Probabilities to the Movement of Gas- Molecules. 253 



U and u from (1) we have Ma = wiA, the required second 

 equation. Thus 



1 .MlP + wm 2 , a . 



exp.-i fr|n . ... (6) 



~ 2,-LTJ F " 2 [T] 



That is, supposing U and u to be measured from the centre 

 of gravity. Otherwise it is proper to substitute for U, 

 (U) — v and for u, (u) — v, where v is the velocity of the 

 centre of gravity supposed constant, and (U), (u) are the 

 velocities measured (as v is) from a fixed point. 



II. A second proof is based on the " H " theorem dis- 

 covered by Boltzmann. " H " may be described as an 

 44 index, " meaning a quantity which by its lnagnilude 

 indicates, but does not measure, the magnitude of some 

 other quantity *. An index may act either positively by 

 always increasing with the increase and decreasing with 

 the decrease of the thing indicated, or negatively by always 

 varying in the direction opposite to that which it indicates. 

 " H" is an index of the latter kind. It is a negative index 

 of a certain probability which may be thus described: — 

 Let f(;v) f be the frequency-function which represents the 

 ultimate distribution of a set of observations numbering n, 

 a large but finite number ; that is, the distribution which 

 would be presented by averaging an indefinitely great 

 number of sets each consisting of n observations, the out- 

 come of unaltered conditions. Let P be the probability 

 that a random specimen of a group numbering- n would 

 diverge from f{x) to the extent at least that /i(#), any 

 particular observed or assigned distribution of n obser- 

 vations, does diverge therefrom. Then " H " is an index 

 of P ; decreasing with the increase of P, and conversely. 

 For u H " is — or differs only by added or multiplied 

 constants J from — the negative of the logarithm of a certain 

 probability <f> which is a positive index of P §. 



The simplest example of " H" occurs when / degrades to 

 a pair of frequencies such as the number of black balls and 

 the number of white balls drawn in the course of n trials 

 from a medley of black and white mixed in proportions p:q. 

 The ultimate frequencies — those which would be presented 



* Compare the use of the term " index-number " in Statistics (e. g. by 

 Bowley in his ' Elements of Statistics ' ). 



■\ Mutatis mutandis when the variation takes place in more than one 

 dimension. 



X Accordingly, expressions for " PI " differing by such constants are 

 given by different writers. 



§ The index of an index is itself an index. 



