A THEOREM OX DOUBLE POINTS IX IXVOLUTIOX 

 By J. W. Lasley, Jr. 



A bilinear relation 



axx' +bx + cx' +d = (1) 



between x and x', in which a, h, c, d are known, may be regarded as a 

 projective transformation of the points .r of a line into the jioints 

 .t' of that line. Thus 



5xx' +2x + 3x' +Q = (2) 



sends the point .r = 1 into the point .r' = — 1 and the point x = — 1 

 into the point x = 2. 



We ask ourselves can it happen that if .r = h is sent into .r' = k\ 

 that x = kis sent into x' = h. Such transformations are said to be 

 of period two. They are called involutions. For example 



xx' + 3ix +x')—7 = (3) 



sends x = — 2 into x' = 13 and x' = 13 into x = — 2. 



One can convince himself that the noticeable difference between 

 (3) and (2), namely that in (3) the coefficients of x and x' are alike, 

 is a property which characterizes an involution. From this point 

 of view an involution is given analytically by 



axx' + b(x + x') +d = (4) 



When for x = h the relation (1) demands x' = k, k is said to cor- 

 respond to h. Ordinarily to k will not correspond /; again. This 

 will be so when, and only when, the projective transformation is an 

 involution. In this event we may say that h and k correspond. 

 The points —2 and 13 are seen to correspond in (3). 

 Consider now the involution 



xx' — 7ix + x')+'S3=0 (5) 



different from (3). Let us see whether (3) and (5) have a common 

 pan* of corresponding points. If so, .t and x' must satisfy both (3) 

 and (5). We have, then, xx', — (.t + x'), 1 proportional to the two- 

 rowed determinants obtained from the matrix 



1 3 -7 



1 -7 33 



(6) 



by deleting in turn the first, second and third columns, that is 

 xx' -.x + x' :1 = — 5:4:1 (7) 



Consequently x and x' are given by 



0-2 — 4x — 5 = (S) 



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