The Imaginary in Geometry 



45 



lutions and precisely the same involutions as are formed by the 

 blue lines through the origin of the pair of conjugate imaginary 

 lines x-/ar -f- yr/b 2 = o. The asymptotes are tangent to the 

 hyperbolas at infinity and cut out on the line infinity involutions 

 which, in Von Staudt's sense, are the intersection of the aforesaid 

 line-pair with the line infinity. We can use as the symbol of 

 these two involutions the red vectors 



b -l b J 



on the line infinity conceived as parallel to the y-axis. 

 The circular rays through the foci 



and 



(.r — ae) 2 -\- y 2 = o 



(x + aey--\-y 2 = o 



are tangent to the ellipse, as can easily be verified in the usual 



Fig. 41. 



way. The thing can be pictured as fig. 41. The blue lines from 

 the four foci F X F 2 FF to I" and /" are tangent at P lf P x , P 2 , P 2 to 



x l _ £. _ 

 a 1 &'- 1 ' 



a supplementary curve. G 1 and G, are foci of the hyperbola. 



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