282 Prof. J. J. Muller on a Mechanical Principle 



quantities dp. j and their variation is to be formed by variating all 

 these quantities simultaneously. Hence the total variation formed 

 under inclusion of the time becomes 



and the question is, to determine the last term -^- • 



ot 



In order to obtain this, let it be remembered that, in the dif- 

 ferentiation according to t, the quantities c k contained in the 

 force-function U have been supposed not to vary. From this it 

 follows that 



dt ~~ "dt dq { dt ' 



and from this we get immediately the partial differential quotient 

 sought 



bt~ dt ^W" U 1+ ^«" 



If we introduce this value into the above equation for 6T, the 

 result is 



Jo Oc k 



-(V-T + lp^Sl. . (3) 



This general equation relative to the variation of motion, which 

 corresponds to the equations 7** and 7 a given by Lipschitz, 

 pp. 122, 123, as well as to Schering's equations [5] and [6], 

 p. 19, containing also the differential equations, is also valid, as 

 soon as a force-function exists, when the force -function and equa- 

 tions of condition explicitly contain the time. For the special case 

 which alone comes into consideration in the following, where the 

 time does not explicitly appear in the force-function and condi- 

 tions, it takes a somewhat simpler form. 



That is, in this case the relation holds, 



T + U=E, 



if E denotes the energy of the system. Hence, if we add and 

 subtract, oi 

 2T, we get 



BV 



subtract, on the right-hand side of the equation for ^— , the value 



ot 



g=E + 2 ft?j -2T. 

 But, with the hypotheses laid down, tk§ vis viva becomes a ho- 



