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



ponents of the force acting upon it, then 



mfdx\^ l v m d^ix 2 ) n , 



or, if (as will always be done in the following) the first differen- 

 tial coefficient according to time be indicated by affixing an 

 accent, 



™ a J*--lXx+- d ^ ) (la) 



2 a? - 2 A,2;+ 4 ^ [La) 



From this results, indicating mean values by drawing a hori- 

 zontal stroke above : — 



!*--5 E - ••••■ » 



m 

 If we name the quantity =■ x H the vis viva with respect to the 



a?-direction, and the quantity — - X*' the virial relative to the 



^-direction, since the x- is any direction we please, the meaning 

 of the equation can be expressed thus : — For each freely movable 

 point, the mean vis viva relative to any direction is equal the virial 

 relative to the same direction. 



If we form for a point the equations relative to the three di- 

 rections of its coordinates and add them up, we get (v denoting 

 the velocity of the point, and / its distance from the origin of 

 the coordinates): — 



If, further, we denote by L the component, in the direction 

 of /, of the force acting on the point, and reckon it positive from 

 the origin of the coordinates onward, the equation (as is readily 

 seen) becomes : — 



m 2 1 T7 m d*(p) 



It is obvious that these equations, which are valid for each 

 individual point, can be extended by simple summation to the 

 entire system of points. We thus obtain : — 



s -*..--is &+ iqS£ (5) 



2 2/4 dt 2 



m 2 * vrv . v . 7 \ i 1 d^lml* 



2 -„ 2= - ?(& + Ty + Zr)+i^L f . . (6 ) 



S^-JSLZ+J^ (7) 



