PROFESSOR IN BERLIN 361 



several of the external forces that act on the system are persis- 

 tently nil, he succeeded, as was indicated above, by the elimina- 

 tion of single co-ordinates, in finding other exact equations for 

 the remainder according to Lagrange's formula, and names the 

 system resulting from the elimination of those co-ordinates the 

 imperfect, in opposition to the original perfect system. These 

 investigations were then specialized for the general case of 

 monocyclic motion, in which several velocities are present, 

 which, however, all depend upon one of the same. Helmholtz 

 imagines fixed associations acting in such a way that they have 

 no influence on such motions as take place of themselves under 

 the play of the effective forces, in correspondence with the 

 equations of the combination, but that they oppose to any 

 incipient deviations such forces as are necessary in order to 

 check this deviation; the forces exerted by fixed associations 

 contribute no work to that done by the forces acting from 

 without. He terms the system after the introduction of these 

 fixed associations, the restricted system. Two equations (ana- 

 logous to the two relations of Carnot-Clausius in the Theory 

 of Heat) are derived from the relation that the work applied to 

 the acceleration of motion in the restricted system is equal to 

 the sum of work expended for the same alterations of velocity 

 in the unrestricted system. The first of these states that the 

 heat that enters the system during a vanishingly small alteration 

 of the absolute temperature and the parameter, as measured 

 by its work-equivalent, is equal to the increase of total energy 

 and of freely convertible work not transformed into heat, which 

 the system gives off externally on alteration of the parameter, 

 provided that alteration of the temperature without alteration of 

 the parameter induces no intake or output of any form of work 

 other than that of heat : the second finds this quantity equal 

 to the product of the temperature and the increment of a 

 quantity which Clausius had called entropy, while Helmholtz 

 terms the factor which here is temperature, or a function of it, 

 the integrating denominator. 



Precisely the same relations obtain for the monocyclic 

 systems, with all the correlative inferences as to limited con- 

 vertibility. Since in the Theory of Heat, temperature, which 

 represents the integrating denominator, is (in accordance with 

 the kinetic theory of gases) proportional to the vis viva of the 



