A Treatise on the Theory of Screws

46 THE THEORY OF SCREWS. [50- Eliminating 3 between the equations for z and p, we obtain (P ~ P$ + z- = m\ 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 *. m is, of course, the radius, and pa 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 2(9 and cos <2.3, and, consequently, tan 3, 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- io 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 7 (fig. ;>) be the origin. Then p„ is the perpendicular ST from the centre S of the circle to the axis PT. The ordinate AP is the pitch of the * The following elegant construction for the cylindroid is given by Mr T. C. Lewis, Messender of Mathematics, Vol. rx. pp. 1-5, 1879. “ 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 axe the pitches of the corresponding screws,” This construction is of special interest in connexion with the represen- tation of the eylmdroid 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.