MOTIONS OF A SET OF NON-HOMOGENEOUS ELASTIC SPHERES. 505 



they become respectively 



jf) 2 + P 8 



- 2c2 - [2 + o'(f+K)]^ -<"- U >') 



&c, &c. 



In completing the process of averaging on these new expressions, the integ- 

 rations 



ff dadSdyffd^ 



affect only the first factors of each of them, and a consideration of the meaning 

 of the factors shows that in each case the integral is a function of the constant 

 quantities A, B, C, a, ft, y and c. 



[Neglecting terms in & as compared with those in c\ the approximate 

 values of the averages of these factors are 



eV <g 2 +y 2 , y 2 +« a , « 2 +/3 2 \ 



3\ A + B + C / ' 



C 2 , /3 2 + y 2 c\ y 2 + g 2 c\ g 2 + /3 2 1 



3 A ' 3 B ' 3 C J 



Hence the three equations 



ABC 



k x k 2 k 3 



are a solution, and therefore must be the solution, of the problem of the 

 "special state." 



The quantity [it — U) 2 finally is affected only by the integrations of the § 21 

 of Prof. Tait's paper already referred to ; and indeed its value may be written 

 down at once from the result of that article. For 



(u-vy=u 2 -tiu+\j 2 -vu=j- 

 The required result then is 



fC-t Kg *G>o 'b 



or, in words : — 



The average energies of rotation of a sphere about each of the three principal 

 axes are equal, and the whole average energy of rotation of a sphere is twice the 

 average energy of translation. 



The forms for the average changes in the rotation-energies at a collision 

 indicate that, if at any time before the special state is attained the three rota- 

 tion energies are equal, they will generally tend to become unequal again; and 



