A Treatise on the Theory of Screws

388] THE THEORY OF PERMANENT SCREWS. 423 Then whence <^>x — 0, + </>2 = 02 + 02^1 Ø’PQ^ØÅ-ØMSt, PQ ^ 0201-^2 8t ’ 02 which is, accordingly, the rate at which P will change its position. If we substitute for 02 and 0.2 their values already found in the last article, we obtain for the velocity of P the expression i ___________ M UfuJfP {A0i + B02\ N being the position of the permanent screw, let p be the length of the chord PN, then the expression just written assumes the form kpØUg where k is a constant. This expression illustrates the character of the screw corresponding to N. If p be zero, then the expression for this velocity vanishes. This means that P has no tendency to abandon Xin other words, that the screw correspond- ing to N is permanent. 388. Another method. It is worth while to investigate the question from another point of view. Let us think of any cylindroid S placed quite arbitrarily with respect to the position of the rigid body. A certain restraining screw r) will corre- spond to each screw 0 on $. As 0 moves over the cylindroid, so must the corresponding screw i) describe some other ruled surface ,S . The two surfaces, S and S', will thus have two corresponding systems of screws, whereof every two correspondents are reciprocal. One screw can be dis- covered on S', which is reciprocal, not alone to its corresponding 0, but to all the screws on the cylindroid. A wrench on this y can be provided by the reactions of the constraints, and, consequently, the constraints will, in this case, arrest the tendency of the body to depart from 0 as the instantaneous screw. It follows that this particular 0 is the permanent screw. The actual calculation of the relations between v and the cylindroid is as follows:— A set of forces applied to a rigid system has components X, Y, Z at a point, and three corresponding moments F, G, H m the rectangular planes of reference.