122 FUNDAMENTAL PRINCIPLES OF MATHEMATICS. 



system were allowed to proceed so slowly that tlie system is never appre- 

 ciably removed from conditions in which it might continuously remain. 

 The motion of heat is not strictly monocyclic, for each atom probably 

 continually alternates in the direction of its motion, and only in that an 

 enormous number of atoms represent all stages of motion is the 

 mechanical character of a monocyclic motion simulated. 



Helmholtz raised the question under what general conditions the 

 known physical characteristics of heat motion could be produced by 

 other known classes of motions, and whether there is any special class of 

 motions understood to be mechanical for which there exist restrictions 

 to the transformation of work equivalents similar to the second law of 

 thermodynamics. He extended the definition of a monocyclic system 

 so as to include, besides those containing only one cyclic coordinate, 

 others in which several such coordinates appear, but all but one of 

 these functions are of another order of magnitude from the one under 

 consideration. The very important and interesting case was discussed 

 in which certain mechanical means are made use of to correlate the 

 velocities in two monocyclic systems, these devices being of such a 

 nature as to exert no influence when the motion proceeds with regu- 

 larity, as desired, but which oppose with appropriate force any devia- 

 tions from regularity. Helmholtz called a system of this sort, as for 

 example two tops whose axes are so connected that they are forced to 

 rotate with equal velocity, fettered, and the condition, the coupling of 

 the system. He recognized in the device of coupling the only means 

 of acting directly on the inner motion of monocyclic systems. Thus 

 in heat motion we are debarred from influencing particular isolated 

 atoms, and are forced to act without distinction on all contained within 

 a given space. When, now, two originally independent monocyclic sys- 

 tems are by suitable regulation of exterior forces caused to assume a 

 state corresponding to the conditions of a rigid connection, such a 

 rigid connection may be inserted without disturbing the motion in 

 progress, which continues in future as restrained by this linkage. In 

 a similar way two bodies at equal temperature may be placed in con- 

 tact without altering their inner motions, so that they retain equal 

 temperatures while slow changes of temperature are made, and the 

 equality is not prejudiced by pressure or action at a distance between 

 the two systems upon coming together. 



With the help of mathematical considerations quite analogous to 

 those employed in thermodynamical investigations, Helmholtz showed 

 in general that when monocyclic systems admit only of such mutual 

 connections that the external forces of each separate system depend 

 only on the momentary condition of the system, and not upon the 

 beginning or ceasing of connection with other systems, then the 

 coupling is a pure coupling of motion, and creates a new monocyclic 

 system. When upon the beginning of interchange of internal motions 

 between two or more systems the equilibrium of internal motion 



