358 HERMANN VON HELMHOLTZ 



myself at one time without coming to any right conclusion ; 

 still I am content, for I see that Boltzmann could not get 

 much farther either.' 



Hamilton had replaced Lagrange's equations of motion by 

 a system of total differential equations of the first order, which 

 present the total differential quotients taken according to the 

 time of the free co-ordinates, and a like number of quantities 

 deduced from vis viva (momentum of motion), as the partial 

 differential quotients according to these magnitudes of the 

 energy supply. Helmholtz, for any given form of the kinetic 

 potential, generalizes the form of the corresponding differential 

 equations of Hamilton. He then proceeds to apply the above 

 theory to the laws of reciprocity that govern the changes in the 

 forward and backward motions consequent on small impacts, 

 after the lapse of a certain time. He terms the motion of the 

 system reversible, when the sequence of positions which it has 

 taken up in its forward motion can also be traversed in the 

 opposite direction without the action of other forces, and with 

 the same time-intervals for each pair of similar positions. He 

 thus arrives at reciprocal laws, of which those which he had 

 long before established for sound and light (though only for 

 systems at rest) are merely special cases. Just as the forces 

 of heat had at an earlier period been referred to the concealed 

 motions of tangible masses, and Clerk Maxwell had recognized 

 in electrodynamic forces the action of concealed masses in 

 motion, so Helmholtz now proposed in general to admit the 

 motion and energy of these concealed masses in the treatment of 

 physical problems, since in the invisibilities that lie behind 

 phenomena, he saw only motion and mass that are incapable 

 of being demonstrated to our senses. And thus he selected 

 the law of least action for the expression of the total motion, 

 since this law admits that the mechanical system the internal 

 forces of which can be represented as the differential quotients, 

 independent of time, of the force-functions of the visible co- 

 ordinates of the system is also affected by external forces that 

 are dependent on time, the work of which must be specially 

 calculated, which therefore do not belong to the conservative 

 forces of motion, but are conditioned by other physical processes. 

 Helmholtz had been led to these universal considerations by 

 his investigation of the form of the kinetic potential, as required 



