212 THE THEORY OF SCREWS. [ 209, 



The following are the pitches of the several points and curves represented 

 Vertices of the Triangle + 992, + 825, - 448, 

 Ellipses + -96, + -92, + 1, - 4, 

 Parabola + 309, 

 Hyperbolas + 748984, + -8300467. 



The locus of the centres of the pitch conies is 

 a^j sin (A - B) sin 2(7 + 2 a 3 sin (B - C) sin 1A + a.^ sm(C-A) sin 25 = 0. 



210. Intersecting Screws in a Three-System. 



If two screws of a three-system intersect, then their two corresponding 

 points must fulfil some special condition which we propose to investigate. 



Let a be one screw supposed fixed, then we shall investigate the locus of 

 the point 6 which expresses a screw which intersects a. We can at once 

 foresee a certain character of this locus. A ray through a can only cut it in one 

 other point, for if it cut in two points, we should have three co-cylinclroidal 

 screws intersecting, which is not generally possible. The locus is, as we shall 

 find, a cubic and the necessary condition is secured by the fact that a is a 

 double point on the cubic, so that a ray through a has only one more point 

 of intersection with the curve. We can indeed prove that this curve must 

 be a cubic from the fact that any screw meets a cylindroid in three points. 

 Draw then a ray ( 200) corresponding to a cylindroid of the three-system. 

 There must be in general three points of the locus on this ray. Therefore the 

 locus must be a cubic. 



As a and 6 intersect we have since d 6a = 



2&amp;lt;3- ea = (p a + p e ) cos 0&amp;lt;9). 



By substituting the values of the different quantities in terms of the co 

 ordinates we have the following homogeneous equation of the cubic : 



= 2 (0!* + 2 2 + 3 2 ) (p^A +P&A + P&A) (a, 2 + 2 2 + a 3 2 ) 



a 3&amp;lt;?3) (ai 2 + 2 2 + a s 2 ) 



We first note that this cubic must pass through the four points of inter 



section of 



= 0,&quot;+ &amp;lt;9./ + 6*, 



= pA* + p&i + pA 2 - 



But this might have been expected, because as we have shown ( 203) each of 

 these four points corresponds, not to a single screw, but to a plane of screws. 



