480 Mr. S. H. Burbury on certain 



12. After collision the / sphere will have component 

 velocities 



u 1 . . . u' + du', 

 v' . . . t/+«fo', 



iy' . . . w [ -\-dw\ 



that is will have passed into the class f, and the F sphere will 

 have component velocities 



U'* . . . U' + rfU', 

 V . . . V' + dV', 



w . . . w+dw, 



that is will have passed into the class F'. Now 



M /8 + ,/» + l(? /2 + XJ/2 + y/1 + W /2 = u 2 + ^2 + 2y 2 + TJ2 + y 2 + ^ 



and 



M 'U' + v'V + w'W = uV + vY + WW. 

 Also <f> remain unchanged, and R is unchanged in magni- 

 tude, but its components are now 



u'-TJ', v'-Y 1 , w'-W. 

 Also by a known theorem 



du' dv ! dio' dW dV' dW = du dv dw dU dV dW. 



13. Exactly in the same way, if about every sphere which 

 initially in the distribution (1) belongs to the class /', we 

 describe an element of volume with the same 6 and $ as 

 before, but with the new direction of R for axis, every F f 

 sphere which is initially within one of these elements will 

 within the time dt after the initial instant collide with the/' 

 sphere to which the element belongs, and after collision f 

 and F' will have passed into the classes /and F respectively. 

 The number of such reverse collisions in the time dt is actually 



k'fW du' dv' div' dU' dY' dW a 2 sin 6 cos d6 dcf> Rdt, 



or, which is the same thing, 



k'f'W du dv dw dU d\ d\N a 2 sin 6 cos dd dcf> Rdt, 



that is k'f'W da dr) dt, and its mean value is f / F / dcr dtf dt. 

 Here k' = 1 on average, as k = l, but k' is independent of k. 



14. Every collision of this second kind increases the 

 number of the class / by one. It follows that, taking into 

 consideration all collisions, 



dt 



That is the actual value. The mean value i 



dt 



% = \§(k'f'F'-kfF)dU dY dW$a* sin 6 cos 6d6d4> R. 



d L = Jjj(/T' -/F) dv ^v dw jjV sin e cos e ae d$ n. 



