ON OUR KNOWLEDGE OF THERMODYNAMICS. 69 



Wlience they show tliat if the motions all occupy a fixed time, so 

 that the initial and final states are connected by the assumed relation 

 <' — <=:constant, then 



8( p/. • ' • K» g/. • • • <?«' )— 1 , , . (1) 



8(iOi, • • • Pn, 9\, ' ' • 9n) 



14. There is another Jacobian relation similar in form but entirely 

 different in meaning, of which the importance seems to have been hardly 

 fully appreciated, if we may judge from the absence of references to it in 

 most wi-itings on the Kinetic Theory. In order that the Boltzmann- 

 Maxwell Law may be definite, it must be independent of the choice of 

 variables as co-ordinates of the systems considered, and for this it is 

 necessary for the multiple differential 



dpidp^ . . . dp^dq^dq^ . . . dq^ 



to be independent of the particular co-ordinates q^, q^, . • ■ q,i chosen to 

 specify the configuration of the system of each instant of time, provided 

 that Pi, ^)o, . . . p„ are the corresponding momenta. 



The proposition may be stated and proved as follows : — 

 Let<7i,(7o, . . . g-n be any generalised co-ordinates defining a dynamical 

 system,with M degrees of freedom ; ;j 1,^2) • • . p„ the corresponding gene- 

 ralised momenta. Let Qi, Q.2, • • • Q,i be any other set of co-ordinates, 

 Pi, P2, . . . P„ the corresponding momenta. It is required to prove that 

 the relation 



a( Qi,Q.2, • • • Q.„Pi • • • Pn )_i , , . (8) 

 8(5-1,5',, . . . q,„pi . . . x>n) 



liolds good at any instant of the motion. 



Let the new co-ordinates be connected with the old by the relations 



Qi=/i ('Zi. 5'2, • • • ^...O. «^c- 

 Then by differentiation 



.8Q,.. ^8Qo^ _u ^9/1 



whence 



=8i^'''s^"'^+ •••+a^ 



/8Q™\ =8Q^n 



aQ„ 



Again, ^^=0,''since'the equations of relation do not depend on the 

 OPa 

 velocities or momenta. 



From the last relations the terms in a quarter-square of the Jacobian 

 vanish ; and, therefore, the Jacobian 



_3(Qi , • ■ ■ Q ,.) X 9(P. , • • • ^n ) 



8(^1, ... q^) d{pu • ■ ■ Pn) 



