74 



KANSAS INI\KRSIIA ( H'A K ri'.Rl .^ 



points of inttn'section of any tangent to K with the hnes 1 and 1' 

 and the two imaginary tangents AA ' BB ' is constant. Hence we 

 have the following theorem which holds for two separate in\^ariant 

 points, whether real or imaginary: 



Theorem 2. — /// c'rvvr /^ro/er/iiw iransfofmaticn -icliieh leai'cs tiv(> 

 distinct f'oiiits i/n'aria/ii. tlic anlianiiojiii ratiii cf any pair of rorrc- 

 spomiiiii:^ points and the t-ti.<o in'oariant points is ronsta/it. 



This is called the cliaractrristic an/iarnumir ratio of the transform- 

 ation T];. It is convenient to designate this constant b)' the same 

 letter k. A transformation Tj^ is then one determined by a conic 

 K whose constant anharmonic ratio determined by the tangents 

 AA,' BB.' 1, r is k: and the residting characteristic anharmonic 

 ratio is also k. 



But when the invariant points of the transformation coincide, we 

 no longer have a characteristic anharmonic ratio for the transform- 

 ation. However another relation is found to hold for pairs of 

 corresponding points, which relation is constant for all jiairs of 

 corresponding points in the transformation. \\"e sliall now proceed 

 to determine this rehition. 



Let K be a h\'perbola haxing one of its as\iii]")totes as AA ' i)er- 

 })endicular to OX. 

 (Fig. 3). Let O'FP' 

 be any tangent cutting 

 1 and 1 ' in V and V .' 

 corresponding points 

 o f t h e projection. 

 Draw KP„ parallel 

 to 1 ' and touching K. 

 Taking A a; EP^, 1. 

 and 1 ' as four fixed 

 tangents, the anhar- 

 monic ratio determ- 

 ined by an\- fifth 



tangent as PP ' is constant. Denoting this b\- k' we have 



O'P O'P' O'P BP' 



PB • p^l3 " crT ' 



O'P AP 



(V^P 

 AP OP 

 A'P'"'TP 



k' (O'BPP- ) 

 Put from the fi<.iure we have 

 hence k ' - 



also 



BP 

 BP' 



OP. 

 PP_ 



A'P' BP 



If the tangent PP ' be moved up infin- 



itesimally near to AA.' we see that the constant anharmonic ratio 



A'I£ 



along the tangent AA' is triven b\' k 



00 HA A' ) 



AE 



From 



