366 DE. PLUCKEll ON FUNDAMENTAL VIEWS REGAEDING MECHANICS. 



by X, Y, Z, L, M, N, as linear functions of the sixth. Accordingly the equation of 



condition, 



LX + MY+NZ = 0, 



may be transformed into an equation of the second degree with regard to the sixth 

 coordinate. 



10. In the complexes hitherto considered, the forces acting along a right line vary 

 in intensity when the point acted upon moves on that line. According to the more 

 usual notion there is, along a given line, one single force of given intensity acting upon 

 any point of the line. In order to represent complexes of such forces, we replace the 

 coordinates (1), made use of hitherto, either by the coordinates 



X, Y, Z, L, M, N (2) 



in admitting the equation of condition 



LX+MY + NX=0, (3) 



or by the coordinates 



X, Y, Z, Yz-Zy, Za;-Xr, Xy-Yo:. . . (4) 



In both systems of coordinates there is no trace left of the point acted upon by the force. 

 The same coordinates belong to right lines, and the homogeneous equation 



AX+BY+CZ-l-DL+EM+FN = O=0 (23) 



represents the same linear complex of lines which was formerly represented by the 



equation 



Q=0. 

 Put 



<i^ = <L^-l = (24) 



All forces, the coordinates of which satisfy this equation, constitute such a new com- 

 plex. It is essential not to confound such complexes with the former ones. 



11. The coordinates w, y, z of any point on the direction of a force are introduced in 

 making use of the coordinates (4). Accordingly the equation of the complex "^ becomes 



AX+BY+CZ + D(Y--Z3^)+E(Za:-Xz)+F(X3/-Ya;)=0. . . . (25) 

 If 



Yz=Zy, 



Zx=Xz, 



Xy=Yx, 



the corresponding forces pass through the origin ; for these forces, belonging to the com- 

 plex ^, we obtain 



AX+BY + CZ=0. 

 Let 



a=az, y=bz 



