280 UNIVERSITY OF COLORADO STUDIES 



trary fixed point B the circles of this system cut out an involution 

 of points whose double-points X and X' are two points of the circular 

 cubic associated with the point B in the Steinerian transformation of 

 the given imaginary quadruple. The points X and X' are also the 

 points of tangency of g with two circles of the given coaxial system. 

 Hence, according to a well known construction, the points X and X' 

 are obtained by finding the point of intersection M of ^ with m, the 

 line joining the finite imaginary points of the quadruple. With M 

 as a center pass a circle K through P and Q which will cut g in the 

 required points. From the figure it is seen that the two points of 

 the cubic on a ray through B are equally distant from m. Hence. 

 taking a ray through B parallel to m, the point at infinity corres- 

 ponding to Q will be in a line a through P parallel to m. In other 

 words, the line a is the real asymptote of the cubic. Considering 

 the pencil of circles through P and Q, the same circular cubic is 

 also produced by this pencil and the pencil of corresponding diame- 

 ters through B. 



II. The Cubic Serpentine. 



This curve is produced by assuming two separate real and two 

 conjugate imaginary points as the fundamental quadruple. In Fig. 

 2 let A, and Aj be the real points and the circular points of the pencil 

 of circles through Aj and K^ the imaginary points. To find the 

 points Y and Y ' where a ray I through B cuts the cubic, let I cut n 

 at N. With N as a center construct the circle L orthogomal to the 

 pencil of circles through Aj and Aj. The circle cuts I in the required 

 points Y and Y'. This cubic appears, again, plainly as the product 

 of a pencil of circles and a pencil of diameters through B. Two 

 points Y and Y' on a ray through B are always equally distant from 

 n. To R corresponds the infinitely distant point of the cubic; con- 

 sequently the asymptote h is parallel to n and its distance SC from 

 til is equal to E.C. 



III. The Cubic Serpentine with Isolated Point. 



The quadruple consists of two distinct points AjA.^ and two co- 

 incident points AjA^. It is assumed that the direction of the line 

 joining A^ with A„ in the limit, i. e., as they become coincident, cuts 



