918 Prof. J. H. Jeans on Non-Newtonian Mechanical 

 Hence 



d S / d 1 W ^ /i in ^W /7 . 



SE~ =/ ' 9 K 1 1 °'"^A= Ei a'»-l)=2 E? • (7) 



since m may be supposed very great. 



If T is the absolute temperature of the gas, and R the gas 

 constant, the value of E x is JmRT, so that the value of 

 5S/BE] becomes */RT. 



If k is taken to be identical with R, then equation (5) 

 becomes 



bs as _ i 



bk, _ si;, t • • • 



(8) 



giving the second law of thermodynamics. 



7. This method of procedure shows the second law of 

 thermodynamics to be more general than any system of 

 dynamical laws ; the same can at once be shown to be true 

 of the theorem of equipartition of energy. For suppose that 

 any other pair of the energy, Bay E 2 , can be expressed in the 

 form given by equation (6), the summation now extending to 

 n terms. The value of AY B can be calculated in the same 

 way as \\\, and, just as in equation (7), we have 



3S Rn 



3E S 2E a 



since / Is now identical with R. Since, by equation (8), 

 3S/dE a must be equal to 1/T, it follows that 



E s =inRT, 



expressing the law of equipartition of energy. The same 

 result is clearly true if E 2 is any quadratic function of n of 

 the co-ordinates. Moreover, if E 2 is a linear, cubic, bi- 

 quadratic or any other homogeneous function of the co- 

 ordinates, the result is still true in a modified sense, provided 

 that E 2 is necessarily always positive. For if 



E 2 =/ 8 (P„ P 2 , . . . PJ, 



a homogeneous function of degree s, then "W B is the volume 

 of the generalized space included between the surfaces 



/ s (P 1 ,P 2 ,...P„)=E 2 ±i6 2 . 

 This is of the form cE 2 



1 BS 



T ~~ BE 2 



\7 "~ JE 2 ~ sE 2 ' 



