160 THE THEORY OF SCREWS. [162- 



The line at infinity 011 the plane of z is of course intersected by all the real 

 generators of the cylindroid, inasmuch as they are parallel to z. The points 

 at infinity on the planes x iy = are each the residence of an imaginary 

 screw, also belonging to the surface. The pitches of both these screws are 

 infinite. 



We may deduce the two screws of infinite pitch on the surface in another 

 way. The equations of a screw are 



y = x tan 6, 

 z = m sin 20, 



while the pitch is 



p + m cos 20. 



If tan 6 be either + i, we find both z infinite and the pitch infinite. We 

 thus see that through the infinitely distant point /, on the nodal line of the 

 cylindroid, two screws belonging to the surface can be drawn, just as at any 

 finite point. The peculiarity of the two screws through / is, that their 

 pitches are equal, i.e. both infinite, and this is not the case with any other 

 pair of intersecting screws. 



It is now obvious why the envelope just considered turned out to be 

 a parabola rather than any other conic section. Every plane section will 

 have the line at infinity for a transversal cutting two screws of equal pitch ; 

 the envelope of such transversals must thus have the line at infinity for 

 a tangent, i.e. must be a parabola. 



163. Chord joining Two Points. 



If 6 and 6&quot; be the angles by which two points on the cubic are defined, 

 then the equation to the chord joining those points is 



where A = 2m cos (0 - a) cos (9&quot; - a) cos (0 + 0&quot;), 



B = hsm/3+m cos j3 cos (6 r + 0&quot;) sin (2a - & - & } 



- m cos sin (0 + 0&quot;) cos (ff - 0&quot;), 

 C = - tan /3 (h - m cot /3 sin 20 ) (h-m cot j3 sin 2(9&quot;). 



If in these expressions we make + 6&quot; = 0, we obtain the equation for the 

 chord joining screws of equal pitch, as already obtained. 



We shall find that, in particular sections, these expressions become con 

 siderably simplified. Suppose, for example, that the plane of section be a 

 tangent plane to the cylindroid. The cubic then degenerates to a straight 

 line and a conic. The condition for this will be obvious from the equation 



