Prof. R. Clausius on different Forms of the Virial. 9 



whence results 



iV?',= -?X+%^ (37) 



Herein, for p' v , we can put an expression to be obtained from 

 (35). For this purpose we will write (35) in the following 

 form : — 



sf^^'A • • • < 38 > 



If now the variables q v <? 2 , q 3 , &c. are each independent of the 

 others, their variations are also independent of each other, and 

 the equation which holds for the sum of all the terms must also 

 hold for each term singly ; we consequently obtain 



dV_dT 



d <l v "" d( U 

 or 



77 ~P' 



, = gT_ff = rf(T-U) 



If we insert this expression for p' v in equation (37), after mul- 

 tiplying it by \ y we get 



1 . W(U-T) ld(pq) 



and when we form the sum of all the equations of this kind, we 

 obtain, in accordance with (36), 



These equations (40) and (41) are two new equations represent- 

 ing generalizations of equations (1) and (6). 



By forming the mean values, new forms of virial-expressions 

 can be deduced from them. In the first place, the expression 

 for the total virial resulting from the last equation is : — 



l y m^rr \difq 



In regard to the last term in this expression a special remark 

 must be made. The variables q v q 2 , q 3 , . . . serve for the deter- 

 mination of the positions of the movable points; and, con- 

 versely, the values of the variables can be determined from the 

 positions of the points. This latter determination, however, 

 may take place in two ways. It may have but one meaning — 

 which is the case for right-line coordinates, the distances of the 

 movable points from one another or from fixed points or the 

 centre of gravity, and for the trigonometrical functions of the 



