PROFESSOR IN BERLIN 363 



inter-associations, in which the two preceding characteristics of 

 heat motion obtain, then the third essential characteristic 

 of heat as expressed in Carnot's Law, its limited convertibility, 

 holds good also, and in accordance with these conditions 

 Helmholtz develops the corresponding characters of the 

 coupling. 



In conclusion he discusses another and hitherto neglected 

 general law, which affects the character of all associations that 

 can be set up by means of ponderable natural bodies in the 

 case of bodies in motion. Wherever, in the older discussions of 

 mechanical problems, fixed associations are referred to, the 

 expression covers only the inalterability of given spatial 

 measurements, but here it implies fixed relations between 

 velocities. At an earlier point Helmholtz only used for the 

 establishment of the equations of the problems associations 

 which had no influence, so long as the motion was already 

 proceeding in and for itself in a way that corresponded with 

 them, and which accordingly neither performed nor destroyed 

 work ; but in considering the question of the cases in which vis 

 viva becomes the integrating denominator of the compound 

 monocyclic system resulting from the associations he was led to 

 distinguish those cases which he terms pure kinematic associa- 

 tions. He refers this distinction back to still more general 

 considerations, arriving inter alia at the interesting proposition 

 that nq kind of attraction of cyclical motions conceivable 

 between physical bodies can avoid the admission of any desired 

 proportional increase of all the velocities, the relations of these 

 velocities to one another remaining unaltered, so long as the 

 value of all the co-ordinates is constant. The problem of 

 finding analytical expressions for such associations as make 

 a polycyclic system monocyclic had been attacked by Kronecker 

 quite generally from a purely analytical point of view, as an 

 appendix to Helmholtz's first paper. Helmholtz here gives 

 merely the integration of the equations of restriction for any 

 physical system, and compares the results with those of 

 Kronecker. 



The simultaneous preparation of the new edition of Physio- 

 logical Optics brought Helmholtz a little respite from these 

 arduous and exhausting labours. Part I appeared in 1885, Parts 

 II and III in the following year, Part IV in 1887, Part V in 



