82 



Journal of the Mitchell Society 



[December 



In the two involutions (4) and 



a'xx' + 6'(x + x') + d' = Q (14). 



the common pair of corresponding points are such that xx', x -\- x'' 

 1 are proportional to their cofactors in 



xx' X -\- x' 1 



a h d (15), 



a' h' d' 



that is, 



xx' : x -{- x' : 1 = {hd' — h'd) : {a'd — ad') : {ah' — a'h). 

 Consequently x and x' are given by 



{ah' — a'h) x^ + {ad' — a'd)x + {hd' — h'd) = (16). 



The double points of (4) are given by 



ax'- + 2hx + d = (17), 



those of (14) by 



a'x2 + 2b'x + d' = (18). 



. The involution determined by the two pairs of points in (17) and 

 (18) is 



X + x' 



— 2b 



— 26' 



Its double points are given by 



x 

 —b 

 —b' 



= 



= 



(19). 



(20), 



or 



(21) 



{ah' — a'6).r2 + {ad' — a'd)x + {hd' — b'd) ^ 



which is identical with (16). We have, then, established the theorem 

 that the common corresponding points in two involutions is the pair 

 of double points in the involution determined by the double points of the 

 given involutions. 

 Chapel Hill, N. C. 



