124 Prof. R. Clausius on a Mechanical Theorem 



In relation to these two magnitudes the following theorem may 

 now be advanced : — 



The mean vis viva of the system is equal to its virial. 



Distinguishing the mean value of a magnitude from its vari- 

 able value by drawing a horizontal line over the formula which 

 represents the latter, we can express our theorem by the follow- 

 ing equation : — 



m-5 



S^te.-tStXtf+Yy+Z*). 



As regards the value of the virial, in the most important of the 

 cases occurring in nature it takes a very simple form. For 

 example, the forces which act upon the points of the mass may 

 be attractions or repulsions which those points exert upon one 

 another, and which are governed by some law of the distance. 

 Let us denote, then, the reciprocal force between two points of 

 the mass, m and m!, at the distance r from each other, by <f>{r), 

 in which an attraction will reckon as a positive, and a repulsion 

 as a negative force ; we thus have, for the reciprocal action : — 



And since for the two other coordinates corresponding equations 

 may be formed, there results 



- \ (X* + Xy 4- Z* + XV + Yy + ZV) = \r<f>{r) . 



Extending this result to the whole system of points, we obtain 



- |2{X# + Yy+lz) = iSrflr), 



in which the sign of summation on the right-hand side of the 

 equation relates to all combinations of the points of the mass in 

 pairs. Thence comes for the virial the expression 



and we immediately recognize the analogy between this expres- 

 sion and that which serves to determine the work accomplished 

 in the motion. Introducing the function <I>(r) with the signi- 

 fication 



we obtain the familiar equation 



- 2 (Xdx + Ydy + Zdz) = dZ®(r) . 

 The sum 2<I>(r) is that which, in the case of attractions and re- 



