6 Prof. R. Clausius on different Forms of the ViriaL 



which by introducing the force-components is changed into 



^_^ = _^_^_J) + |^^. (21) 



Just such equations hold for the other two coordinate-directions ; 

 and we will add up these three equations. Therein the distance 

 between the two points shall be denoted by r ; and their relative 

 velocities, consequently the quantity 



S (x' v -0 2 + (y' v -y\f + (z\-z'J\ 



we will call u. Lastly, of the forces acting on the mass-points 

 m„ and m M , let the components which fall in the direction of r 

 be denoted by R v and R /Jt , and at the same time let the direction 

 of force from each point to the other point be reckoned positive. 

 We can then put : — 



X v fa — x v ) + Y„ (z/ M — y v ) +Z v fa — z v ) = R„r, 



X„(#„ — x^) + Y^(y„— y M ) + Z li .(z v — Zp) = 'R fl r. 



Accordingly the equation resulting from the above-mentioned 

 addition takes the following form : — 



*-(£ + &WJ*£5 m 



\m m J 2 dt z 

 Into this we will introduce another simplifying symbol, putting 



5n + 5^ =& ; (23) 



m v m„ 

 the equation will then read : — 



dHr^ 



u *=* r+ w ^ 34 ) 



m m 

 Multiplying this equation by ■ " M and extending it to the 



entire system of points, we get 



m^ m ' m ^=m^ m ^ Mr+ m^w^' * (35) 



wherein the three sums refer to all the combinations of two each 

 of the given mass-points. 



5. Between the sums which occur in this equation and the 

 sums previously considered, there are simple relations, which 

 can be discovered by means of a general formula of transforma- 

 tion. For, besides the masses m lt m 2 ,...m n , given two other 

 groups of quantities belonging to them, which shall provisionally 



