A Treatise on the Theory of Screws

128 THE THEORY OF SCREWS. [138- If A and B be any two impulsive screws, and if A' and B' be the corre- sponding instantaneous screws, then the chords AB and BA' will always intersect upon the fixed right line XY. This right line is called the homographic axis. It intersects the circle in two points, X and Y, which are the double points of the homographic systems. These points enjoy a special dynamical significance. They are the two Principal Screws of Inertia, and hence— The homographic axis intersects the circle in two points, each of which possesses the property, that an impulsive wrench administered on that screw will make the body commence to move by twisting about the same screw. The method by which we have been conducted to the Principal Screws of Inertia shows how there are in general two, and only two, of these screws on the cylindroid. The homographic axis is the Pascal line, for the Hexagon AA'BB'CC, and thus we have a dynamical significance for Pascal’s theorem. 139. Determination of the Homographic Axis. The two principal screws of inertia must be reciprocal, and must also be conjugate screws of inertia (§ 84). The homographic axis must therefore comply with the conditions thus prescribed. We have already shown (§ 58) the condition that two screws be reciprocal, and (§ 135) the condition that two screws be conjugate screws of inertia, and, accordingly, we see— 1°. That the homographic axis must pass through 0, the pole of the axis of pitch. 2°. That the homographic axis must pass through O', the pole of the axis of inertia. The points 0 and O' having been already determined we have accordingly, as the simplest construction for the homographic axis, the chord joining 0 and O'. 140. Construction for Instantaneous Screws. The points 0 and O' afford a simple construction for the instantaneous screw, corresponding to a given impulsive screw. The construction depends upon the following theorem (§ 81):— If two conjugate screws of inertia be regarded as instantaneous screws, then the impulsive screw corresponding to either is reciprocal to the other. Lot A be an impulsive screw (fig. 22); if we join AO we obtain H, the screw reciprocal to A ; and if we join HO' we obtain A', the conjugate screw