614 Mr. J. H. Jeans on the 



molecules of which the velocities lie between u and u + du, &c. 

 is N/(n, v, w) du dv dw. We find that H has a minimum 

 value for the region denned by equation (26), and we find 

 that the propositions of § 24 apply equally, mutatis mutandis, 

 to this case. 



The minimum value of H and the corresponding law of 

 distribution can be found by the variation of equation (27), 

 keeping /subject to 



ffifdudvdw=l. (2S) 



$im{u* + v 2 + iv 2 )fdudvdiv=~. . . (29) 

 The resulting equation is 



ffi ( 1 + lo s/ + x + 2 p™ ^ + v ~ + ?y;2 ) W dH dv dw = °> • ( 30 ) 



where A, i*> are indeterminate multipliers. The solution is 



l + log/+A+iA«»(^ + »' + ti7 8 )=0; . . . (31) 



or, changing X. /jl for new constants, 



f=Ae- hm ^ 2+l - 2+K ' 2 \ (32) 



the well-known law of Maxwell and Boltzmann. 



§ 28. This law gives /=0 when u, v, orw becomes infinite. 

 There is therefore the difficulty that if we divide all possible 

 velocities into " cells " in the manner of § 19, the number of 

 molecules in some of these cells cannot be treated as infinitely 

 great. The difficulty is best met by taking a definite velocity 

 V, such that the molecules of which the velocities clo not 

 satisfy 



ii <V, v<Y, ic<V, .... (33) 



form an infinitesimal fraction of the whole. If the velocities 

 which satisfy (33) can be partitioned into cells in the manner 

 of § 19, so as to satisfy the conditions of § 25, there is no 

 further difficulty, and equation (32) gives the law for velocities 

 which satisfy (33). The law has no meaning for velocities 

 which clo not satisfy (33) . It is obvious, for instance, that 

 the law given by equation (32) does not impose any upper 

 limit whatever on the possible values of 21, v, and w for a 

 single molecule, whereas in point of fact such a limit is 

 definitely imposed by equation (26). 



§ 29. We have investigated the law of partition of the 

 coordinates %, y, z and of the coordinates u, v, w separately. 

 They could have been quite easily investigated together as 

 follows : 



