75-77] Equipartition of Energy 67 



and two similar equations with zu sew and xv yu respectively instead of 



yw zv. 



Instead of being given by equations (145) the minimum value of ^ is now 

 given by a system of equations of the form 



1 + log/a + \E a + fj, a + S^raaW + 2#i m a (yw zu) = 0) 



...... (149), 



1 + log f ft + \Ep + fj.ft + ^p 1 m^u + ^q^mp (yw zu} = O 



in which Pi,p 2 ,p3, (?i> ?2> #3 are new undetermined multipliers derived from the 

 six equations of which (147) and (148) are typical. In the case in which any 

 of the six momenta are not known to be constant, the corresponding multi- 

 pliers must be put equal to zero in equations (149). 



The solution (146) must now be replaced by the more general solution 



, 

 etc.) .............. 



in which the unknown quantities p lt p 2 , p 3 , q 1} q 2 , q 3 are to be deduced from 

 the corresponding momenta when these remain constant, and put equal to 

 zero when they do not remain constant. Obviously the solution expressed by 

 equation (146) is a special case of that expressed by (150). 



This generalisation of equations (146) is, however, of abstract interest 

 only. In future we shall suppose the law of distribution to be expressed by 

 equations (146), assuming in so doing that the momenta are zero, so that the 

 gas as a whole has neither translational nor rotational motion. 



The Partition of Energy. 



77. Of the 2n a coordinates (136) of which E a is a function, let , 2 ---n 

 be positional coordinates, and n+1 ... %. 2n be momenta, the suffix a now being 

 omitted, as we are only going to consider one type of molecule. In general 

 E a will consist of two parts, the kinetic energy L which will be a quadratic 

 function of w +i... 2 n w ^ tn coefficients involving ^...^ n , and secondly the 

 potential energy V which will depend only on f, ... n . We therefore write 



where 2 = a n 2 , l+1 + 2o w n+1 n+a + 



It is known that by a linear substitution of the form 



(152) 



. , etc.) 

 the function L may be reduced to a sum of squares of the form 



52 



