A Treatise on the Theory of Screws

336 THE THEORY OF SCREWS. [310, Two conjugate points to conic (i) denote two reciprocal screws. Two con- jugate points to conic (ii) denote two screws at right angles. Let A be any point representing an instantaneous screw. Take the polar of A, with respect to conic (i). Let P be the pole of this ray, with respect to conic (ii). Then P will correspond to the impulsive screw, while A corresponds to the appropriate instantaneous screw. For this is clearly a homography of which A and P are two correspondents. Further, the double points of this homography are the vertices of the common conjugate triangle to conics (i) and (ii). If A lie on (i), then its polar is the tangent to (i); and as every point on this polar will be conjugate to P, with respect to conic (ii), it follows that A and P are conjugate, with respect to (ii)—that is, J. and P are correspondents of a pair of screws at right angles. As the pitch of the screw, corresponding to A, is zero, we have thus obtained the solution of our problem. 311. Fundamental Problem with Free Body. We now give the geometrical solution of the problem so fundamental in this present theory which may be thus stated : A perfectly free body at rest is struck by an impulsive wrench upon a given screw. It is required to construct the instantaneous screw about which the body will commence to twist. The rigid body being given, its three principal axes are to be drawn through its centre of gravity. The radii of gyration a, b, c about these axes are to be found. On the first principal axis two screws of pitches + a and — a respectively are to be placed. Similarly screws of pitches + b, —b, and + c, — c are to be placed on the other two principal axes. These are, of course, the six principal screws of inertia: call them Ao, Alt Aa, A3, Ait zl5. We then draw the five eylindroids AuA2, -AøAgj A0A4, AøA5. It is always possible to find one screw on a cylindroid reciprocal to any given screw. In certain cases, however, of a special nature, more than a single screw can be so found. Under such circumstances the present process is inapplicable, but the exceptional instances will be dealt with presently. Choose on the cylindroid A0A4 a screw 0t which is reciprocal to the given impulsive screw y, which is, of course, supposed to lie anywhere and be of any pitch. In like manner, choose on the other four cylindroids screws 0.lt 03, 04, 05, respectively, all of which are also reciprocal to y.