ON OUR KNOWLEDGE OF THERMODYNAMICS. 99 



the assumption that dQ, has an integrating divisor ; or, in other words, 

 that the solution of the equation 



dQ=0 



can be expressed in the form of a single primitive. Under such circum- 

 stances, the proof shows that the kinetic energy of the system must 

 necessarily be one of the integrating divisors of dQ. But, on the other 

 hand, there may be cases in which the equation cZQ.=0 does not possess 

 a solution in the form of a single primitive, and Helmholtz's investiga- 

 tions are not applicable to such cases. 



In fact the theory of differential equations shows that the equation 



dQ='Sq,,ds^=0 



does not in general lead to a single primitive of the form (33) 



r(S(,)= constant. 



In order to obtain an integral of (32) it is therefore in general necessary 

 to assume certain functional relations between the variables. In other 

 words, we must assume the existence of certain geometrical equations 

 connecting the different parts of the system, and this is equivalent to 

 imposing certain constraints whereby the number of degrees of freedom 

 of the system is reduced. Helmholtz finds that the kinetic energy T 

 will be an integrating divisor of dQ, provided that the assumed geometri- 

 cal equations are purely kinematical, and in this category are included all 

 forms of constraint which are possible in a perfectly conservative dyna- 

 mical system. 



There are, however, as Helmholtz has shown, certain cases in which 

 (32) has for its integral a single primitive of the form (33), and in these 

 cases it is not necessary to assume the existence of geometrical equations 

 representing constraints on the system. Such a polycyclic system possesses 

 properties identical with those of a monocyclic system, and, although the 

 gyrostatic coordinates are independent, the kinetic energy is always an 

 integrating divisor of dQ. 



It is probable that Helmholtz's geometrical equations can be interpreted 

 thermodynamically as the conditions that the different parts of the body 

 may be all at the same temperature. Unless this condition is satisfied we 

 know from purely physical considerations that dQ has not in general an 

 integrating divisor. 



25. The limitations, as well as the meaiaing of ' purely kinematical ' 

 geometrical conditions, are, however, more clearly shown in Helm- 

 holtz's second paper,' in which he deduces the analogue of the Second 

 Law by means of an application of the principle of similitude, as follows : 

 The geometrical conditions are considered purely kinematical when they 

 allow the rate at which the system is moving to be varied without vary- 

 ing the relations between the coordinates of tlie various parts. Thus 

 corresponding to any state of motion of the system we may obtain another 

 possible state of motion of the system by supposing all the velocities of 

 the system increased n fold, provided that proportional alterations be 

 made in the external forces (P) of the system. In the new motion the 



' Crellf, Journal, vol. scvii. pp. 317-322. 



H 2 



