110 Lord Rayleigh on the Law of 



error in assuming that by choosing suitable coordinates the 

 expression for the vis viva could always be made to contain 

 only the squares of the momenta/'' This is precisely the 

 objection which I supposed myself to have already answered 

 in 1892. I wrote, " It seems to be overlooked that Maxwell 

 is limiting his attention to systems in a given configuration, 

 and that no dynamics is founded upon the reduced expression 

 for T. The reduction can be effected in an intinite number 

 of ways. We may imagine the configuration in question 

 rendered one of stable equilibrium by the introduction of 

 suitable forces proportional to displacements. The principal 

 modes of isochronous vibration thus resulting will serve the 

 required purpose." 



It is possible, therefore, so to choose the coordinates that for 

 a given configuration (and for configurations differing infi- 

 nitely little therefrom) the kinetic energy T, which is always 

 a quadratic function of the velocities, shall reduce to a sum of 

 squares with, if we please, given coefficients. Thus in the 

 given configuration 



T=i? l 2 +i^-H-.-+i?„ 2 ; .... (3i) 



and, since in general p — dT/dq, 



Pl='l, P2 = <j2, &C., 



so that 



T=ii> 1 2 + i/>/ + ...+^ n 2 (35) 



Whsther the coordinates required to effect a similar re- 

 duction for other configurations are the same is a question 

 with which we are not concerned. 



The mean value of p r 2 for all the systems in the given con- 

 figuration is, according to (38), 



jV ■ F{ V + jr/V + . . . + \ Pr ?)dp l . . . dp n ^ (36 ^ 

 $F{V + ^ + ....+ip n *\d Pl ...dp n 



The limits for each variable may be supposed to be + oc ; 

 but the large values do not really enter if we suppose F(E) to 

 be finite for moderate, perhaps for nearly definite, values of 

 E only. 



It is now evident that the mean value is the same for all the 

 momenta p; and accordingly that for each the mean value of 

 \p* is 1/n of the mean value of T. This result holds good for 

 every moment of time, for every configuration, for every 

 value of E, and for every system of resolution (of which there 



* Confer Bryan, loc. cit. 



