532 Mr. S. H. Burbiuy on 



sphere m, u, v, ic its component velocities. In Jeans' notation 

 the 6N coordinates and velocities of the N spheres (which if 

 given define the state of the system) are the " coordinates '' 

 (in a wider sense) of a " point " in generalized space of ()N 

 dimensions. That way of regarding the problem may be more 

 advantageous or less so than Boltzmann^s, probably in som.e 

 respects more, in others less. But the difference is only in 

 mathematical method, not in the physical conditions. 



Every physical assumption which is necessary in the one 

 method is equally necessary in the other. 



Of course the conservation of energy is assumed in both 

 cases. But Boltzmann concludes that conservation o£ energy 

 in such a system is not alone sufficient to insure the existence 

 o£ Maxwell's law. We require, he says, in addition a special 

 assumption. And he^ as Jeans and I think, in effect makes 

 assumption A. If that assumption is really necessary to 

 Boltzmann, it is equally necessary to Jeans. But Jeans is at 

 liberty to show that some other assumption is available 

 instead of A. He does not, however, show this^ does not in 

 fact make any assumption except {37 d] . And that by bringing 

 in the infinite rarity of the medium, renders assumption A 

 legitimate in the limiting case. 



9. Notwithstanding that Jeans" consistency is thus pre- 

 served by (37<i), I think he has made assumption A unawares, 

 as great writers have done before him. I will explain my 

 reasons, freely admitting that I may be mistaken. 



Jeans, like Boltzmann, has 6K (in a sense) independent 

 variables, that is, 6^ independent equations are required to 

 define the state of the system. It does not, however, neces- 

 sarily follow that all combinations of the o^ coordinates, given 

 the 3N velocities, are equally probable, or occur with equal 

 frequency (as is asserted in assumption A). Out of a number 

 of such combinations, each of which is possible and sometimes 

 occurs,, some may occur more frequently than others. But if 

 you make no special assumption at all, you do, ipso facto. 

 make all such combinations equally probable. You make, 

 therefore, assumption A. And that Jeans has done by 

 implication. If he is not against assumption A, he is for it. 



10. Again, in art. (27) the number of molecules whose 

 velocities are between u and tt + chi^ v and v + dv, ic and w + dw 

 is /'(?/, v, ic) dii dv div, or fdu dv dw. Then Jeans says the most 

 probable, and therefore the actual, distribution is found by 

 making ^/log/ minimum subject to the conditions 



2/=N, (a) 



and 



2/^(^2 + r2 + 2^^^) = 2E, . . . . . . {h) 



