /_ — -^. ; 



it follows that ftR„R,!=- . {n + r) {n + r-1) {n + r-2)...{n-r). 



The reciprocal functions a„, a„' may be obtained by putting r = 

 in R„ and RJ ', similarly, if we put r = l, we get b„, b„', &c., and thence 

 we obtain the reciprocal functions relative to double integration, namely, 



dP d^P d^P 



S,'=:ao'{tt'Y ^n + «.'(«')^^'^ + «^'(«T^"^-f" + «3'(«')^-"^", &c. 



In the same manner if we vary the constant a while r remains constant, 

 we obtain the reciprocal functions 



Vol. V. Part II. S 



