284 Prof. J.J. Miiller on a Mechanical Principle 



the vis viva, which has acquired by far the greatest importance 

 in Lagrange's method. 



For that purpose, the already indicated presupposition is made, 

 that in all motions henceforward to be examined time does not 

 occur explicitly, either in the force-function or in the equations 

 of condition ; so that Hamilton's equation takes the form 



~$V = 2,p i Sq i -2p°8g°-('?^8c k dt--m. ... (4) 

 Jo O c k 



Making use of the well-known substitution given by Euler, and 

 employed also by Hamilton and Jacobi"*, 



V=-W + E/, 

 from which 



SV=-SW+E8*-t-/SE, 



this equation of motion changes into 



Herein 



SW=^<8g f -^yfyJ-f s|2&fc<& + fSE (5) 



Jo V C/i 



n 



W=-V+E/=r(T-U)^+(T + U)^=r(T-U)^ 



a.-'o «^0 



h Jo 



«^o 



and is therefore nothing else but the quantity known under the 

 name of the expenditure of force. It is to be understood as a 

 function of the quantities q. t q®, E, c k ; and the time t, which in 

 the integral in equation (5) remained over, is to be replaced by 

 the equation 



BE~' j 



so that / and E in equations (4) and (5) occupy a perfectly ana- 

 logous position, in such sort that the one quantity may always 

 replace the other. If now the two relations (4) and (5) be com- 

 pared, there comes 



_ r 8 y_ S |I^- g &1 =SW-2|^- |J SE. (6) 



L & k k B* J Bc fc ^E w 



In this equation the variations are still quite undetermined. 

 One of the infinitely many systems of virtual variations will now, 

 under the suppositions made, be the system of the variations 

 which enter with the actual change of motion during the minute 

 portion of time dt. Referred, however, to these actual variations, 



* Compare the general transformations of Lipschitz and Sobering. 



