16 Mr. S. B. McLaren on Hamilton's Equations and the 



applied. It may be, however, that there is true material 

 energy as well as electromagnetic ; and then Maxwell's 

 reasoning can be resumed, with its old paradoxical con- 

 clusion of equipartition. 



To many it has always seemed that the method of sta- 

 tistics builds much, and to little purpose, on a very unsure 

 foundation. 



I recall its postulates before deducing from them the 

 results just described. 



§ 2. The General Dynamical Method. 



(1) The laws of heat are dynamical. 



This is fundamental and at once raises the question of 

 reversibility. Fire may freeze a kettle instead of boiling it, 

 only the law of chances favours the more familiar process 

 (Jeans). Let there be a dynamical system with a very 

 large number of coordinates. The equations ot motion are 

 in Hamilton's form : 



<hr_an dp r ___dH 



dt - d P ; dt~ d q ; i r -^ ■•••»;. 



The p's represent momenta ; the q's are coordinates of 

 position. 



Suppose the initial values are regarded as mere matter of 

 chance. Then it may be shown (Jeans' ' Dynamical Theory 

 of Gases ') that in the vast majority of cases the distribution 

 of energy and momentum is near what may be called the 

 temperature distribution. It is therefore very long odds 

 that, if we start with any arbitrary heat distribution, the end 

 will be equality of temperature. With this we are brought 

 naturally to the second assumption. 



(2) Any set of values of the coordinates and momenta is 

 possible provided it is consistent with the constancy of 

 energy. 



(3) All these configurations are of equal probability. 

 For what is possible by (2) must have attached to it a 



definite measure of probability. The successive configura- 

 tions assumed by any one system are equally probable, and 

 so far as we know they have only one thing in common — 

 the fact that their energy is the same. Hence (3), and 

 hence without further assumption the law of equipartition. 

 The only condition to be satisfied by R has already been 

 stated. 



The truth of (2) is certainly not at all obvious. When 

 the number n is finite it may indeed be allowed very 



