84 Proceedings of the Royal Irish Academy. . 



12. If in the equation 



[G{t)f(tz)'dt = g(x), 



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we set t = t- T , x = e-i, we get 



[*" c-r G (err) f{e-{*+S))dT = # (c-«), 



which may be written in the form 



f H(t)f(t + x)dt = h(x), 



where | H (t) | is integrable from to oo. We may also suppose that 

 | f(x) | and | h (re) \ are integrable from to cc ; if they were not, we should 

 merely have to multiply them by e-P x (0 < p< 1), and to change Hit) into 

 ept H{t,. 



Setting then 



$(x) = ^/(x)dr, 



we obtain, on integrating by parts, 



H (0) $ (x) + j H'(t) <p{t + x) dt = h (x), 



which we may write, supposing H(0) =f 0, as follows: — 



ty (x) + K(t) (t • + x) dt = t}j (x). 

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K(t) and i£ (a.) are absolutely integrable from to oo, and we get as the 



solution for < x < x. 



t - f c~(*-y)^(y)dy 



^U 1+ ^ e atK(t)dt 



to which may be added solutions like 



e&*(b a + b^ + . . . + bj>), 

 where (S is a root (of multiplicity & + 1) of the equation 



1 - [ c~<K(t)dt = 0. 

 The solution of the equation 



f H(t)f(t+z)dt = h(x) 

 may now be found in the form 



2mdxlz> r c . tH (t)dt 



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