178 M(ijo7^ MacMahov and Mr Darling, Reciprocal Relations 



Reciprocal Relations in the Theorij of Integral Equations. By 

 Major P. A. MacMahon and H. B. C. Darling. 



[Received 1 February 1918. Read 4 February 1918.] 

 1. Let f{oc)K{ajt)dx = ylr,{t) 



J a, 



and f2{cc)/c(a;t)da; = yjr^{t); 



J a., 



then, if we suppose the functions f^,f and k to be such that the 

 order of integration is indifferent, we have 



fbi rbo rb, 



/ /i (•^) fa (^t) da; = dy \ f {x)/., {y) k {xyt) dx 



= \\Uy)i^i{yt)dy, 



or, as it may be written, 



/ A(oo)yjr,(xt)dx= f,(x)yfr,(xt)dx (1). 



*i J a.2 



In the Messenger of Mathematics, May 1914, p. 13 Mr Rama- 

 nujan has employed this result to deduce a number of ' interesting 

 relations between definite integrals. The method is very suggestive 

 and appears capable of considerable extension. For example, if 



f{x)K[e{x,t)\dx = ^lr,{t)\ 



[b. \ (2), 



and / fM'c{e{x,t)]dx = ^lrM 



*^^n \j^ (•^) ts [0 {a; 01 dx = ^J, {x) f, {0 (x, t)} dx . . .(3), 



provided that {x, 6 {y, t)]=- d [y, 0{x, t)\ (4). 



The functional equation (4) is satisfied by 



0{^,t) = cl,-^f(x) + cl>(t)\ (5), 



where / and are arbitrary functions ; which is a general form of 

 solution and includes among others such solutions as 



H^>t) = c}>-^{f(x).cl>{t)} (6), 



^ ^ \f(^) + cP{t)\ ^'>' 



^{^,t) = cf^~^f(x) + cf,{t)+f(x)cf,(t)] (8). 



