ON OUR KNOWLEDGE OF THERMODYNAMICS. 97 



These coordinates are therefore r/i/rosfatic or, as J. J. Thomson calls 

 them, ' speed ' coordinates. The present hypothesis seems to assume that 

 the molecules exert no mutual forces except those due to impact or un- 

 yielding constraints. At any rate, if there be any other molecular forces 

 they can only depend on the controllable coordinates of the system. 



(ii.) When the state of the system is changed the changes take place 

 very slowly, so that the velocities of the controllable coordinates are small, 

 and so also are the acceleratious of the molecular or gj-rostatic coordiuates. 

 (This corresponds to the second assumption in § 11 ) 



21. Let the generalised coordinates of a polycyclic system be denoted 

 by ^, the generalised velocities by (7, the generalised momenta by s, and 

 the generalised force components exerted by the system, in the direction 

 of jj increasing, by P ; also, let the suffix a refer in each case to the con- 

 trollable coordinates, and b to the molecular coordinates of the system. 

 Let T=kinetic energ}-, V= potential encrgj-, H=V— T, so that H is the 

 Lagrangian function with its sign changed. 



The general equations of motion give 



'^ dt' ^ a^ c)(/ 



dt \ dq J dp 



(20) 



la consequence, however, of the assumptions (i.) and (ii.) we have 



= 0, qa=0, S„ = r~=0 . . (21) 



"whence the generalised equations for the polycyclic system become 



" ^^« I . . . . (22) 



^ dt dt LBt'i, J 



Hence if (ZQ is the total energy communicated through the gyrostatic 

 coordinates j,, in time dt, we have 



A^=-^V,q,dt=+^q^:^\lt=^q,d^, . . . (23) 



Also, if the Lagrangian function has not been modified, or if, in other 

 words, no gyrostatic coordinates have been ignored, T is a homogeneous 

 quadratic function of the quantities q,^, and hence in this case 



2T=^q,/, (24) 



22. The simplest form of monocyclic syatem is that containing only 

 one gyrostatic coordinate 5,, ; here 



dq=q,ds, (25) 



Thus q,, is an integrating divisor of dQ, and by § 2 the product of q^ 

 with any function of s,, is also an integrating divisor of dQ. In par- 

 ticular 



2T=g,.., (20) 



.-. '^=2di\ogs,) (27) 



1891. K 



