A Treatise on the Theory of Screws

24 THE THEORY OF SCREWS. [17- 17. Form of the Cylindroid in general. The equation of the surface contains only the single parameter pa~ Pß, consequently all cylindroids are similar surfaces differing only in absolute magnitude. The curved portion of the surface is contained between the two parallel planes z = + (pa —pß), but it is to be observed that the nodal line x = 0, y = 0, also lies upon the surface. Ihe intersection of the nodal line of the cylindroid with a plane is a node or a conjugate point upon the curve in which the plane is cut by the cylindroid according as the point does lie or does not lie between the two bounding planes. 18. The Pitch Conic. It is very useful to have a clear view of the distribution of pitch upon the screws contained on the surface. The equation of the surface involves only the difference of the pitches of the two principal screws and one arbitrary element must be further specified. If, however, two screws be given, then both the surface and the distribution are determined. Any constant added to all the pitches of a certain distribution will give another possible distribu- tion for the same cylindroid. Let be the pitch of a screw 6 on the cylindroid which makes an angle I with the axis of x\ then (§ 13) Pt = cos31 +pß sin21. Draw in the plane x, y, the pitch conic p«^ +pßy'2 = H, where H is any constant; then if r be the radius vector which makes an angle I with the axis of x, we have whence the pitch of each screw on a cylindroid is proportional to the inverse square of the parallel diameter of the conic. This conic is known as the pitch conic. By its means the pitches of all t e screws on the cylindroid are determined. The asymptotes, real or imaginary, are parallel to the two screws of zero pitch. 19. Summary. We shall often have occasion to make use of the fundamental principles demonstrated in this chapter, viz.,