778 PROCEEDINGS OF THE AMERICAN ACADEMY 



From the triangle ABD we get 



^ BD + DA + AB 

 o = • 



BD 



Since in the isosceles triangle ABC, AB = CB = c = — a , and since 

 DA = 0, being measured on a line passing through F, the above relation 

 can be written 



g ^ BD + c ^ BD-a ^ CD ^ (CB|DR) = (AF|DP). 

 BD BD BD ^ 



Thus the invariant 8 is the cross ratio determined by the point A, 

 the point F and the points in which FA cuts the given line BC and 

 the line f. 



From the fundamental relations of the triangle we have 



a + h^c _ A + B+ C 



a ~ A ' 



Therefore 



^^AJ,B±C^ (21) 



which is the same function of the angles that the former is of the dis- 

 tances. It is to be noted also that if A describes a line passing through 

 R, 8 will be unchanged. 



The group of collineations which leaves F, f fixed leave S invariant 

 although distance and angle are not invariant. 



36. Metrical Illustration. When we throw the line f to infinity the 

 system takes an interesting metrical form. The equation 



AB = const. 



gives for fixed A a line b parallel to FA (Figure 26). 



For points B on this line the distance AB is then a constant times 

 the Euclidean area of the parallelogram ABF. Further, our distances 

 along the line AB are proportional to the Euclidean distances and there- 

 fore to the area of the parallelogram. Hence distance from a point A 

 is proportional to the area of the parallelogram ABF, or 



AB = kn ABF 

 where ^ is a function of A. 



