A Treatise on the Theory of Screws

108 THE THEORY OF SCREWS. [117- From the condition of reciprocity we must have: Pi'll («1 cos I + /3j sin I) + ... +p,iyi (a6 cos I + ßti sin I) = 0, or, cos I + sin I = 0. From this tan I is deduced, and therefore the screw 3 becomes known (§ 26). In general if be the virtual coefficient of any screw y and a screw 3 on the cylindroid, we have cos I + sin I; whence if on each screw 3 a distance be set off from the nodal line equal to the virtual coefficient between y and 3, the points thus found will lie on a right circular cylinder, of which the equation is; «2 + y1 = + vr^y. Thus the screw which has the greatest virtual coefficient with y is at right angles to the screw reciprocal to y, and in general two screws can be found upon the cylindroid which have a given virtual coefficient with any given external screw. 118. Relation between Two Cylindroids. We may here notice a curious reciprocal relation between two cylindroids, which is manifested when one condition is satisfied. If a screw can be found on one cylindroid, which is reciprocal to a second cylindroid, then conversely a screw can be found on the latter, which is reciprocal to the former. Let the cylindroids be (a, ß), and (X, p). Il a screw can be found on the former, which is reciprocal to the latter, then we have: PiK (ctj cos I + Ä sin I) + ... + pnXn (a,t cos I + ßn sin I) = 0, Pifh (a, cos I + ß1 sin I) + ... 4- pnpn (an cos I + ßn sin Z) = 0. Whence eliminating I, we find:— ^aX^ßp ' ^ßx^ap — 0. As this relation is symmetrical with regard to the two cylindroids, the theorem has been proved. 119. Co-ordinates of Three Screws on a Cylindroid. The co-ordinates ol three screws upon a cylindroid are connected by four independent relations. In fact, two screws define the cylindroid, and the third screw must then satisfy four equations of the form (§ 20). These relations can be expressed most symmetrically in the form of six equations, which also involve three other quantities.