resulting from Hamilton's Theory of Motion, 277 



the amplitude of the principle of the vis viva. An attempt by 

 Szily* to get the proposition out of Hamilton's treatment of the 

 subject comes nearer to the above notion ; only, not to mention 

 that, on account of a limitation adhering to the form in which it 

 has hitherto appeared, it could not lead to the general deduction 

 required, it does not approach more closely the physical side of 

 this method. 



It resulted from the investigation that the new treatment 

 satisfies a general principle similar to that satisfied by Lagrange's; 

 for perfectly coordinate with the proposition of the vis viva is 

 the following ; — In a motion whose equations of condition and 

 force-function do not explicitly contain the time, let the primary 

 function and expenditure of force respectively be denoted by V 

 and W, so that 



~V= f (T-U)^, W= f*2T<&, 



Jo Jo 



understanding by T the vis viva } and by U the force-function ; 

 and let it be assumed that V may be represented as a function 

 of the initial and final coordinates and the time, W as a function 

 of the initial and final coordinates and the energy; then for 

 every change of motion occurring during an element of time dt 

 the relation 



d(V+W) r 3(Y + W)i 



dt a* J 



holds, in which the symbol d signifies the whole of the alteration 

 which is connected with change of motion, while "d denotes all 

 alterations of V+W not produced by variations of the coordi- 

 nates. Therefore, in every motion whose equations of condition 

 and force-function do not explicitly depend on /, the change of 

 the primary function and force-expenditure produced by the 

 variation of the coordinates alone is =0. The two quantities W 

 and V are here capable of a physical interpretation similar to that 

 of T and U. The former has already been designated by Ha- 

 milton as the vis viva accumulated in the motion ; the significa- 

 tion of the latter results from a peculiarity of the entire Hamil- 

 tonian theory of motion: namely, while la mecanique anahj- 

 tique prefers to introduce the forces into the equations of motion, 

 Hamilton's treatment involves the introduction of the momen- 

 tary impulses — indeed, so that the place of the forces is taken by 

 those impulses which at each instant are capable of producing 

 the velocities actually present. Now, in a group of motions, 

 these impulses can, analogously to the forces, be represented as 

 negative partial differential quotients of a function of the coor- 



* Pogg. Ann. vol. cxlv. p. 295; vol. cxlix. p. 74. Phil. Mag. S. 4. 

 vol. xliii. p. 339 ; vol. xlvi. p. 426. 



