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



there occur only the variations $g., Sq ! ., Sc k , and we have 



By partial integration in the second part of the right-hand 

 side there hence results, if the values of the various quantities 

 for the time /=0 be denoted by the index 0, 



and if we put the differential quotients of the vis viva, taken ac- 

 cording to q ! ., 



3T _ bT°_ 



-bq'r Pi '~dq'\~ Pi3 



according as they are referred to the time t or to the initial time 

 0, we get 



. $ v, w „-x, W -f(4*-M^.] s , i)a 



-J>f>)- ; • m 



This is an equation of motion of the most general kind, similar 

 to one to which prominence is given by Jacobi* and to another 

 by Scheringt ; but it has the peculiarity that the quantities c ht 

 not contained in the latter equations, occur in general in the 

 force- function likewise. 



All the quantities in equation (1) are presumed to be functions 

 of t and 2/x + v arbitrary constants, of which the first 2juu have 

 arisen from the integration of the differential equations of the 

 motion. The quantities g., #? can now, by means of the integral 

 equations, be expressed by the arbitrary constants and t ; but 

 by the same integral equations the 2/j, arbitrary constants can 

 also be represented by the quantities q., #?, and t. Let the latter 

 be presupposed. Then V becomes a function of / and 2jjl quan- 

 tities q P q\ ; but it contains in addition the arbitrary constants c k 



* Vorlesungen iiber Dynamik, pp. 143, 356. 

 t Hamilton- Jacobi' 'sche Theorie, p. 19. 



