98 KEPORT — 1891. 



Moreover, if E=T + V is the total energy of tbe system, 



dq=dE + ^{FJp,:) .... (28) 



so that clQ, is the analogue of the quantity of heat communicated to a 

 body. 



Hence equation ('27) is analogous to the Second Law of Thermodynamics 

 as given by equation (2), on the assumption that the kinetic energy T takes 

 the place of the temperature. 



If S is the quantity corresponding to entropy in (27), we have on 

 integration 



S=2(log S(,— log A), where A is a constant. 



This may also be put in the form 



S=logT + log(^j .... (29) 



Here s,,/qi, is of no dimensions in time ; hence Si,lqi, is a function of length 

 only, and the expression for S is exactly analogous to the corresponding 

 formula for a perfect gas — 



S=cJoge + {c,-c,)]ogv + G . . . (30) 



If (ji, is of the nature of angular velocity, so that qj. is of no dimensions 

 in length, s,,t will be of dimensions [L]'^, and therefore Sjjq,, will be of 

 dimensions [L]'^. But v is of dimensions [L]^, hence by comparing the 

 dimensions of the quantities in (29), (30), we must have (c^j — Cj,)/c^=§, 

 .•. c^,^=^Ci„ and this is the relation between the specific heats of a mon- 

 atomic gas. 



23. Helmholtz next considers the more general case in which there 

 are several velocity coordinates q^,, and he investigates the relations con- 

 necting them on the assumption that dQ has an integrating divisor. 

 Writing 



dQ='^qi,dsi^=\d(T .... (81) 



it is evident that the required conditions will be satisfied by assuming 

 that the equation 



dq=0 (.32) 



has an integral of the form 



F(s^)=o-=constant .... (33) 



and that 



8F 

 2^=^97, (^'i> 



The conditions that the kinetic energy should be an integrating divi- 

 sor are also found. If the Lagrangian function has not been modified, 

 Helmholtz finds that the kinetic energy is in every case an integrating 

 divisor of dQ, provided that the geometrical relations between the motions 

 of the various coordinates are ■purely kinematical, or such as could exist 

 in nature. 



24. It has, however, been pointed out by Boltzmann, in his remarks on 

 Helmholtz's paper,i that Helmholtz's proof of this theorem is based on 



1 Boltzuiann, 'Ueberclie Eigenscliaften monocyclischer Systeme,' Crelle, Journal, 

 xcviii. p. 86 ct seq. 



