ON THE THEORY OP INTEGRAL EQUATIONS. 423 



Bryon Heywood l and Poincare, 2 while the applications to problems in 

 the conduction of heat are very numerous. 



Lord Eayleigh 3 has obtained an integral equation of the first kind 

 when studying a dynamical problem in illustration of the kinetic 

 theory of gases. He considers a number of equal masses which are 

 not free to wander indefinitely, but are moored to fixed points by similar 

 elastic attachments so that they perform vibrations of a certain type. 

 Let r be some quantity by which the amplitude of the vibration is 

 measured, and suppose that the number of masses whose component 

 velocity in the direction of the axis of x lies between u and du is 

 represented by 



(p(r,u)du 



where f is a known function, which depends upon the law of vibration. 

 Then if \{r)dr denotes the number of vibrations for which r lies between 

 ?• + dr, the law of distribution of u is given by 



f(u) = L{r)t(r,u)dr. 



If this law of distribution remains permanent when there are collisions 

 between the different masses, we must have 



• /(«) = -7- e '"' 



and the problem is to determine the law of distribution of the ampli- 

 tude — i.e., to determine x(>") — so fchat"/(%) may have this value. 



Lord Eayleigh considers in particular the case when u = r cos and 

 all angles U are equally likely. The integral equation is then 



/(«)=i-f- x ^l. 



U 



and is of Abel's type. The solution for 

 is given by 



X(r) = 4N/w e~ 



hr" 



28. Non-linear Integral Equations. 



The method of successive approximations has been applied to the 

 study of non-linear integral equations by Block, 4 Orlando, 5 d'Adhemar G 

 Schmidt, 7 and others. The method indicates that the solution of an 

 equation of the type 



1 Thesis, 1908, Paris ; Liouville's Journal, 190S 



2 Lectures at Oottingcn, 1909 ; Teubner, Leipzig, 1910. 



3 Phil. Mag., 1891, vol. sxxii., pp. 424-45. 



4 Arkirfor Math. u. rhys., 1907, Stockholm, Bd. 3, Hiifie 3-4. 



5 Send. Lincci, 1907, vol. xvi., 2nd ser. 



6 Bull, de la Sod&U Math, do France, 1908, t. xxxvi. 

 ' Math. Ann., VMS, Bd. G5. 



