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UNIVERSITY OF COLORADO STUDIES 



incident and A2, A3 conjugate imaginary, a cubic with a double- 

 point, or node, at A,A^ arises. In Fig. 4, a vertical line through 

 A,A^ represents the limiting direction of the line joining the two 



points. As conies of the pencil through the fundamental quadruple 

 take the pencil of circles tangent to each other at A,A^ and to the 

 vertical line. A2 and A3 are then represented by the circular points 

 at infinity. To construct the cubic associated with an arbitrary 

 point B, draw rays through B. On each of these rays the pencil of 

 circles cuts out an involution whose double-points are points of the 

 cubic. These points are also the points of tangency of circles of the 

 pencil. Hence, to find the points where a ray <j through B cuts the 

 cubic, take the point M where g cuts m as a center of a circle 

 K passing through A,A^, K cuts </ in the required points X 

 and X'. From this it is seen that this cubic is also the product 

 of a pencil of circles with coincident limiting points and a pencil of 

 diameters through B. As X and X' are equally distant from m, the 

 asymtote is parallel to in at a distance to the left of m equal to BA, 

 (BA, _L m for the sake of symmetry). 



