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 # 1} x z , x 3 , x 4 has for its polar, with regard to the infinite quadric, 

 the plane, 



dU dU dU dU A 



X 1 3 h X 2 -j h X. A -j h # 4 -7 = 0. 



dxi dx 2 dx 3 dxi 



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. 



7T 



The intervene from an object to any object in its polar extent is equal to -^ . 



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, Y, corresponding 

 thereto in the system B. The correspondence is to be simply of the one-to- 



