occurring in the theory of radio-active transformations. 427 



differential equation (10) which is simpler than (9), inasmuch as 

 it depends upon fewer independent variables, 



Tn many cases the solution of the integral equation 



n (s) = I e-«' V(t) dt 

 Jo 



may be calculated by means of the inversion formula * 



where c is a contour which starts at — oo at a point below the 

 real axis, surrounds all the singularities of the function u (^) and 

 returns to - oo at a point above the real axis, as in the figure. 



The conditions to be satisfied by u (^) in order that this 

 inversion formula may be applicable have not yet been expressed 

 in a concise form. 



The formula may be used to obtain the solution of a problem 

 in the conduction of heat when we require a solution of 



dx^ ~ dt' 



which satisfies the boundary conditions 



F= when x=0 and x = a, 



V =f(x) when ^ = 0. 



The solution found in this way is identical with the one given 

 in Carslaw's Fourier-' s Series and Integrals, p. 383. 



* A particular case of this formula has been given by Pincherle, Bologna 

 Memoirs, 10 (8), 1887. 



