406 REPORTS ON THE STATE OF SCIENCE. 



and Ponhhaminer. The last two authors have enriched the theory by 

 th3 introduction of double circuit integrals. 



Pincherle ' has obtained a symbolic expression for Euler's trans- 

 formation E„ in terms of Laplace's transformation L and the opsration 

 M n of multiplying the dependent variable by x n . This formula 



E V = L- 2 M_L 



suggests the existence of inversion formulae of the type 



t)dt 



^(t) = \[(x-t) — 1 f(x)cl 



where X is some constant. 

 The equation 



f( s )= (\_jm ,. >0 



JK! J (1 - 2ts + s")" 

 -i 



,>(s) = i 1 - 8 *) "* ("[^(a.) + a./i(a.)j s i n =— ada 



where x = s + * yl — s' 2 cos a provides an interesting example. 



Abel's integral equation is a particular case of a formula of this 

 kind. Other formulae have been considered by Pincherle 2 and the 

 author. 3 



The transformations associated with kernels of the forms k(x — t), 

 t.(xt) have been discussed by Mellin. 4 These transformations may be 

 conveniently studied in connection with the multiplication formulae. 

 Transformations of other types have been considered by Picard and 

 Cunningham, but at present these seem to belong to the theory of 

 differential equations rather than that of integral equations. 



Transformations of difference equations into differential equations 

 lead to integrals involving Gamma Functions. These integrals were 

 discovered by Pincherle, 5 and have been studied subsequently by 

 Mellin ° and Barnes. 7 The theory of these integrals is connected with 

 the inversion formula of Eiemann and Mellin. 8 



22. Applications to the Partial Differential Equations of 

 Mathematical Physics. 



Integral equations frequently occur in the solution of problems of 

 mathematical physics when a definite integral solution of a partial 



1 Tincherle and Amaldi, I.e operazioni distributive (Bologna), 1901. 



2 Acta Math., 1887; Bologna Memoirs, 1907. 



' Proc. Lond. Math. Sw.,'l907 ; Cambr. Phil. Trans., 1909. 



* Acta Societatis Fennictr, 189fi, t. 21, No. G. A paper by Cailler (Darboux 

 Bull., t. 23) may also be referred to. ' 



5 Bend. lAncei, ser. 4, vol. iv., pp. G94-7O0, pp. 792-799. 



6 Acta Math,, 1891, vol. xv., pp. 317-384. 



7 Proc. London Math. Soc, ser. 2, vol. vi. (1907). 



9 Acta Math,, vol. xxv. (1902) ; Math. Ann., Bd. 68 (1910). 



