for a System of Correlated Variables. 17 



Every proof of the theorem in its ordinary form, i. e. with 

 Q = %ms 2 :> either is based necessarily on the assumption, 

 express or implied, that there is no correlation, or else it 

 would prove the absence of correlation as a necessary fact, 

 that is, that no physical system in which the motions of the 

 parts are correlated can exist in stationary motion. 



28. The theorem as above stated fails, or the proof fails, 

 if Q can become negative ; and therefore the proof fails if D, 

 or as the case may be A, becomes negative. It fails also, 

 and this is important, if one of the variables on which Q 

 depends becomes discontinuous, which may happen by the 

 variation of external conditions. 



The Law of Maximum Probability. 



29. Since Q contains products as well as squares of the 

 variables r i ...r n or u l ...u a , we can effectively make e _Q 

 maximum, subject to the constants of the system, and the 

 kinetic energy may be one of such constants. By making 

 £~ Q maximum we obtain the most probable, or normal state 

 of the system, subject to the constants. 



30. Whether we should use r l ...r n , or Ui...u n for indepen- 

 dent variables is, as above stated, a question of convenience. 



ML 2 



The constant kinetic energy has dimensions ■ . It is 



therefore convenient to use the r's, if r has dimensions ^ , and 



ni r 1 



u has T ~; and the us, if u has dimensions ~, . I will assume 

 then that the kinetic energy, or 2nE. 



= m l u l 2 + nuu.y 1 + ... + m n Un* 



and Q=ia 1 u l *+b 12 u 1 u 2 +... + $a 2 u 2 2 + etc. 



The kinetic energy, Swim 2 , if expressed in terms of i\...r n , 

 would be a quadratic function of fi...r«, with coefficients 

 functions of a, /3 of (20). 



31. I assume also that E is either constant, or varies very 

 slowly with the time, while u x ..,u n in general vary very 

 rapidly, so that they may go through cycles of changes 

 while E is sensibly constant. E belongs to the class of 

 variables which Max Planck calls rt langsam veranderKch," 

 while Si...s n belong to the class " schnell veranderlich. v 



Also I assume that the />, or correlation coefficients, are 

 Phil. Mag. Ser. G. Vol. 17. No. 97. Jan. 1909. C 



