434 Loi;d Rayleigh on Dynamical Problems in 



in the unit of time in the path of each mass. The chance of 

 a collision for a given mass in time dt is thus represented by 

 v dt. The number of collisions by which masses are expelled 

 from the range du in time dt isf(u)du.vdt. The number 

 which enter the range from the two sides is 



\f(u-2qv) +f (u + 2qv) }du . Jv dt, 



so that the excess of the number which enter the range over 

 the number which leave is 



\if(u-2qv) + \f(u + 2qv) -/(«) }du . v dt, 

 and this is to be equated to • ' — dt. Thus 



rift =¥^-2qv)+\f{u + 2qv)-f{u)=2qV^, (23) 



the well-known equation of the conduction of heat. When 

 t = 0,f(u) is to be zero for all finite values of u. The Fourier 

 solution, applicable under these conditions, is 



_A 



Vt' 



A^t')=ih,e- u2lu \ 



where t' is written for 2q 2 v 2 vt. The total number of masses 

 being N, we get to determine A 



so that 



(* + «> 



N= /(«,i , )rftt=2 N /7r.A; 



If n be the whole number of collisions (for each mass), n = vt, 

 and we have 



At' = 4,q 2 vK2n (25) 



If the unit of velocity be so chosen that the momentum (2qv) 

 communicated at each impact is unity, (24) takes the form 



'w-dk)***' • • • • (26) 



which exhibits the distribution of momentum among the 

 masses after n impacts. In this form the problem coincides 

 with one formerly treated* relating to the composition of 

 vibrations of arbitrary phases. It will be seen that there is a 



* Phil. Mag. August 1880, p. 73 



