HAVE ANALOGOUS PROPERTIES. 41 



ensemble of compound systems after the addition of the sup- 

 posed infinitesimal forces will differ infinitesimally from one 

 which would be in statistical equilibrium. 



The case is not so simple when some of the normal types of 

 motion have the same periods. In this case the addition of 

 infinitesimal forces may completely change the normal types 

 of motion. But the sum of the partial energies for all the 

 original types of vibration which have any same period, will 

 be nearly identical (as a function of phase, i. e., of the coordi- 

 nates and momenta,) with the sum of the partial energies for 

 the normal types of vibration which have the same, or nearly 

 the same, period after the addition of the new forces. If, 

 therefore, the partial energies in the indices of the first two 

 ensembles (101) and (102) which relate to types of vibration 

 having the same periods, have the same divisors, the same will 

 be true of the index (103) of the ensemble of compound sys- 

 tems, and the distribution represented will differ infinitesimally 

 from one which would be in statistical equilibrium after the 

 addition of the new forces.* 



The same would be true if in the indices of each of the 

 original ensembles we should substitute for the term or terms 

 relating to any period which does not occur in the other en- 

 semble, any function of the total energy related to that period, 

 subject only to the general limitation expressed by equation 

 (89). But in order that the ensemble of compound systems 

 (with the added forces) shall always be approximately in 

 statistical equilibrium, it is necessary that the indices of the 

 original ensembles should be linear functions of those partial 

 energies which relate to vibrations of periods common to the 

 two ensembles, and that the coefficients of such partial ener- 

 gies should be the same in the two indices.f 



* It is interesting to compare the above relations with the laws respecting 

 the exchange of energy between bodies by radiation, although the phenomena 

 of radiations lie entirely without the scope of the present treatise, in which 

 the discussion is limited to systems of a finite number of degrees of freedom. 



t The above may perhaps be sufficiently illustrated by the simple case 

 where n = 1 in each system. If the periods are different in the two systems, 

 they may be distributed according to any functions of the energies : but if 



