784 PKOCEEDINGS OF THE AMERICAN ACADEMY. 



of two points it should be kept in mind that the magnitude of the 

 sector may be zero without the product necessarily being zero, e. g. if the 

 lines intersect on f In this case the magnitude of the sector is zero, 

 but the product is not zero. If we now take the two lines a and b as 

 represented by segments starting at the point of intersection, thus 



a = BC b = AC 



where A, B, C are unit points, then 



AB - AC == (ABC) A. (32) 



Since the magnitudes of the lines a, b are represented by the lengths 

 of the segments, from the definition the product would be BC • AC • Aj 

 where A^ represents the sector determined by the two lines. But A^ 

 can be written &s A -A where A means a unit sector and the product 

 takes the above form. 



The product of three unit lines is defined to be the angle area of 

 the triangle determined by the three lines, and the product of any three 

 lines is this product multiplied by the product of the magnitudes of 

 the lines. When the three lines pass through a point the angle area 

 of the triangle is zero and conversely. Hence, TTie necessary and suffi- 

 cient condition that three lines a, b, c (not zero lines) should pass through 

 the same point is 



abc = 0. 



Since the angle area is also a directed quantity 



abc = boa = cab = — bac = (33) 



43. The product of a line a and a point A will be defined as the 

 invariant 8 of the point and line multiplied by the product of the 

 magnitudes of the point and line. The invariant vanishes when and 

 only when A lies on a. Therefore if neither a nor A are zero elements, 

 the necessary and sufficient condition that A lies on a is 



Aa = 0. 



If the line a is represented. by a segment of length a determined by two 

 unit points B and C, then irom the above definition 



ii.2i = aAb = aABc (34) 



= a (A Be) 

 = A {acB). 



That is, the product of a point and a line (represented by one of its 

 segments) is equal to the line area of the triangle having the point for 



