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



of the proposition of the action when there is no force-function, 

 and consequently no primary function, gives occasion to bring 

 out its position to another well-known equation in mechanics, 

 which relates to the above-mentioned momentary impulses. 



If, namely, the components of the impulses, formed according 

 to the axes of the rectangular coordinates, are 3 HZ, and the ve- 

 locity-components induced by them are riy's 1 , the equation of 

 motion is 



X [(S - mm}) Sx + (II - my 1 ) By + (Z - mat) hz\ = 0. 



If now as a system of virtual variations the actual alterations of 

 the coordinates be introduced, there results 



X{Bx t + Uy , + Zz l )dt=Xm(x h2 + y'* + z ! *)dt=2TdL 



Integrated over the given motion, there comes 



J o Jo 



and from this results, by variation, 



8('z(nrf + 'Ky' + Zz')dt=&\ t 2Tdt. ... (9) 



Jo Jo 



This is the equation which, in the general case assumed, takes 

 the place of the action-equation ; its terms have a similar mecha- 

 nical meaning to that of the terms of the latter. That is to say, 

 the sum of the left-hand side is nothing else but twice the me- 

 chanical work which the sum of the forces constituting an im- 

 pulse perform during the same. The equation, therefore, imme- 

 diately passes into 



2Tdt; (10) 



Jo Jo 



and the action-proposition also can be easily brought into this 

 form; for, according to (7), 



=ap2 



Jo 



which, inserted in its equation, .furnishes immediately the form 

 (10). The difference between the two cases consists only in this, 

 that in the case of a force-function the terms of the equation are 

 functions of the coordinates, in the other case they are not so — 

 relations analogous to which occur likewise with the proposition 

 of the energy* 



§3. 



In order to illustrate the principle found, which represents a 

 characteristic property of the variations of motion of all systems 

 which satisfy the oft-insisted-on conditions, a simple example, 



