203] PLANE REPRESENTATION OF THE THIltD ORDER. 203 



we shall obtain 



*- Pi)* (Pi- Pi)* 



- (PI-P)(P*-P) 



and as these values satisfy the equation of the quadric, the theorem has been 

 proved. 



The family of quadric surfaces are therefore inscribed in a common tetra 

 hedron, and they have four common points, as well as four common tangent 

 planes. For, write the two cones 



#- + if + z- = 0, 



pj? + ihy&quot; + ihz&quot; = o. 



These cones have four generators in common, and the four points in which 

 these generators cut the plane at infinity will lie on every surface of the 

 type 



(Pi ~ P) ? + (P* ~ P) V&quot; + (Pa ~p)2- + (pi - p) (p* - p) (p 3 -p) = 0. 



We now see the distribution of the screws in the imaginary planes. In 

 each one of these planes there are a system of parallel lines ; each line of this 

 system passes through the same point at infinity, which is, of course, one of 

 the four points just referred to. Every line of the parallel set, when it 

 receives appropriate pitch, belongs to the three-system. 



It thus appears that the ambiguity in the pitches of the screws in the 

 planes is only apparent. The system of screw co-ordinates which usually 

 defines a screw with absolute definiteness, loses that definiteness for the 

 screws in these planes. Each plane contains a whole pencil of screws, 

 radiating from a point at infinity, but the co-ordinates can only represent 

 these screws collectively, for the three co-ordinates then represent, not a single 

 screw, but a whole pencil of screws. As the pitches vary on every screw of 

 the pencil, the co-ordinates can only meet this difficulty by representing the 

 pitch as indeterminate. 



The proof that only a single screw of each pitch is found in the pencil is 

 easily given. If there were two, then the same hyperboloid would have two 

 generators in this plane of equal pitch ; but this is impossible, because, from 

 the known properties of the three-system, only one of these generators 

 belongs to the three-system, and the other to the reciprocal system. 



