-)=-l^1W « 



2 Prof. R. Clausius on the Relations between the 



are X, Y, Z ; then from the differential equation of motion, 



can very readily be derived the following equation : — 

 m/dx\*__ l v _ , m d^(oc 2 ) 

 2\dt 



This equation 1 have applied to the case in which the motion 

 is stationary — that is, where the coordinates and velocities do 

 not continually change in the same direction, but only vary 

 within certain limits*. In this case the mean value, referred to 

 one period of the motion or to a very long time, of the differen- 



tial coefficient of the second order J ) 2 =0. Hence, if we in- 

 dicate the mean values of the two other quantities occurring in 

 the equation by putting a horizontal stroke above their symbols, 

 the equation becomes 



-M=-l^ ■ « 



Equations (1) and (2) hold, of course, in the same form, also 



for both the other coordinates. If, further, instead of a single 



freely movable point, we have a whole system of such points, the 



same equations hold for each point of this system. We can 



therefore forthwith extend them so that they shall refer to all 



three coordinates and the whole system of points. If, namely, 



we employ the letter r for the radius vector of a point, and the 



letter v for its velocity, which letters can have different indices 



for the different points, the two equations will be transformed 



into : — 



v m 2 1 V/Y v „ 1 d 2 (^mr 2 ) 



^jv q =-^%(Kx-hYy + Zz) + - v ^ 2 - y , . (la) 



m 



2^t;*:=--2(Xtf + Yy + Zs) (2a) 



Equation (2), in its amplified form (2 a), is the first of my 

 above-mentioned two equations. The quantity found on the 

 right-hand side of (2a) I have named the virial of the system, 

 and expressed the sense of the equation in the following propo- 

 sition : — The mean vis viva of the system is equal to its virial. 

 But at the same time I remarked that the proposition is true not 

 merely for the whole system of points, and for the three coordi- 



* Sitzungsberichte der Niederrhein. Gesellschaft fur Natur- und Heil- 

 kunde 1870, p. 114; Pogg. Ann. vol. cxli. p. 124; Phil. Mag. S.4. vol. xl. 

 p. 126. 



