A Treatise on the Theory of Screws

444 THE THEORY OF SCREWS. [403, The intervene between the objects X and - X is iHir. Nor is this inconsistent with the fact that X = zero denotes two coincident objects, as does also X= +infinity. In each of these cases the coincident objects are at infinity, and the intervene between two objects which coalesce into one of the objects at infinity has an indeterminate value, and may thus, of course, be iHir, as well as anything else. 404. Intervenes on Different Ranges in a Content. Let us suppose any two ranges whatever. There are an infinite number of objects on one range, and an infinite number on the other. The well- known analogies of homographic systems on rays in space lead us to inquire whether the several objects on the two ranges can be correlated homo- graphically. Each object in either system is to correspond definitely with a single object in the other system. We determine an object on a range by its appropriate X. Let the corresponding object on the other range be defined by X'. The necessary conditions of homography demand that for each X there shall be one X', and vice versa. Compliance with this is assured when X and X' are related by an equation of the form PXX -f- QX + Rx + $ = 0. Let Xj, X2, X3, Xt be any four values of X, and let X/, Xj, Xj, Xj be the cor- responding four values of X’, then, by substitution in the equation just written, and elimination of P, Q, R, S, it follows that A-, Å-3 Xo X4 X4 X3 X% — X4 X3 X3 Åq X3 X, — Xt We now introduce the following important definition:— By the expression, anharmonic ratio of four objects on a range, is meant the anharmonic ratio of the four values of the numerical parameter by which the objects are indicated. We are thus enabled to enunciate the following theorem :— When the objects on two ranges are ordered homographically, the an- harmonic ratio of any four objects on one range equals the anharmonic ratio of their four correspondents on the other. Three pairs of correspondents can be chosen arbitrarily, and then the equation last given will indicate the relation between every other X and its corresponding X'. Among the different homographic systems there is one of special im- portance. It is that in which the intervene between any two objects in