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



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Jo V dft 3 ft 9?.-/ 



On the other hand, if u and « denote the components (formed 

 according to the coordinate q { ) of the velocity before and after 

 the impulse P 2 , the mechanical work done by the forces during 

 the impulse is 



u + v 



If now a system of /x impulse-components P t - be presupposed, 

 which shall bring forth the whole of the velocities q\ from the 

 state of rest of the system, so that the velocities preceding them 

 are all =0, but the velocities after them those q ! i, then the me- 

 chanical work done by the forces during the impulse becomes 



But the equivalent vis viva 



therefore 



and 



_|i.p,.. ...... w 



that is, the negative partial differential quotients of the primary 

 function, formed according to the coordinates, represent the 

 components of the momentary impulse which is capable of bring- 

 ing about the velocity existing at the time in question. 

 If, from («), the value of P* be inserted in 



there results further 

 and 



dt 



Bfc dt 



~*^i*rfO*l 03) 



that is, the negative partial differential of the primary function, 

 formed according to the whole of the coordinates, represents the 

 product of the time- element dt into twice the mechanical work 

 done by the impulses which bring forth from the state of rest 

 the actual velocities q f im 



If, finally, the value of this partial differential, from (/3), be 



