FUNDAMENTAL PRINCIPLES OF MATHEMATICS. 121 



tion ±n neighboring masses of iron, all remain at every point unaltered. 

 Hence the motion which we regard as the cause of the phenomenon 

 must in a sense be so far stationary that as soon as a particle leaves its 

 place there must, within an infinitesimal time, be substituted another 

 moving in the same direction with the same velocity, so that, in spite 

 of the continual motion, there is at no point in space any apparent 

 change. Helmholtz designated motions, such, for example, as the 

 motion of a rotating top or that of a frictionless liquid in a circular 

 canal, as cyclic. When all the motions of a system of bodies are cyclic, 

 the system is said to be a cyclic system. Cyclic motions are generally 

 hidden motions, since they do not alone cause a change in the appear- 

 ance of the distribution of masses, and, conversely, hidden motions are 

 usually cyclic. A coordinate is called cyclic when the whole condition 

 of the system is not altered by changes in this coordinate. Since the 

 kinetic energy of the system remains unchanged, it is not a function of 

 a cyclic coordinate, but in general its differential is, since the kinetic 

 energy is greater the faster the cyclic motion progresses. The condi- 

 tion of a system may be determined through other than cyclic coor- 

 dinates, which Helmholtz called the slowly varying coordinates or 

 parameters. These change so slowly that their differential coefficients 

 with respect to time may be neglected, and the kinetic energy is there- 

 fore a function of the parameters, but not of their differential coefficients. 

 When the parameters remain constant for a long period of time the 

 motion taking place during the interval is cyclic. Systems are classi- 

 fied, according to the number of their cyclic coordinates, as monocyclic, 

 etc., and are in general polycyclic. 



The condition for the existence of a cyclic system can be fulfilled 

 with any degree of approximation whenever the system possesses 

 chiefly cyclic coordinates, provided the parts of the energy which are 

 due to the velocity of change in the parameters are small in compari- 

 son with the parts which depend on the cyclic intensity, or in other 

 words, provided the velocity of change of the parameters is negligible 

 compared with that of the cyclic coordinates. The forces of a cyclic 

 system are by definition independent of the velocity of change of its 

 parameters, as follows immediately also from the Lagrange expression 

 through the kinetic energy. It follows also that when no forces operate 

 on the cyclic coordinates of a cyclic system the whole cyclic move- 

 ment of the system, determined by the product of the mass by the veloc- 

 ity, is constant. In this case the motion is defined as adiabatic. The 

 motions characterized by Helmholtz are defined in accordance with 

 their properties as such that the potential and actual energy of the 

 system are independent of a certain number of coordinates which 

 would be necessary to completely determine the position of the system, 

 but which are represented in the values of the energy only by their 

 differential coefficients with reference to the time. This would also be 

 the case with motions not strictly stationary when the changes in the 



