ON THE THEORY OF INTEGRAL EQUATIONS. 361 



where the quanitties a h are the roots of the transcendental equation 



i 



x (x) =f K(s)e iX! ds=l. 



o 



The solving functions K(s), Q(s) are solutions of the equations 

 k(t) = K(t) — K(t — s)k(s)c!s — oo < t < + 00 



1 



;(t) = Q(t) — f Q(t — s)k(s)(U < t 



< + 00 



respectively, and the solutions of the two equations (1) and (2) are given 

 by the formulas 



t(t) =f(t) + (f(t-s)K(s)ds .... (I)' 



-00 



<)'(t) =7(0 + [ f(s)Q(t-s)ds .... (II) 



respectively, where f(t) is written for h[<f>(t + o) + <i>(t— 0)] 

 There is also an expansion theorem : — 



1 1 



f(t + 0) + f(t - 0) = 2i S -$$\ [ e T <t>(s)ds [ K-(n)e "''dr 



;,=i \ (o„) J J 



Herglotz's paper contains many other interesting results. 



An interesting integral equation which is reducible to an equation of 

 Poisson's type was obtained and solved by Beltrami ' in some researches 

 on the conduction of heat. The equation is 



/(0 = *(*)-AJ ^t-^e-'dd 



tfi 



and its solution depends upon the fact that the operation 



Of 



possesses the property. 



f„,..(0=p„ + ..(0 



i.e., the effect of repeating the operation is simply to add the arguments. 

 This result may be deduced from Borel's multiplication theorem given 

 in Section 16. 



1 Bologna Memoirs, ser. 4, t. 8, pp. 291-326. 



