Prof. R. Clausius on different Forms of the Virial. 7 



be denoted by p lt p& . . .p n and q v q 2 , . . . q ni then the following 

 identical equation holds : — 



Xm v m lx (p v -p IJ )(q v — q^=l i mX7npq — ^mpl < mq; . . (26) 



in which the sum on the left-hand side refers to all combinations 

 of two masses each, while the sums on the right-hand simply refer 

 to all the masses. A conviction of the correctness of the equation 

 can be obtained by carrying out the multiplication on the left- 

 hand side, and suitably arranging and collecting the terms then 

 contained in the sum. We will now apply this equation to our 

 case by attributing successively different significations to the 

 quantities/? and q. 



First let us put p = q=x; the result is:- — 



Xm v m ll {oo v —x i i) 2 = ^rrilimx 2 — (Zmx) 2 = M^mx Q — M 2 a£. 

 We then put p = q=cc ! , and obtain in a corresponding manner 



2m v m lx {a; , v -x'rf = WZmx' 2 - M V*. 



Lastly, we put p = — and q = %; then comes 



= MSX#— MXcffc 



Just such equations are valid for the other two directions of 

 coordinates ; and if we form the sum of each three belonging to 

 one another and divide it by M, we obtain the equations ex- 

 pressing the relations sought, namely : — 



^Xm,7 V 2 = 2m/ 2 -M/^ .... (27) 

 jjjSm„m^w 2 =2»i« 2 — Mi??, .... (28) 



M Wz^ = £L/-L c / c . .... (29) 



Combining these with equations (16), (17), and (18), we get 

 the following very simple equations : — 



•^jSm v m jX r 2 = SmX 2 , ...... (30) 



^Xm v m f j^ = y Zmw 2 , (31) 



^^m v m H ^r=l t A\ (32) 



