ON THE THEORY OF INTEGRAL EQUATIONS. 359 



X 



u(x) = f p tu(£)di 



of aa equation which possesses a discontinuous solution 



U(x) = x'- 1 . 

 W. Kapteyn l has shown that a solution of the integral equation 



X 



/(®)=J J (p-y)t(y)dy 



o 



can be expressed in the form 







It is interesting to compare this result with the formula given on 

 p. 416, § 26. 



6. Physical Applications of the Equations loith Variable Limits. 

 Problems of Heredity and Hysteresis. 



Volterra has made use of an integral equation with variable limits 

 in some researches on the equilibrium of a rotating mass of fluid, 2 and 

 there are numerous applications of equations of this type to partial 

 differential equations of the hyperbolic and parabolic type ; 3 but the 

 most interesting applications appear to be to problems of heredity in 

 which a physical system exhibits phenomena of memory or hysteresis. 



It has been remarked by Picard, in his article on ' La Mecanique 

 classique et ses approximations successives,' 4 that mechanics can be 

 divided into hereditary and non-hereditary mechanics. The latter deals 

 with cases in which the future of a system depends only upon its 

 actual state and the states which precede it but are separated from 

 the momentary one only by an infinitesimal interval of time. The 

 former deals with the case when any action leaves an impression on 

 the system and the actual state depends upon all the preceding states. 



The theory of mechanical problems of this type was commenced by 

 Poisson in his Memoir on magnetism already referred to. He repre- 

 sented the components of magnetisation at time t by expressions of the 

 type 



A(0 =[f(t-0)a(ti) dd, 



and arrived at the integral equation with variable limits cited in 



1 Liege Memoirs, 1906 (3), t. 6. 2 Acta Math., 1903. 



8 See, for instance, the researches of Le Roux, Holmgren, Goursat, Annates de 

 Toulouse, 1904; Mason, Math. Ann., 1908, Bd.lxv., p. 570; Myller, Math. Ann., 1910, 

 JJil. lxviii. 



4 Rivista di Scienza, vol. i., Bologna (1907). 



