380 MATHEMATICAL APPENDICES 14* 



This mathematical property of the function F is the expression 

 of the fact that the occurrence of a speed u, the probability of 

 N^ which is U(u)du, is independent of the values of the simultaneously 

 occurring velocities v and w of the same pa.rjjftle., thp. probabilities 

 of which are V(v)dv and W(w)dw ; a fact the correctness of 

 which is so evident of itself that Maxwell chose it as an obvious 

 axiom for the foundation of his first * proof of the law found by 

 him. 



15*. Determination of the Constants 



Since the probability that some one of all the conceivably 

 possible values of the speeds may occur is a certainty, and has 

 therefore the value 1, it necessarily follows from the above inter- 

 pretation of the functions U, V, Wthat the sum of the probabilities 

 of all possible values must be 1, or 



poo 



L 



and that for the two other functions two corresponding equations 

 must hold. By simple substitutions all three formulas give the 

 same result 



1 = B j dre-**"* = B*/(ir/kni), 



whereby the constant B, and thus the constant of integration A } 

 is determined. 



We further arrive at a knowledge of the constants a, /?, y by 

 calculating the mean values of the components of velocity, or, in 

 other words, as the first equations show, the components a, b, c 

 of the motion of the centroid. From the equations so obtained, 



CO 



duu/e~** i *~** 



CO 



dvve -**' 



- CO 



c =B 



co 



we obtain simply 



= a, = b, y = c, 

 since 



(w - a) "' = dr(r + a 



f 



J 



co 



1 Phil. Mag. [4] xix. 1860, p. 22 ; Scientific Papers, i. p. 377. 



