, Prof. R. Clausius on different Forms of the Final. 5 



2L>=-^X f+ -^. . . . (15) 



All the equations above derived for the ^-direction, of course 

 hold good in a corresponding manner for the other two direc- 

 tions of coordinates ; and when each three equations thereby 

 arising are added together, a new system of equations is ob- 

 tained. In order to write these conveniently, let us introduce 

 the following symbols. We will name the distance of the centre 

 of gravity from the origin of the fixed coordinates l c ; and the 

 distance of a mass-point from the centre of gravity, X. Let the 

 velocity of the centre of gravity be called v c , and the relative 

 velocity of a mass-point about the centre of gravity, consequently 

 the quantity \/g* + if + f ' 2 , be called w. Further, of the force 

 whose components in the coordinate-directions are X c , Y c , Z c , 

 the component in the direction of l c may be denoted by L* c ; and 

 of the force acting on a mass-point, let the component in the 

 direction of \ be denoted by A. Then the equations will be 

 written as follows : — 



^mP = M% + 2m\*, ...... (16) 



Zmv 2 =Mvl + X?nw% (17) 



2L7=L c / + 2AX, (18) 



2 c 2 cc+ 4 ^ 2 



~ m 9 1 „ A N , 1 d 2 l<m\ 2 . , OA . 



S 2 w= 3 SAX+ 4~^- ' * ■ (20) 



4. We will now turn to the kind of transformation which I 

 communicated in the Comptes Rendus, and which depends on the 

 introduction into the formulae of the mutual distances and rela- 

 tive velocities of each two material points. 



First, if v and ja represent any two of the indices 1, 2, 3, &c, 

 and accordingly m v and m^ are any two of the given mass-points 

 with the coordinates x vi y v} z v and x^ y^, z^, we can form, corre- 

 sponding to the above, the following identical equation, 



l ^[K-^)*] - pfe-ay) ! 2 , ( _ d*(x v -x„) 

 2 dt~ L dt* J + ^ ^ dt* 



or, differently arranged and written, 



U -J\t--/ r r )( d ***_ d **A , l d*\.{x v -**)*] 



