Law of Distribution of Energy. 153 



described about the centre of disturbance varies as -, — ™, or 

 jL 47rR 27 



varies as ^, if N=47rR 2 and the coordinates #, of which 



. . 1 



that energy is a quadratic function, vary as —p=- If, now, we 



may further assume that any number of such waves may 

 coexist, so that the disturbance x at any point at any instant 

 is the algebraic sum of the disturbances due to all the waves 

 which are passing the point at that instant, the redistribution 

 of S will go on in the physical system in exactly the same 

 way as in our analytical system — and with the same result, 



that the distribution denoted by Ce 2T will be permanent. 

 The disturbances x x . . . x ? in any system will be respectively 



_ ,— times the sums of the values of the corresponding 

 disturbances in systems distant E from the one in question, 

 as they were t seconds ago, -• being the velocity of wave- 

 motion. The potential energy, %, of the internal forces of a 

 system we suppose not to be capable of transmission in that 

 form. But if the law of distribution be e~ h(x+s \ we know 

 that this distribution will not be affected by the internal 

 forces ; neither will it be affected, on the above hypothesis, by 

 the transmission of S. It will therefore be permanent. 



But if we are not allowed to assume the coexistence of 

 waves in the sense above stated, the law cannot be permanent, 

 except in the known case of binary encounters. 



Paet III. 



(13) Our investigation hitherto has been based on the 

 assumption that the N associations are independent of one 

 another ; that is, that the chance of the variables x± . . . x n in one 

 association having assigned values is independent of the 

 values of a± . . . x n f the variables in another association, or, if 

 / (x) be the chance of any one of the first set having assigned 



value, , , • = for each a/. The condition of complete 

 ax 



independence is that the expression for the energy contains 



no term of the form bxx f , where x and x' belong to different 



associations. This condition may not be satisfied if our 



associations are material systems very near to one another. 



It is necessary to replace the stringent condition by one 



which can be satisfied more easily, as follows : — In the first 



place it will be shown that our method is applicable even 



