748 PROCEEDINGS OF THE AMERICAN ACADEMY. 



Since A, B, C are linearly independent, if 



A + /i + v = 



then X is on f. 



From the above equations 



(A + /i + v) X - yuB = (A + v) a. 



Hence 



A + n -r V 



where P is the point in which BQ cuts f. Similar relations involving 



A, V can be obtained. Since A, fx, v are proportional to uniquely de- 

 termined cross ratios the expression for X is unique. 



§ 2. Distance and Angle. 



13. We have considered two kinds of addition, either of which is 

 expressible in terms of the other. The vector addition gives for the 

 points of the plane a non -homogeneous, two-dimensional representa- 

 tion, the point addition, a homogeneous three-dimensional. The latter 

 is more satisfactory for descriptive problems and will be assumed as 

 the fundamental addition throughout the remainder of this paper. It 

 has been remarked that expressions of the form B — A, when A and 

 B are unit points, then combine according to the vector addition. 

 Through a study of their expressions we shall derive a sort of distance 

 that is intimately connected with the subject under discussion. 



14. Vectors. We have seen that B — A represents a point of zero 

 magnitude on f We might infer that, if 



B - A =- D - C, 



the lines AB and CD pass through a common point on £ This is 

 proved by writing the equality in the form 



A + D = B + C. 



The line f thus has the same harmonic E with respect to A, D and 



B, C. Let AD and BC cut f in P and Q. Then from the intersection 

 of AB and CD the points A, E, D, P and B, E, C, Q are perspective. 

 Since f joins two corresponding points, it must pass through the inter- 

 section of AB and CD. Writing the equality in the form 



A-C=B-D 



