A Treatise on the Theory of Screws

142 THE THEORY OF SCREWS. [153- 153. Law of Distribution of va. As we follow the screw a ai’ound the circle, it becomes of interest to study the corresponding variations of the linear magnitude va. We have already found a very concise representation of pa and wa by the axis of pitch and the axis of inertia, respectively. We shall now obtain a similar representation of va by the aid of the axis of potential. It is shown (§ 102) that must be a quadratic function of the co- ordinates; we may therefore apply to this function the same reasoning as we applied to uA (§ 134). We learn that is at each point proportional to the perpendicular on a ray, which is the axis of potential. Thus, if A (Fig. 32) be the screw, the value of va2 is proportional to AP, the perpendicular on PT; if 0" be the pole of the axis of potential, then, as in § 59, we can also represent the value of i>„- by the product AO". A A'. 154. Conjugate Screws of Potential, In general the energy expended by a small twist from a position of equilibrium can be represented by a quadratic function of the co-ordinates of the screw. If, moreover, the two screws of reference form what are called conjugate screws of potential (§ 100), then the energy is simply the sum of two square terms. The necessary and sufficient condition that the two screws shall be so related is, that their chord shall pass through 0". Another property of two conjugate screws of potential is also analogous to that of two conjugate screws of inertia. If A and A' be two conjugate screws of potential, then the wrench evoked by a twist round A is reciprocal to A', and the wrench evoked by a twist around A' is reciprocal to A.