A Treatise on the Theory of Screws

152 THE THEORY OF SCREWS. [160 § 160. Equation to plane section of Cylindroid. Each generator of the cylindroid is the abode of a certain screw, and accordingly each point in a plane section will lie on one screw, and generally on only one. We may, accordingly, regard the several points of the cubic as in correspondence with the several screws on the cylindroid. It will often be convenient to speak of the points on the section as synonymous with the screws themselves which pass through those points. We must first investigate the equation* to the cubic curve produced by cutting the cylindroid by a plane situated in any arbitrary position. Let OX and OY (Fig. 35) be the two principal screws of the cylindroid of which Oil is the nodal line. Let X EQ be the arbitrary plane of section. The position of this plane is defined by the magnitudes h, a, ß, whereof h is the length of the perpendicular from 0 on XY, a is the angle between OR and OX, and ß is the angle ORO, or the inclination of the plane of section to the principal plane of the cylindroid. Draw through fl the line flTV parallel to XY; then we shall adopt ON as the new axis of æ and OR as the new axis of y, so that if P be any point on the surface, we have PN = y and ON = x. The dotted letters, x, y', z' refer to the original axes of the cylindroid. Let fall PT perpendicular on the plane of OXY, and TM perpendicular to XY. Then we have MN = RO, whence y + z cosec ß = h sec ß ......................(i), * Transactions of the Royal Irish Academy, Vol. xxix. p. 1 (1887).