120 FUNDAMENTAL PRINCIPLES OF MATHEMATICS. 



expresses a particular characteristic of the natural forces in considera- 

 tion not included in the fact that they are conservative forces. The 

 derivation of the value of the kinetic potential from the energy intro- 

 duces arbitrary quantities which are homogeneous functions in the first 

 degree of the velocities. This fact is of significance in that it shows 

 that it is not possible with a complete knowledge of the relations of 

 the energy to the coordinates and the velocities to find the kinetic 

 potential and with it the laws of motion of the system, assuming that 

 the principle of least action is followed. It is necessary in addition to 

 these facts to discover the linear functions of the velocities which cor- 

 respond to the hidden motions. 



After developing some general correlated relations between the 

 forces exerted by a system in different directions, as, for example, the 

 thermodynamic law that if with rising temperature the pressure of a 

 material system increases then compression will cause a rise of tempera- 

 ture, Helmholtz was able to show, at least for a restricted number of 

 coordinates, that, conversely, the principal of least action is applicable 

 when these correlated relations exist. Finally he derived both the 

 total and partial differential equations of motion of Hamilton for the 

 generalized form of the kinetic potential. From them he obtained a 

 series of results for reversible motions of a system; that is, for such 

 motions that the series of positions assumed in a positive motion should 

 be reassumed in a return motion without the action of exterior forces, 

 and with the same time intervals intervening. 



We shall presently discuss further applications of the principle of 

 least action as generalized by Helmholtz, and it need here only be 

 remarked that Hertz discovers another generally valid law at the basis 

 of this principle, which describes the motions of all systems directly. 

 This law asserts that where the connections of a system can be dis- 

 solved for an instant all the masses of which it is composed will part 

 asunder in rectilinear and uniform motions, but when such a dissolu- 

 tion is impossible the system will approximate to these preferred 

 motions as nearly as possible. 



The derivation of the characteristics of motion from the principle of 

 least action involved great mathematical and physical difficulties and 

 led Helmholtz to investigations described in "Studies upon the statics 

 of monocyclic systems" (1884), and "Principles of the statics of mono- 

 cyclic systems" (1884). These mark a distinct advance in the method 

 of treatment of mathematical and physical problems, which has already, 

 in the hands of Boltzmann, reached a commanding place in the theo- 

 retical physics. 



When a system of bodies is affected with motion there is in general 

 a change either in the position of the system in space or else in the 

 condition of the bodies. This, however, is not necessary, as is exem- 

 plified in the passage of a long-continued current of electricity through 

 a wire. In this case the position, the temperature, the magnetic condi- 



