64 PKOCEEDINGS OF THE AMERICAN ACADEMY. 



where /? is a plane and F a point. Dual considerations show that /? is 

 fixed under all the coUineations, i. e. coincides with </>. Hence b}' a 

 proper choice of units we have 



— ^ _ {4>r-Fs) ^ (cl>s-Fr) ^ ^ ^^^^ 



{p r) {p s) (pr) {ps) 



The angle between two lines is zero if they cut a line through F in the 

 plane ^. The angle is infinite if one of them belongs to the complex 

 p and they are not cut by a line of the plane (/> passing through the 

 point F. 



14. We have seen that 



aH/ = X 



sets up a correlation. To x and y correspond planes 



X(0 x) 4> — p X, 



X(0 y) (f, — py. 



To X y corresponds the intersection which can be written 



X [(t)-xyF] + (p X y) p — h (p p) [x y] 



Hence to lines r and s correspond lines 



X [0 r • F] + {pr) p — r, 

 \[(t)S-F] + (j)s) p — s. 



The angle between these lines is 

 (0 jX[<Ag-i^]+ {ps)p-s\ F\\[(i>rF]+ {p r) p - r]) 

 (p IX [<pSF]+ {p s)p-s\) {p IX [c/> r F] + Cp r) p-r\) 



_ (cj) s . F r) 

 (p s) (p r) 



Hence the angle between two lines is equal to that between the lines 

 corresponding to them through the correlation. 



xy = \ 



In particular when X = we see that the angle between two lines 

 is equal to that between their polar lines with respect to the complex p. 



15. Distance from point to plane. We wish to determine a 

 function ^1 a of the coordinates of a point and plane such that if 

 either is fixed the other satisfies a linear relation and such that 



