H& 



168 Mr. J. Buchanan on a Law of Distribution of 



(8), to be satisfied simultaneously. From these equations it 

 is possible to eliminate four of the variations : the remaining 

 variations are quite arbitrary. For the elimination, multiply 

 the equations (3), (4), (5), and (6) respectively by 2k, —2ka, 

 — 2k/3, —2ky, as indeterminate coefficients, and add the results 

 to equation (8). This gives 



= -2*J(«&»+/886+78e)SOT, (9) 



Let the indeterminate multipliers be so chosen — being arbi- 

 trary constants — that the right-hand member of equation (9) 

 vanishes. At the same time they can be chosen so as to make 

 each of the multipliers of the arbitrary variations on the left- 

 hand side of the equation equal to zero. Hence 



J* («8fl + / 88& + 7$c)2w n =0. . . . (10) 



For the other quantities, we may write, as a typical set of 

 terms, 



I J 'CI 



r ^+2M«-«)=0 ) (11) 



■I ^"Cl 



r _+2M«-/3)=0, (12) 



^ + 2M,- 7 )=0 (13) 



These last give, by integration, 



F(w, v, w) = G«-*»(<"-«> t +0M"P+<»-y)') t 



Thus the probability that the component velocities lie be- 

 tween u and u + du, v and v+dv, w and iv + ihc respectively 

 may be written 



F(i l ,v,zo) = B s .e-^^-^+^-^+(--yy)dudvdw. . (14) 



The probabilities for the existence of the components are 

 therefore respectively 



Be-*™**-*? da, 



Be- km ( w -v) 2 dw. 



And since each component velocity must lie somewhere 

 between — co and + cc , we have 



