60 THE THEORY OF SCREWS. [66- 



66. Screws of Zero Pitch. 



A screw of zero pitch is reciprocal to itself. The tangent at a point 

 corresponding to a screw of zero pitch, being the chord joining two reciprocal 

 screws, must pass through the pole of the axis of pitch. This is, of course, 

 the same thing as to say that the axis of pitch intersects the circle in two 

 screws, each of which has zero pitch. 



67. A Special Case. 



We have supposed that the axis of pitch occupies any arbitrary position. 

 Let us now assume that it is a tangent to the representative circle. This 

 specialization of the general case could be produced by augmenting the 

 pitches of all the screws on the cylindroid, by such a constant as shall make 

 one of the two principal screws have zero pitch. 



The following properties of the screws on the cylindroid are then 

 obvious : 



1. There is only one screw of zero pitch, 0. 



2. The pitches of all the other screws have the same sign. 



3. The maximum pitch is double the radius. 



4. The screw is reciprocal to every screw on the surface, arid this 

 is the only case in which a screw on the cylindroid is reciprocal to every 

 other screw thereon. 



68. A Tangential Section of the Cylindroid*. 



Let the plane of section be the plane of the paper in Fig. 17, and let the 

 plane contain one of the screws of zero pitch OA. Let OH be the projection 

 of the nodal axis on the plane of the paper. Then OA being perpendicular to 

 the nodal axis must be perpendicular to OH, Let P be the point where the 

 second screw of zero pitch cuts the curve. Then since any ray through P 

 and across AO, meets two screws of equal pitch, it must be perpendicular 

 to the third screw which it also meets on the cylindroid ( 22). Hence 

 PH is perpendicular to the screw through H, and as the latter lies in the 

 normal plane through OH it follows that the angle at H is a right angle. 



Any chord perpendicular to AO must for the same reason intersect two 

 screws of equal pitch, and therefore APHO must be a rectangle. 



If tangents be drawn at A and P intersecting at T, then it can be shown 

 that any chord TLM through T cuts the ellipse in points L and M on two 

 reciprocal screws. 



* For proofs of theorems in this article see a paper in the Transactions of the Royal Irish 

 Academy, Vol. xxix. pp. 132 (1887). 



