58 M. C. Cellerier on the Distribution of 



designate this number by rnf for all the groups united, and the 

 total number of collisions by m; we shall have 



m=*2^V, m' = k¥(u,v)dz, F(«,r) = 2^; . (7) 



the sums are extended to all the elements co of the sphere by 

 suppressing in the second the terms for which the angle of 

 the velocities would not satisfy the condition YY'>z. Sup- 

 posing the radius I of the sphere parallel to the velocity u, 

 we can take for &> the narrow zone comprised between two 

 circles having I for their pole, and and + c/0 for their an- 

 gular distances from that point. Its altitude being sin d0, 

 we shall have co = 2-tt sin cW; the angle 0, constant throughout 

 its extent, will be that of the velocities; V, V' will also be the 

 same for the whole of the zone. The sums 2 will become in- 

 tegrals from to ir. We shall thus have for the indefinite 

 integrals, from the values (7) and (4), 



m = U (V sin 0d0=J- V* 

 buv 





dd W_ 



' txiv 



For 0=tt, Y 3 = (u + v) 3 , for 0=0, Y z = (u-v) z or (v-u) 3 , 

 according as«> or < v. Consequently 



m = hf(u,v) (8) 



on putting 



/(m,i-) = m4 ~ if u>v, f( u ,v) = v+^- if v>u. . (9) 



This function is continuous in value, but not in form. 

 Let us remark next that, from values (4) we get 



W = v / (?r + r 2 ) 2 -4 ? rV J cos' r 0, 

 an expressio n varying with between the limits u 2 + v 2 and 

 \/{u z — v 2 )-, or the numerical value of u 2 —v 2 . emust be sup- 

 posed <u 2 + v 2 , since the collision does not alter the sum of 

 the squares of the velocities. 



(1) If, moreover, z is <\Z(u 2 — v 2 f, the condition z<YY f 

 being constantly satined, in the value of F(u, v) it will be ne- 

 cessary to integrate from 0=0 to = ir, which gives 



consequently 



FO,t)=- l£u>v; FO,r)=*ift-> W . . . (10) 



