46 THE THEORY OF SCREWS. 



Eliminating 6 between the equations for z and p, we obtain 



[50- 



Let p and z be regarded as the current co-ordinates of a point. Then 

 the locus of this point is the circle which forms the foundation of the plane 

 representation*. 



7?i is, of course, the radius, and p is the distance of the centre from a 

 certain axis. Any point on this circle being given, then its co-ordinates p 

 and z are completely determined. Thus sin 20 and cos 20, and, consequently, 

 tan 0, are known. We therefore see that the position of a screw and its 

 pitch are completely determined when the corresponding point on the circle 

 is known. To each point of the circle corresponds one screw on the cylin- 

 droid. To each screw on the cylindroid corresponds one point on the circle. 

 This may be termed the representative circle of the cylindroid. 



51. The Axis of Pitch. 



Let T (fig. &quot;&amp;gt;) be the origin. Then p Q is the perpendicular ST from the 

 centre 8 of the circle to the axis PT. The ordinate AP is the pitch of the 



P T 



Fig. 5. 



* The following elegant construction for the cylindroid is given by Mr T. C. Lewis, Messenger 

 of Mathematics, Vol. ix. pp. 15, 1879. &quot; Suppose that a point P moves with uniform velocity 

 around a circle while the circle itself rotates uniformly about an axis in its plane with half the 

 angular velocity that P has around the centre. Then the perpendiculars from P on the axis of 

 rotation trace out the cylindroid, while the lengths of those perpendiculars are the pitches of the 

 corresponding screws.&quot; This construction is of special interest in connexion with the represen 

 tation of the cylindroid by a circle discussed in this chapter. The construction of Mr Lewis shows 

 that if the circle rotate around the axis of pitch with half the angular velocity of the point P 

 around the circle, then not only does P represent the screw in this circle but the perpendicular 

 from P on the axis of pitch is the position of the screw itself. 



