176 THE THEORY OF SCREWS. [175- 



rays from the centre of the quadric to the vertices of this triangle are the 

 three principal axes of the quadric. 



We thus prove again that if a and /3 be any two screws of a three-system, 

 the centre of the pitch-quadric must lie in the principal plane of the 

 cylindroid through a and /?. For the common perpendicular to any two 

 screws of equal pitch on the cylindroid will be bisected by the principal 

 plane and therefore any hyperboloid through these two screws of equal 

 pitch must have its centre in that plane. 



176. Screws through a Given Point. 



We shall now show that three screws belonging to S, and also three 

 screws belonging to 8 , can be drawn through any point x , y , z . Substitute 

 x, y , z , in the equation of 17 5 and we find a cubic for k. This shows that 

 three quadrics of the system can be drawn through each point of space. 

 The three tangent planes at the point each contain two generators, one 

 belonging to S, and the other to S . It may be noticed that these three 

 tangent planes intersect in a straight line. 



From the form of the equation it appears that the sum of the pitches of 

 three screws through a point is constant and equal to p a +pp + p y - 



Two intersecting screws can only be reciprocal if they be at right angles, 

 or if the sum of their pitches be zero. It is hence easy to see that, if a 

 sphere be described around any point as centre, the three screws belonging 

 to S, which pass through the point, intersect the sphere in the vertices of a 

 spherical triangle which is the polar of the triangle similarly formed by the 

 lines belonging to S . 



We shall now show that one screw belonging to S can be found parallel 

 to any given direction. All the generators of the quadric are parallel to 



the cone 



(p a - k) x* + (p ft - k) f + (p y - k) z* = 0, 



and k can be determined so that this cone shall have one generator parallel 

 to the given direction ; the quadric can then be drawn, on which two gene 

 rators will be found parallel to the given direction ; one of these belongs to 

 S, while the other belongs to S . 



It remains to be proved that each screw of 8 has a pitch which is propor 

 tional to the inverse square of the parallel diameter of the pitch quadric*. 



* This theorem is connected with the linear geometry ol Plucker, who has shown (Neue Geometric 

 des Ratlines, p. 130) that k l x- + L 2 y- + k, t z 2 + k 1 kJ{ 3 ) is the locus of lines common to three 

 linear complexes of the first degree. The axes of the three complexes are directed along the 

 co-ordinate axes, and the parameters of the complexes are fcj, ._,, k 3 ; the same author has also 

 proved that the parameter of any complex belonging to the &quot; dreigliedrige Gruppe&quot; is propor 

 tional to the inverse square of the parallel diameter of the hyperboloid. 



