A Treatise on the Theory of Screws

 442 THE THEORY OF SCREWS. [401- I whence , dF de dF p? — 2p cos 3 + p2 VW- dfdX~ de (p - X)2 ’ , dF de _ dF Å,2 — 2X cos 3 + p3 ’’ W ~ ~ dp “ ~de Jff^Xf ’ (X2 — 2X cos 3 + p2) </>' (X) = (/? - 2p cos 3 + p2) cf (p) = p sin 3 suppose. Hence we have </> (X) = tan-1 p sin 3 \ p cos 3 — X) ’ and thus we get for the intervene with a suitable unit ' p sin 3 ' g cos 3 — p. ' p sin 3 ,p cos 3 — X 8 = 402. On the Infinite Objects in an Extent. On each range of the extent there will be two objects at infinity, by the aid of which the intervene between every two other objects on that range is to be ascertained. We are now to study the distribution of these infinite objects over the extent. Taking any range and one of its infinite objects, 0, construct any other range in the same extent containing 0 as an object. This second range will also have two infinite objects. Is 0 to be one of them? Here we add another attribute to our, as yet, immature conception of the intervene. In Euclidian space we cannot arrive at infinity except we take an infinitely long journey. This is because the point at infinity on one straight line is also the point at infinity on any other straight line passing through it. Were this not the case, then a finite journey to infinity could be taken by travelling along the two sides of a triangle in preference to the direct route vid the third side. To develop the analogy between the conception of intervene and that of Euclidian distance, we therefore assume (in Axiom iv.) that an infinite object has an infinite intervene with every other object of the content. In consequence of this we have the general result, that If 0 be an infinite object on one range, then it is an infinite object on every one of the ranges diverging from 0. The necessity for this assumption is made clear by the following con- sideration :—Suppose that 0 were an infinite object on one range containing the object A, but were not an infinite object on another range OB, diverging also from 0; then, although the direct intervene OA is infinite, yet the intervenes from A to B and from B to 0 would be both finite. The only- escape is by the assumption we have just italicised. Otherwise infinity could be reached by two journeys, each of finite intervene.