394 REPORTS ON THE STATE OF SCIENCE. 



It is clear that any multiplication theorem for integrals may be used 

 in a similar way. 



The multiplication theorems for integrals of the types 



i 



f(x) = ^>(t)dt (2) 





 CO 



m=y~H{t)dt (3) 



o 



have been given by Mellin. 1 



If </>,(/), f_,(/) are the functions corresponding to /,(x), f 2 (x), the 

 function \j,(t) corresponding to the product is given for integrals of 

 type (2) by. 



i 



i 



and for integrals of type (-3) by 



CO 



o 



There are also a number of multiplication theorems for Fourier's 

 integrals. For instance, if 



r 



/(as) = J cos xtf(t)dt, g(x) = ("cos xU{t)dt 



o 



and <j>(t) is an even function of t, then 



CO 



f(x)g(x) = \cos xt x (t)dt, 



o 



CO 



where x (t) = \ j [f(t - T ) + ^ + T )]^(r)dr. 



o 



These theorems have not been thoroughly investigated. 

 Integral equations of the type 



fix) = f *(x - l)<p{l)di 







are of very frequent occurrence. The following hydrostatical problem 

 may be taken as an example : — 



_ A solid in the shape of a surface of revolution is immersed with its 

 axis vertical in a heterogeneous fluid, and weights are placed on the top 



' Acta Sccichdis Fennh-ae, 1806, t. 21, No. 6 ; Math. Ann., Bd. 68, 1910 See also 

 Bromwich's Infinite Series, ' oeeaiso 



