32 Prof. R. Clausius on the Theorem of the Mean Ergal, 

 one coordinate-direction, takes the form 



S(H-| tf*)=mtf*Slo%i+^8c. . . * (12) 

 Further, according to the theorem of the virial, 



Combining this with (11), we get 



H=-Z 2 *, (13) 



n K ' 



dR x~ n -75-1 , I/n 



By insertion of these values equation (12) is changed into 

 n-2 



2n 



S (mx h2 ) = mx' 2 8 log i — ma/ 2 8 log c 



which equation, after putting 



n 2 7i-— 2 n ~ 2 



2 $ (ma, 1 *) = -~ — ma 12 S log (mat 2 ) = mx 12 8 log (mx 12 ) 2n , 



can be brought into the form 



8\og\-(mx h2 ) 2n \=0; .... (15) 



and from this it follows that 



i — rt ~ 2 



~(mx n ) 2n =C, (16) 



in which C is a constant the value of which depends on n. 

 This equation can be written thus, 



2— n 



i = Cc(mx 12 ) 2 » ; 



* I here take the opportunity to remark that the relation expressed in 

 (13), between the mean ergal and the mean vis viva, has a much more 

 general validity. If, namely, for any system whatever of material points 

 the ergal (abstraction made of the arbitrary additive constants, which can 

 be supposed equal to nil) can be represented by a homogeneous function 

 of the coordinates of the points, then (n signifying the degree of the ho- 

 mogeneous function) the virial, and consequently also the mean vis viva, is 



equal to the mean ergal multiplied by -. 



