ON THE THEORY OF INTEGRAL EQUATIONS. 405 



dx n 



„d"'v 



h x {u) = ?a nm x n ^ 



M t (v) = Za m ,r dv 



The equation M,(v)= is then the Laplace transformed equation. 



It was shown by Petzval ' in 1859 that Laplace's transformation is 

 periodic, the second transformation yielding the original equation except 

 for a change of sign of the independent variable. This, of course, was 

 indicated by Fourier's integral formula. The result is important, as it 

 indicates the existence of inversion formulae of the type 



f( x ) = [ e-*Wt)dt, </>(t) = ~ f e"/(x)dx. 



i • 



These formulas have been studied by Pincherlc. 2 A particular 

 inversion formula 



CO CO 



f(z) = ~ f(v)fi*-*dv, f(v) — f(k + is)c-' f ' ds 



o — co 



z = x + iy 

 x < k 



was given by Cauchy 3 long ago. 



The general theory of periodic transformations has been developed 

 by Pincherle 4 and Levi Civita. 5 Those in which the original differential 

 equation is reproduced after a single transformation are of special 

 interest, as they connect themselves naturally with homogeneous integral 

 equations of the type 



^(s) = X j k(s,t)f(t)dt. 



An important class of equations which are transformed into 

 themselves by a transformation of Laplace has been discovered by 

 M. Abraham, and has been extended by the author. 7 The equation of 

 the elliptic cylinder belongs to this class of equations. 



The transformation in which the kernel is of the form 



(x-t)- 1 

 is of special interest, and is called after Euler, since he first used 

 integrals of the type 



b 



[(x- t)"- 1 i'(t)dt 



a 



to solve linear differential equations. The theory of the transformation 

 has been developed chiefly by Heine, Pincherle, Schlesinger, Jordan 



1 Integration der linearcn Different! algleicliwn gen, pp. 472-473. Simple deriva- 

 tions of Petzval's result have been given by Pincherle (Le operazioni distributive, 

 Bologna, 1901) and the author, Proc. Lond. Math. Soc, 1907,. ser. 2, vol. iv., p. 487. 



2 Bologna Memoirs, ser. 4, t. 7-8, 1886-87 ; ser. 6, 1907, vol. iii., p. 143. 



3 Compte.t rendus, Paris, 1851, t. 32, p. 215. 4 Acta- Math., 1892, t. 16. 



5 Torino Aiti, 1895, t. 31 ; Lomh. Hind., 1895, t, 28. 



6 Math. Ann., 1901, Bd. Iii., p. 81. » (Jamlr, Phil. Trans , 1909. 



