Mechanical Equation to the Motion of Material Points. 323 



The sum 



dV. , rfU. dV. 

 ax ay dz 



must not at once be regarded as the variation of the ergal, and 

 hence, if the signification of U be extended so that it shall re- 

 present the ergal not only for the original, but also for the altered 

 motion, cannot at once be denoted by SU. The preceding equa- 

 tion holds good, namely, as already intimated, also for the cases 

 in which the force operating on the point has undergone a varia- 

 tion which can be supposed to be mathematically expressed 

 thus : — that one or more quantities contained in the ergal, con- 

 stant during one stationary motion, have different values in the 

 two motions. In such a case, of course, in the determination of 

 the variation SU the difference of the constants must be taken 

 into account in addition to the difference of the coordinates. 



But we will now assume that, in the two motions which we 

 have now to compare, such a difference does not occur — the ergal 

 in both being represented by one and the same function of the 

 coordinates, with unaltered constants. In this case the above 

 sum is the complete variation of the ergal and can be denoted 

 by SU ; and accordingly the left-hand side of equation (2) is the 

 mean value of the variation of the ergal, or (which is the same 

 thing) the variation of its mean value, which is represented by 

 SU. For this case, therefore, equation (2) passes into 



StJ^-^S^+jn^Slogi (3) 



To this equation we will now give a somewhat simpler form. 

 We will first transform it in the following manner : — 





= mv 2 (p log v 2 + 8 log i) 



=zmv 2 Slog [i^/v 2 ). 

 For the product here standing under the logarithm, we will 

 introduce a single sign, putting 



\=i\/^. (4) 



Then our equation is transformed into 



8U=mi5*81ogX. ..... (5) 



As the left-hand side of this equation is a variation, the right- 

 hand side must be so too. Hence it follows that mv 2 is a func- 

 tion of X ; and accordingly U must then be also a function of X. 



Y2 



