DE. PLtJCKER OX FUNDAMENTAL VIEWS EEOAEDING MECHANICS. 369 



take place, and besides the following two : 



LX +My +NZ =0, 

 L'x'+M'y'+N'z'=0. 

 The last equation may be developed thus, 



(L-L)(X-x)+(M-M)(Y-y)+(N-NXZ-z)=0, 

 and reduced by means of the preceding one as foUows : 



Lx+My+Nx ■ 



+Xl +Ym +Zi\ I (31) 



=LX+MY+NZ. . 



If the coordinates of the dyname be regarded as constant, x, y, z, l, m, n as variable, 

 this equation represents a linear complex of forces. By interchanging the two forces we 

 meet again the same equation. Hence 



A dyname may he resolved into ;pairs of forces, the forces of all pairs constitute a linear 

 complex. 



We must desist from entering into any further detail. 



15. Any number of dynames being given, the coordinates of the resulting dyname are 

 obtained by adding the coordinates of the given ones. If the six sums are equal to zero, 

 equilibrium exists. 



16. Dynames (P, R) the coordinates of which satisfy the linear equation 



^=AX+BY+CZ+DL+EM+FN-1=:0, (32) 



constitute a complex of dynames. In supposing P and E, and therefore the coordinates 

 of the dyname, infinite, the last equation becoming homogeneous, 



AX+BY+CZ+DL+EM+FN=0 (33) 



represents a complex of two variable lines. 



Dynames the coordinates of which satisfy simultaneously two linear equations, 



T=0, -^'=0, 

 constitute a congruency of dynames. In eliminating the constant term, the resulting 

 equation, ijr "^i^zQ 



represents a complex of two variable lines. 



II. 



1. We determine a force producing repulsion or attraction by means of two points 

 in space, one of which is the point acted upon. In quite an analogous way we may 

 represent a rotation, or the force producing it, by means of two planes, 



ix+v;y+ilz=i, -y 



tx -^uy -\-vz =1, \ 



MDCCCLiVI. 3 E 



