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



introduce a movable system having for its origin the centre of 

 gravity of all the material points, and parallel to the fixed system, 

 and if we name the coordinates of the centre of gravity in rela- 

 tion to the fixed system x c , y C) z Ci and the coordinates of any one 

 of the material points in relation to the movable system %, 77, f, 

 then is 



X = X c + %, yz=y c + V} z = Z c +£ 



If we now form the equation 



and consider that we may put 



"27?ix c tj = xi£m% = 0, 



we get, if M denotes the total mass of all the material points, 

 consequently the sum %?n, the equation 



Sroa^MffJ + Swil 9 (U) 



In precisely the same manner we obtain 



Imx't^Mx'l + Zmg* (12) 



Finally, the mere substitution in 2X# of x c +f; for the coor- 

 dinate x } X c denoting the sum 2X, gives 



SX*=X c * <J + 2Xf ..... (13) 



If now we form for the centre of gravity the identical equation 

 which for a single material point has served for the derivation 

 of (1), viz. 



2 dt 2 



/dx c \* d 2 x c 



M 



which, after multiplication by — , can be written thus, 



M , 2= _M i*g. % Md*{xl) 

 2 e 2* c dt*~+ 4~dF~' 



and suppose herein 



we then obtain 



_ J ,. =? .^-.V. + __ r (14) 



With the aid of this equation in conjunction with (11), (12), 

 and (13), the following equation can be immediately derived from 

 (&):- 



