J. 



of the Mechanical Theory of Heat. 175 



dx 

 m of the point) the product m-j- 8x, in which 8x, as before, sig- 

 nifies the difference between a value of x in the original path 

 and the value of x at the corresponding place in the altered path. 

 This product changes its value periodically during the motion 

 of the point, so that at the expiration of every time of revolution 

 i it returns to its former value. We can hence construct the 

 following equation : — 



-rr (m — 8x)dt = 0. 

 dt \ dt ) 



This equation can be further simplified when we consider not 



merely one material point, but an entire group which move 



equally and hence have the same time of revolution i. The 



dx 

 quantity m-j-8x changes its value according to the phase in 



which the point is. As, however, at a fixed time the points 

 belonging to the group have different phases, and the number of 

 the points constituting the group is so great that at every time 

 all the phases may be considered to be proportionately repre- 

 sented, the value of the sum 



v dx ^ 

 Zm -77 ox, 

 dt 



referred to all these points, will not perceptibly vary. The same 

 holds good for every other group of points of like kind and with 

 equal motion; and hence we can at once refer the preceding sum 

 to all the points of our system and likewise regard as constant 

 the sum so completed. We thus obtain the equation 



i s -S & =° ^ 



We will now carry out the differentiation herein indicated : 



d v dx <. v d^x ~ „ dx d(8x) ,_ _ x 



df lm dl Sx= * m W Sx+ * m W-dt- ■ (26) 



d(8x) 

 In the expression -^— , in which the quantity x is first va- 



riated, and then differentiated according to /, the order of these 

 two operations must not be reversed. Probably, however, this 

 may be done when the differentiation refers not to the time t, but 

 to the phase (j>. Hence we form the following equation, 



d(8x) __ d(8x) dcf> 

 dt dcj> dt 



or by replacing (in accordance with equation 5) the differential 



