statistical mecl)anic8 31 



where 



(50) □ (/a) = III du^ dv^ dtv, (./;" f.," — w) . 



00 



As to V(,/'i) and V (/o) their values are, according to (37) 



V (./i) = ///// du^ dv., dw^ c?(j> dh (,/;' — ./^ ./g) ft lo , 



and 



V {.Q = ///// du^ dv^ dw, rf^ dh (// ./;' — ,Q h (0 . 



25. Total number of collisions. In calculating tlie total number of colli- 

 sions we may use either the expression for the number of stars »thrown out» or 

 that for the number of stars »thrown in», integrating in either case over all values 

 of the components of velocity between + co . Per unit of time and space (stellar 

 year and cub. siriometer) we get ' 



(51) Total number of colHsions = TUf^^ zfs^ Js^/^ co . 



Denoting the mean value of a function — // — of u^, \\, iv^; n^, v,^, vk^ by 

 M{y), so that 



and putting 



= jAs,f\ = n, 



so that n is equal to the number of stars per cub-siriometer. Then we have 

 and hence 



(51*) Total number of collisions = Jtc?^ i)f(oj) . 



M{u>) is the mean value of the relative velocity of the stars. If the frequency 

 function / is known, this mean value can be computed. Take, for instance, 



m,{u^'' -\- v,^ -\- Wj^) 



f = Jill e 



2o« 



where I suppose the masses of the two bodies unequal, though our expressions for 

 the collision function is only valid for equal masses. 



To evaluate the integral M{iy>) we make an exchange of variables, putting 



v^=U, 



w„ = V, + 7 



' + 



