178 Prof. R. Clausius on the Second Axiom 



supposition and only retain the assumption that the motion is 

 stationary. 



Since with motions which need not be in closed paths the no- 

 tion of time of revolution is, in the literal sense, no longer appli- 

 cable, the question arises whether another, corresponding, can be 

 put in its place. 



For this purpose let us first consider only those components of 

 the motions which refer to one determined direction — for example, 

 the components in the x direction of our system of coordinates. 

 We have then to do simply with motions alternately to the posi- 

 tive and the negative side; and if, in particular, in relation to 

 elongation, velocity, and duration manifold varieties occur, there 

 yet is in the notion of a stationary motion the prevalence of a 

 certain uniformity, on the whole, in the way that the same states 

 of motion are repeated. Accordingly it must be possible to ex- 

 hibit a mean value for the intervals of time within which the 

 repetitions take place with each group of points that are alike in 

 their motions. If we denote this mean duration of a period of 

 motion by i, we can unhesitatingly assume as valid also for the 

 motion now considered equation (28), namely 



Corresponding equations can be formed also for the y and z 

 directions; and in fact we will assume that the motions in the 

 various directions of the coordinates so far agree that, in each 

 group of points, we may ascribe a common value to the quantity 

 8 log i for all three coordinate-directions. By then treating the 

 three equations so formed as we have equations (28), (28a), and 

 (28 b) we arrive again at equation (31): — 



BL=^S{v Q )+%mv^\ogi. 



14. In the further treatment of this equation a difficulty 

 occurs, because in the different groups of points both the velocity 

 v and the duration of a motion-period i may vary, and hence 

 these two quantities occurring under the last sign of summation 

 cannot at once be separated. Nevertheless if we take advantage 

 of a near-lying supposition, the separation becomes possible, and 

 we thereby arrive at a very simple form of the equation. 



As the various material points belonging to our system reci- 

 procally operate upon each other, the vis viva of one group of 

 points cannot be altered while the vis viva of the others remains 

 unaltered ; but the alteration of the one supposes the alteration 

 of the others, because a certain equilibrium between ths vires vivce 

 of the various points must be restored before the new state can 

 remain stationary. We will now suppose that, in the motion 



