180 Major MacMalion and Mr Darling, Reciprocal Relations 

 so that, provided r is not zero, we have 



X sin {x — a) + r cos {x — a)\ e"'^'^ 



J r- + x^ 



X sm X -{-r cos .r , 



r^ + ic- ^ 



an identity which may be verified by differentiation with respect 

 to a. Putting x = r tan ^ and then replacing ^ by a;, (9) becomes 



•*^'^~' «/'• cos(^ + a- r tan .^■) „ tan x 



'^-^"^'^ ' "^ ■_::^-J g a tan x ^^ 



J cos X 



.tan 1 a/;- ^^g (^ _ ^ ^^^ ^A 



= ^ ax (10), 



J cos X 



which admits of ready verification by differentiation with respect 

 to a. The identities (9) and (10) hold generally, provided that 

 the constants are finite; we have seen that r must not be zero. It 

 will be noticed that both (9) and (10) are of the form 



Jo Jo 



where the upper limits of integration involve a. 



2. As another illustration of how the method admits of genera- 

 lisation, let 



fAx)'c{0{x,t)]dx = y\r,{t). 

 J «, 



rb, 

 and f2{x)K{d {x, t)] dx = yjr, (t) : 



J 0.2 



fbi . fb, 



then I /i (x) v/^a {\ (x, t)} dx=\ fo {x) f, {\ (x, t)} dx 



J a, J a, 



when \ {x, t) = 4>^-^ {/(x) + (j>, (t)} 



and e(x,t)=g{f(x) + cl>,it)}, 



f, g, (pi and (f>2 being any functions. It should be observed that A. 

 becomes 6 when (f)^ = ^2 and g = 02~^. Other corresponding pairs 

 of functions are 



\(^,O = </>rM/(^')-0i(O), 



0(x,t)^g{f(x).cj>,{t)}> 



and M^^0 = <^r^J4^r\^4^|, 



'f(x)cf>,{t) + l] 



e{x,t)=g 



f{x) + (f>,{t) 



