for a System of Correlated Variables, 21 



On the Equipartition of Energy. 

 37. In the first equation of (19) 



Ul = ~jf ri+ jf r ' 2+ &c -' 



multiply both sides by r l5 and take mean values of both sides, 

 remembering that since 1\ . . r n are possible values of Si . . f n , 

 r{ 2 = s^ = a 2 . Similarly r 2 2 = a 2 &c. and T^r q = b pq &c . 

 That gives 



D 



^ D =l 

 D 



Similarly vv^ = 1 and rjWj = ?vh 2 &c. = r n u n . Note that this 

 result would fail if s[, s~ 2 &c. were not zero, for then ?y -^= a x &c. 



Evidently r^=M!^ &c. and r 1 u 1 =r 1 ~^ &c. Whence the 

 law of equipartition of energy takes the general form 





dQ dQ f 



?<j = w 2 = &C. 



«««! ' du 2 



Let us now apply the results of making Q minimum above 

 investigated. We have, since 



- x _ dQ % <ZE <ZQ - rfE 



Q = \nh, -.— = An = , or a- — = \nu -— , 

 ^ «m ate aw an 



for each 1*. Whence, if we may assume that 



dQ </Q 



"1 - 



&c. 



du l ' du 2 



for the normal or most probable state, we have 



dE (IE c 



it l ; = l'-y-, <KC. 



at/] aw a 



And if 2/iE = Swim 2 , m^ii 2 = »? 2 W 2 2 & c « It mav perhaps be 

 objected that the law 



U, 7— = Wo 7 & C " 



a« x aw a 



is proved only when the means are taken over all values of 

 the variables, consistent with the constancy of E, and may 



mo3 



