60 PROCEEDINGS OF THE AMERICAN ACADEMY. 



f intersecting (/) on the polar plane of .r with respect to p. The corre- 

 spondence between x and ^ is a correlation. From the equation 



— ^ (p -^ y) ^ ^ 



"^ ^ {ck -v) (<!> y) 

 or X (<^ x) (0 y) — (p X y) = 0, 



it is seen that the locus of y is 



^ = X (0 .,-) (/, - [p x] . . . (13) 



Similarly the locus of planes rj making a given angle ^ with the plane ^ 

 is a point such that the line connecting it to F passes through the 

 polar point of | with respect to p. The locus of planes making with 

 ^ an angle — A. is 



z= -\(F^)F-M. 



Substituting in this the value of | from (13) we get 



z=\{Fpx)F -\(<j> x) [p ct>] -\-[p-p x] 



since (F (f)) = 0, F being a point of 0. Using the conditions 



[F p] = 0, [(j>p] = F, and [p-px] = ^ (pp) x = x 



we get z = X 



Hence the correlations determined by a distance ^ and by an angle 

 — X are inverse. Now the correlation set up by an angle — X is 

 inverse to that determined by an angle X. Hence the equations 



Jy = \ 



fVi = \ 



where .r and ^ are given, y and v variable, set up the same correlation. 

 Through a correlation 



xy = \ 



to Xi and Xo correspond the planes 



\ {(j) Xi) (I) — p Xi, 



X ((/> .T2) — p .r2. 

 The angle between these planes is 



ip [>^ {4> xi) (i> — p .ri] [X (<^ .T2) <i> — p XoJ[) 



{F[\ {4>x,) (/) -px,]\ \F[K {ci>x-2) 4> -pxo\: 



