PHILLIPS AND MOORE. — ALGEBRA OF PROJECTIVE GEOMETRY. 747 



lent to zero. The line f plays the role of the line at infinity, and the 

 point P of zero magnitude is analogous to zero times infinity. 



11. By duality we obtain a definition for the sum of two lines Aa, 

 /jib of magnitudes A, fi respectively. For this addition we assume a 

 fundamental point F and 



Xa + fjib 



is a line c of magnitude A + /t* such that 



(CF|ab)=-^. 



The point F is an exceptional point, i. e. all the lines passing through 

 F are exceptional in the same sense in which the points on f were ex- 

 ceptional. The difference a — b is the line of zero magnitude joining 

 the point F to the intersection of a and b. Here also the Hue of zero 

 magnitude passing through F is not algebraically equivalent to zero 

 but indeterminate. It is analogous to zero times infinity. The com- 

 plete fundamental system then consists of a line f and a point F. 



12. Starting with the definition of point addition the vector addition 

 could be derived from it. Thus take a fixed point and represent 

 any point A in the plane by the difference — A. The addition of 

 these quantities will lead exactly to the vector addition with which we 

 started. The two additions are then related in such a way that either 

 can be derived from the other. 



From the definition of point addition as derived from the vector ad- 

 dition it follows that the point addition obeys the same algebraic laws 

 as the vector addition. 



Vectorially any point in the plane can be expressed in terms of two 

 independent points A, B, where the line AB does not pass through 0. 

 Then choosing A, n properly any point in the plane can be represented 

 by A A -I- /xB. For the point addition, however, AA + //B represents 

 only the points of the line AB, where neither A nor B is on f. In this 

 addition, however, any point X of the plane can be expressed as 



where A, B, C are three non-collinear fixed unit points not on f. That 

 is, the numbers A, ft, v can be so determined that this relation holds for 

 any point whatever of the plane. To show this connect X to B. This 

 line will cut AC in Q, which can be expressed in terms of A and C. 

 Then X can be expressed in terms of Q, and B. Thus 



(A + I/) a == AA -h . C 

 {\ + H + v)X = (k + v) a + mB = ^A -f ^B + vC. 



