10* MAXWELL'S LAW 371 



occurrence of the possible case that the particle w t should move 

 with a velocity made up of the components u lt v it w l . 



Although the motion of a particle is not independent of the 

 motions of the other N 1 particles, yet the function F { will be 

 determined only by the three arguments u lt v lt w l if the NZF^ 

 cases are so counted that account is taken in them only of the 

 state of the one particle Wj, and not of the states of the other 

 particles also. 



For the same reason, if the particles of the group are all like 

 each other, the probability of the event that a second particle m 2 

 has the components u 2 , v%, io 2 is determined by the same function 

 with different arguments, viz. 



From these two values of the probability-function we easily 

 obtain by a known law the expression for the probability of the 

 occurrence of both sets of circumstances, viz. that the particle w, 

 should have the components u lt v } , w\, and ra 2 the components 

 %, v 2 , w 2 . It is necessary to assume only that the number Z, 

 which we look upon as very large, may be approximately taken 

 as infinitely large, and that the function F is determined in 

 correspondence with this assumption ; in this case the two events 

 of m l possessing the components u lt v lt w l and of w 2 having 

 u^ ^2> W 2 are independent of each other, and the probability of 

 their simultaneous occurrence is therefore expressed by the 

 product of the two functions, and therefore by l 



F,F 2 = F(u } , v lt io } )F(u 2 , v< 2 , w 2 ). 



If we also consider a third particle m 3 which may have the 

 velocities u 3 , v a , w s , a fourth with components %, v 4 , w 4 , and so 

 on for all the N particles which form the group, we have in the 

 product of the N factors 



F^F 2 ...F N = F(u lt v } , wdFfa, v 2 , w 2 ) F(u s , %, W N ) . 



1 This formula and those which follow later would contain numerical 

 factors which would have to be formed according to the rules of combinations 

 if we did not fix a definite series of the particles. If we sought the probability 

 that one of two particles m^ and ra 2 had the components u^, v n w v and the 

 other the components u v v v w v this would be twice as great as in the 

 case we have taken. Since these factors have no influence on the result, 

 it would be superfluous to complicate the formulas by inserting them. The 

 factors, furthermore, disappear when the number of particles considered approxi- 

 mates to infinity. ( E n c k e , Astron. Jahrb. filr 1834, p. 256.) 



B B 2 



