HELMHOLTZ IN BERLIN 



potential energy depend on the uncontrollable co- 

 ordinates themselves (that is, co-ordinates incapable 

 of being affected by the action of any external 

 system), but only on the velocities corresponding 

 to them. Further, he assumes that the velocities 

 corresponding to the controllable co-ordinates are 

 always very small in comparison with the others, 

 and further, that the accelerations of the uncontrollable 

 or cyclic velocities are also small. These assumptions 

 imply that no forces depend on the cyclic co-ordinates. 

 Changes in the cyclic velocities may change the energy 

 of the system, but changes in the cyclic co-ordinates 

 cannot do so. 1 Helmholtz also studied the effects of 

 one monocyclic system on a neighbouring one, and of 

 two or more monocyclic systems held together by a 

 common band. The chief interest of these investiga- 

 tions arises from the fact, that thermodynamic pheno- 

 mena (more especially those related to the second law) 

 show the peculiarities of monocyclic systems. One 

 may also say that the value of the work of Helm- 

 holtz in this department is, that his conceptions of 

 monocyclic and polycyclic systems have such general 

 application. 



Helmholtz showed that the principle of least action 

 enunciated in 1744 by Maupertuis, and expanded by 

 William Rowan Hamilton, the inventor of quaternions, 

 is even a greater and more comprehensive generalisa- 



1 For a discussion of these difficult matters, see Gray, Treatite on 

 Electricity and Magnetism, vol. i., chap. 7, p. 180. 



235 



