A Treatise on the Theory of Screws

454 THE THEORY OF SCREWS. [413- We hence deduce the following important result, that— The intervene between any two objects is proportional to the logarithm of the anharmonic ratio in which the straight line joining the corresponding points is divided by the infinite quadric. We similarly find that— The departure between any two ranges in the same extent is proportional to the logarithms of the anharmonic ratio of the pencil formed by their two corresponding straight lines, and the two tangents in the same plane from their intersection to the infinite quadric. 414. Poles and Polars. The point xlt x3, x3, x4 has for its polar, with regard to the infinite quadric, the plane, dU dU dü dU + ^2 t---------F x3 -j--h x4 —— = 0. aæ1 (LXi> clx^ Thus we see that an object corresponding to the point will have a polar extent corresponding to the polar of that point with regard to the infinite quadric. The following property of poles and polars follows almost imme- diately. The intervene from an object to any object in its polar extent is equal to ~ . -J We have hitherto spoken of the departure between a pair of ranges which have a common object: we now introduce the notion of the departure between a pair of extents by the following definition :— The departure between a pair of extents is equal to the intervene between their poles. 415. On the Homographic Transformation of the Content. In our further study of the theory of the content we shall employ, instead of the objects themselves, their corresponding points in ordinary space. All the phenomena of the content can be completely investigated in this way. Objects, ranges, and extents, we are to replace by points, straight lines, and planes. Intervenes are to be measured, not, indeed, as distances, but as logarithms of certain anharmonic ratios obtained by ordinary distance measurement. Departures are to be measured, not, indeed, as angles, but as logarithms of anharmonic ratios of certain pencils obtained by ordinary angular measurement. I now suppose the several objects of a content to be ordered in two homographic systems, A and B. Each object, X, in the content, regarded as belonging to the system A, will have another object, F, corresponding thereto in the system B. The correspondence is to be simply of the one-to-