946 Prof. J. H. Jeans on Non-Newtonian Mechanical 



of the calculus to be legitimate we are not compelled to choose 

 any one particular basis Bor the calculation of probabilities. 

 We may select any basiswe please, and the use of the calculus 

 of probabilities will be legitimate provided we retain the same 

 basis throughout the whole investigation. 



In the present investigation we shall agree to say that the 

 probabilities of a system being in states A or B are in the 

 ratio of W A to W B if the regions of the generalized space 

 occupied by points representing systems in states A or B 

 are in the ratio of \\ A to W r ,. Or, in simpler language, in 

 estimating probabilities, we think of the system as being 

 selected at random from all systems in the generalized space, 

 equal volumes of the space having equal chances of selection. 

 This way of estimating probability leads at once, as we shall 

 gee, to Boltzmann'a relation between entropy and probability. 



Lei the points representing systems in different states 

 A, B, C ... occupy regions which are in the ratios 

 \\" \ : Wb : Wo... • Then, if a system is selected at random, 

 the probabilities oE its possessing characteristics A, B, C ... 

 are in the ratio- W \ : Wb : W c ... • From the steadinessof 

 the bydrodynamical motion, it also Follows that if the system 

 is selected at random and allowed to follow its natural motion 

 for any time /, the probabilities of its possessing characteris- 

 tics A. B, 1 1 ... at the end of this time will be in the ratios 

 \V A : \V|. : \\ c •• • And if the system is not initially 

 selected at random, but starts from a known state, and moves 

 for an indefinite time under its laws of motion, the probability 

 of its possessing characteristics A, B, C ... at the end of this 

 time will iti genera] also be in the ratio W A : Wb "• Wc .... 

 But this requires obvious modifications if the system is so 

 started that at the end of infinite time it must inevitably have 

 characteristics X, Y, Z ... . The statement is then only true 

 if Wa : Wb : Wc ... measures the ratio of those parts of the 

 space in which the characteristics AXYZ, BXYZ, CXYZ, ... 

 obtain. 



Let A, B, C ... now be characteristics of different parts 

 of the system, such that the co-ordinates involved in the 

 specification of any one characteristic are not involved in any 

 of the others. Then the whole system may possess two or 

 more of the characteristics simultaneously, and the probability 

 that it possesses them all is of the form 



W = KW A W B W c (2) 



where K is a constant. The value of W is obtained by pure 

 multiplication of Wa, Wb, Wc . • . because the co-ordinates 

 are orthogonal ; it is in no way necessary to suppose that 



