Prof. R. Clausius on different Forms of the Virial. 3 



In the formation of the mean values, in all these equations 

 just as in (i), the last term on the right-hand side falls away; 

 and the expression then remaining on that side represents the 

 virial. 



2. The first method of transformation of these equations is 

 based on the fact that when the points are acted on by forces of 

 different sorts which we wish to consider singly, the force-com- 

 ponents can be separated into as many summanda as the kinds 

 of force that are to be distinguished, whereby the virial is divided 

 into just as many parts. 



If, for instance, the above-mentioned distinction be made be- 

 tween the forces which the points of the system exert on each 

 other, and those which act upon the system from without, and 

 this be denoted by the indices i and e } we can put X = X t - + X fl ; 

 and the same holds for the components Y, Z, and L. It is 

 readily seen how the above equations are changed by the inser- 

 tion of these sums. Equation (6), for example, thereby changes 

 into 



2 1 v *= ~ 1 2 ^ + Y * + M - 1 S(X^ + Y e y + Z e z) 



1 i*SmP 



+ I^~ • (8) 



When more special assumptions are made concerning the 

 nature of the forces, the expressions also take more special forms, 

 of which I will briefly cite two which are exhibited in my first 

 memoir. When, namely, the internal forces consist of reciprocal 

 attractions or repulsions, which, according to any law, depend 

 on the distance, so that for two points whose distance is r the 

 force (which as an attraction is reckoned positive, and as a repul- 

 sion negative) can be represented by a function <£(r), we can put 



-|2(X^ + Y,.y + V) = l2r^(r), ... (9) 



in which the sum on the right-hand side refers to all combina- 

 tions of two mass-points each. When the system of points is 

 further considered as a body on which the only external force 

 acting is a symmetrical pressure^? normal to the surface, we can put 



-\Z(X e x + Y e y + Zz) = lp\r, .... (10) 



in which V denotes the volume of the body. 



3. Another mode of transformation depends on the separation 

 of the coordinates of the points into summanda. 



To tbis belongs the transformation effected by Yvon Villar- 

 ceau. If, namely, besides the fixed systems of coordinates, we 



B2 



