58 PROCEEDINGS OF THE AMERICAN ACADEMY. 



Then either 



F, {A, B) = 

 or 



F.2 {A, By = F2 (A, A) F.2 {B, B). 



This last equation shows that Fo {A, B) factors into a function of .1 

 times a function of B. CaUing this function (</> A), and writing 



Fi {A, B) = I [Fi U, B) - F, {B, A)] = {a A) (/3 B) - (a B) (/3 A), 



we have 



^^ (g A) (^ B) - (g B) {/3 A) 



(<t> A) {<!> B) 

 Using the identity (3) this takes the form ^° 



(a^-AB) _ (q-AB) 



AB 



((/,.4) (</.5) {4>A) {<l>By 



where we put q in place of the t\\o rowed matrix (g /3]. 



10. Angle between two planes. We define the angle between 

 two planes as such a function a (3 of their coordinates that if the angle 

 is given and one of the planes fixed, the other passes through a point 

 and for three planes of a linear pencil 



'a^ + (S7 + 7a = 0. 



By the same argument as for the distance between two points we 

 obtain for the angle 



"^ {F a) {F 13) 



where ^ is a fixed complex and F a fixed point. 



Distance is a relative invariant under the group of collineations 

 that leave the complex q and the plane ^ fixed. Similarly angle is a 

 relative invariant under the group leaving p and F fixed. In order 

 that fixed relations may exist between distances and angles we wish, 

 if possible, these groups to be the same. We assume that the complex 

 q does not degenerate into a line. Then the only complex and point 

 determined by q and (f> is the complex q itself and the polar point of (f) 

 with respect to it. Hence we have 



p = q, 



F=\{<i> p]. 



10 We consider (a p.A B) as a regressive product (a. p. A B), in which we 

 expand the product (P.A B) and then multiply by a. 



