A Treatise on the Theory of Screws

20-23] RECIPROCAL SCREWS. 27 For a body only free to twist about y would be undisturbed by wrenches on 0 and <f>; but a wrench on any screw y/r of the cylindroid can be resolved into wrenches on 0 and </>; therefore a wrench on y/r cannot disturb a body only free to twist about y ; therefore y/r and y are reciprocal. We may say for brevity that y is reciprocal to the cylindroid. y cuts the cylindroid in thx-ee points because the surface is of the third degree, and one screw of the cylindroid passes through each of these three points; these three screws must, of course, be reciprocal to y. But two intersecting screws can only be reciprocal when they are at right angles, or when the sum of their pitches is zero. The pitch of the screw upon the cylin- droid which makes an angle I with the axis of x is cos2? +Pß sin2Z. This is also the pitch of the screw ir -1. There are, therefore, two screws of any given pitch; but there cannot be more than two. It follows that y can at most intersect two screws upon the cylindroid ol pitch equal and opposite to its own; and, therefore, y must be perpendicular to the third screw. Hence any screw reciprocal to a cylindroid must intersect one of the generators at right angles. We easily infer, also, that a line intersecting one screw of a cylindroid at right angles must cut the surface again in two points, and the screws passing through these points have equal pitch. These important results can be otherwise proved as follows. A wrench can always be expressed by a force at any point 0, and a couple in a plane L through that point but not of course in general normal to the force. For wrenches on the several screws of a cylindroid, the forces at any point all lie on a plane and the couples all intersect in a ray. The first part of this statement is obvious since all the screws on the cylindroid are parallel to a plane. To prove the second it is only necessary to note that any wrench on the cylindroid can be decomposed into forces along the two screws of zero pitch. Their moments will be in the planes drawn through 0 and the two screws of zero pitch. The transversal across the two screws of zero pitch drawn from 0 must therefore lie in every plane L. We hence see that the third screw on the cylindroid which is crossed by such a transversal must be perpendicular to that transversal. 23. Reciprocal Cone. From any point P perpendiculars can be let fall upon the generators of the cylindroid, and if to these perpendiculars pitches are assigned which are equal in magnitude and opposite in sign to the pitches of the two remaining