3G0 REPORTS ON THE STATE OF SCIENCE. 



Section 5. A somewhat similar procedure was adopted by Boltzmann ! 

 in his theory of elastic afterworking, and by Korn ' 2 in the study of the 

 action of selenium cells. 



A general theory of problems of this type has been developed by 

 Vol terra. 3 In 1887 he gave a general expansion theorem for a quantity 

 which depends on all the values of another function in a given interval, 

 and in his recent researches he has developed the theory of integro- 

 differential equations, 4 which appear to be the type of equations usually 

 required for the treatment of problems of hysteresis in electrodynamics 

 and the theory of elasticity. It is found that in electrodynamics equa- 

 tions of Poisson's type are of special importance. 



In problems of memory, when the state of a system is represented 

 by an equation such as 



<p{t) =f(t) + X f '(WWW, 



it is interesting to determine the sets of values of t and 6 for which the 

 function k (t, 0) becomes infinite. If these are determined by an equa- 

 tion of the form 



F(t,H) = o, 



then the events which occurred at the particular times given by this 

 equation are those whose effects will be emphasised at the given time t. 

 The emphatic events will, however, generally change as t varies, unless 

 it happens that the last equation is independent of t. 



An interesting problem of heredity connected with the motion of an 

 electron has been discussed by G. Herglotz s and P. Hertz. 6 The 

 integral equations are in this case of the forms 



f(t) = f(t) _ j ,(s)<t>(t-s) 







k(s) = 3-6s 2 £s < 1 

 and f(t) = <j>(t) — [ t(t—s)u(s)d& 







K (s) = 5 - 30s 2 + 30s 4 o <^s < 1 



respectively, and are solved by a method of successive approximations. 

 An interesting feature is the occurrence of solutions of the homogeneous 

 equations of the form 



-«., ft 



1 Ann. Phys. Chem. (Poggendorll), Ergzgsbd. 7 (1878); Sitzungsberichte, Vienna 

 (1874). ' Phys. ZeiUchr. (1909), p. 793. 



3 Bend. Lincei, 1909, vol. xviii., pp. 203, 295 : ' L'inversion des integrates d6- 

 finies.' ' Lectures at Clark University ' (Teubner). 



4 Certain types of integro-differential equations have been considered by Picard, 

 Lauricella, and Burgatti. 



5 GUt. Nachr., 1903 ; Math. Ann., 1908, Bd. 65. 



6 Dissertation Gbttingen, 1901; Math. Ann., 1908, Bd. 65. An account of the 

 work is also given in a paper by G. F. C. Searle, Proc. Roy. Soo. A., vol. lxxix., p. 550. 



