322 Prof. R. Clausius on the Application of a 



infinitesimally different, stationary motion, which may take place 

 in a different but likewise closed path, with another velocity, and 

 under the influence of another force. In comparing these two 

 motions, we will name the difference between a quantity relative 

 to the original motion and the corresponding quantity relative 

 to the altered motion the variation of the quantity, and denote it 

 by a prefixed 8 ; so that, for instance, Si will be the variation of 

 the time of revolution i. But with respect to the quantities which 

 are variable in the course of every motion, it is necessary to settle 

 which values are to be regarded as corresponding values of the 

 quantity. This may be done in the following manner. We first 

 take, in the two paths^ two places infinitely near each other as 

 corresponding places, and reckon the periods of motion from the 

 moments when the point passes through these places. Then, 

 denoting by t the time of motion till any other place in the path 

 is reached., with the original motion we put 



t-=i(j), 



and with the altered motion we put 



in which (p is a variable which I have named the phase of the 

 motion, and, in both motions, increases from to 1 during a 

 revolution. If, now, <£ has the same value in both these equa- 

 tions, t and t + $t are corresponding values of the times of mo- 

 tion. Thence follow further the corresponding places in the two 

 paths, and the corresponding values of all the other quantities 

 which refer to the two motions. 



From these explanations the following equation (which is the 

 simplest form of the one above mentioned) will be intelligible : — 



-(TL8x + Y$y + Z8z)=j$v* + mv*8\ogi. . . (1) 



When the force which operates on the moving point has an ergal 

 — in other words, when the components of the force can be repre- 

 sented by the partial differential coefficients of a function of the 

 coordinates of the point taken negatively, which may be denoted 

 by U, the equation is transformed into 



* In my above-cited earlier memoir I have investigated under what cir- 

 cumstances the left-hand side of this equation may be valid as the expres- 

 sion of the mechanical work done in the transition from the one stationary 

 motion to the other. Into that question we need not enter here, since, for 

 our present investigations, we have not to trace the passage from one sta- 

 tionary motion to another, but only to compare two given stationary mo- 

 tions differing infinitely little the one from the other. 



