742 



PROCEEDINGS OF THE AMERICAN ACADEMY. 



(0,f|AA,A)=X, 



and every point on the line OA can be represented in the form XA. 



7. The point 



AA + fjiB, 



where A and B are not collinear with 0, is constructed in the same 

 way as the point A + B, and has the following relations to the points 

 A and B. 



Figure 3. 



(1) The point H where the line joining to XA + /tB cuts the line 

 AB is such that 



(Hf|AB)=-^ 



A 



AA + mB = (A + /x)H. (f) 



In order to prove the above relations we shall first prove the fol- 

 lowing lemma : 



^ two lines intersecting in and cutting / in Q and R are cut hy 

 two other lines {intersecting f in P, S) in A, C and B, D respectively, 

 and if through any arbitrary line is drawn cutting AB in H, CD in 

 K, and fin T, then 



(OQiAC) _ (HP[AB) 



(ORIBD) (KSICD)* 



