1921] A Theorem on Double Points in Involution 81 



and are — 1 and 5. We find upon testing these in (3) and (5) that 

 they correspond in both involutions. 



Let us inquire whether in an involution, like (3), there are points 

 which correspond to themselves. If so, these are given by 



x2 + 6x — 7 = (9) 



obtained from (3) by putting x = x'. These are here —7 and 1, 

 and are called the double points of the involution. Similarly the 

 double points of (5) are 3 and 11. 



The question naturally arises as to what extent a choice of cor- 

 responding points determines an involution. Equation (4) appears 

 to have three arbitrary constants, but it will become evident upon 

 reflection that only the ratios of these are important. Clearly, if 

 we knew two pairs of corresponding points, they would by (4) give 

 us two homogeneous linear equations, which are usually just ade- 

 quate for determining the ratios a : b \ d. For instance, let us arbi- 

 trarily assign — 7, 1 and 3, 11 as two pairs of corresponding points, 

 and ask for the involution determined by them. The pair — 7, 1 

 in (4) gives 



7a+6b — d = (10); 



the pair 3, 11 in (4) gives 



33a + Ub + d = 0. (11). 



From (10) and (11) we have a : — b : rf as the second order determin- 

 ants obtained from the matrix 



7 6—1 



33 14 1 



by deleting in turn the first, second and third columns. In this way 

 we obtain a:b:d = 1: 2: — 5. Consequently, the involution de- 

 termined is 



xx' — 2{x + x') — 5 = (13). 



whose double points are — 1 and 5. 



We have seen that the involutions (3) and (5) have just one pair 

 of corresponding points in common, the pair — 1,5 given by (8). 

 The double points of (3), namely — 7, 1 and of (5), namely 3, 11, 

 taken as two pairs of corresponding points determined (13), and in- 

 volution different from (3) or (5). The common pair — 1, 5 of cor- 

 responding points in (3) and (5) turned out to be the same as the 

 double points of (13). We shall now show that this is not an acci- 

 dent of the particular involutions chosen, but follows from the nature 

 of involutions. 



(12), 



