DE. PLUCKER ON FUNDAMENTAL VIEWS EEGAEDING MECHANICS. 363 



This equation not being admitted, the corresponding dyname (P, R) depends upon six 

 linear constants (2), independent of each other. There is no relation admitted between 

 the direction of the force (P) and the direction of the axis of the moment (R). 



In denoting the angle between both directions by <p, we have 



LX + MY + NZ ^ ._. 



PR = COS<P (7) 



3. A linear complex of right lines* is represented by a homogeneous equation of the 

 first degree, 



A(a:-a;')+B(//-y)+C(z-5')+D(ys'-yz)+E(2a/-z'^)+F(.ry-y2/)EEQ=0, . (8) 



between the six line-coordinates 



{x-a^), {y-y'), {z-z'), {yz'-y'z), {zai-z'x), (xy'-x'y), . (1) 



regarded as vai'iables. These quantities are simultaneously the coordinates of a force. 

 Let us replace the homogeneous equation (8) by the general one of the first degree, 



O-l-S=0 (9) 



Forces the coordinates of which satisfy this equation constitute a linear complex of forces. 

 The six coordinates of the two points (x', y', z') and {x, y, z) by which the force is deter- 

 mined may likewise be regarded as variables replacing the coordinates (1). 



In order to get the forces of the complex H acting upon any given point of space 

 {al, y, 2'), we must regard the coordinates of this point as constant. On this supposition 

 the equation of the complex, which may be written thus, 



(A+Fy-Es')a? ^ 

 + (B-FA-'+D2')y 

 +(C+Ea/-D/> 



=Aa,^+By + Cs'+l, 



(10) 



represents a plane. Therefore the geometrical locus of the second points {x, y, z), by 

 which the forces acting on the given point (x', y', z') are determined, is a plane. This 

 plane may be called conjugate to the point acted upon. 



In a linear complex there are acting upon each point of space forces in all directions, 

 the intensity of each force being the segment on its direction hetioeen the point acted upon 

 and its conjugate plane. 



4. In supposing the forces, and consequently their coordinates, to be infinite, the equa- 

 tion (9) of the complex S becomes 



Q=0. 



This equation, therefore, representing a complex of right lines, indicates the position of 

 those forces of the complex H the intensity of which is infinite. 



* See geometrical Paper, p. 734, Philosophical Transactions, 1865. 



3d2 



