8 Prof. It. Clausius on different Forms of the Virial. 



It scarcely needs to be mentioned that in equations (8), (14), 

 (15), (19), (20), and (25), as well as in the earlier corresponding 

 equations, with the formation of mean values the last term 

 (which is a differential coefficient according to time) drops oat, 

 and the terms then remaining on the right-hand side are forms 

 for virials, the special signification of which is readily seen in the 

 individual cases. 



6. Having thus far been occupied in introducing special quan- 

 tities of various kinds for the determination of the virial, we will 

 finally derive some equations which, in relation to the variables 

 to be employed, are perfectly general. 



Given any variables serving to determine the positions of the 

 points, and denoted by q {) q 2 , q 3 , &c, then the coordinates of 

 the points, and all the quantities determined by them, are to be 

 regarded as functions of these general variables. The velocities, 

 and the quantities determined by them, can accordingly be re- 

 presented as functions of these variables and of their coefficients 

 of differentials according to time. Let us now assume that the 

 forces acting in our system have a force-function or ergal U, we 

 can treat this as a function of q v q^ q 3 , &c, and at the same 

 time the vis viva T of the system as a function of q ]} q^ q B , &c. 

 and g' v q' q> q'^ &c. Between these two functions there subsists, 

 according to Lagrange, the following equation, 



in which the sum refers to the variations of all the variables 

 Qu q 2 , q s , &c If, for abbreviation, we introduce the symbols 

 P\> P 2 > P$> &c v and put 



dT 



*~w; (34) 



v signifying any one of the indices l3 2 , 3 , the preceding equa- 

 tion becomes : — 



SV = %(^-p')Sq. . . ■ . (35) 



Besides, according to Lagrange, the following easily derived 

 equation holds for the vis viva T : — 



T=|2W (36) 



If we now differentiate according to time the product/?,,, q v , 

 we have 



it Fvl " "' ' 



