142 



THE THEORY OF SCREWS. 



[153- 



153. Law of Distribution of v a . 



As we follow the screw a around the circle, it becomes of interest to study 

 the corresponding variations of the linear magnitude v a . We have already 

 found a very concise representation of p a and u a by the axis of pitch and the 

 axis of inertia, respectively. We shall now obtain a similar representation 

 of v a by the aid of the axis of potential. 



It is shown ( 102) that v a 2 must be a quadratic function of the co 

 ordinates; we may therefore apply to this function the same reasoning as 

 we applied to u a 2 ( 134). We learn that v a 2 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 v a 2 is proportional to AP, 

 the perpendicular on PT\ if 0&quot; be the pole of the axis of potential, then, 

 as in 59, we can also represent the value of v a ~ by the product AO&quot;. 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&quot;. 



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. 



