14 Colorado College Studies. 



origin. In finding other points, the construction must 

 proceed somewhat differently for the two branches. For 

 the infinite branch, take on the line BG a distance BK 

 greater than BH and less than OE. Draw BC and OK. 

 and call their point of intersection F. Draw HF, inter- 

 secting the imaginary axis at L, and then draw LM paral- 

 lel to the real axis and meeting the semi -circumference, 



(of radius- r ] at M. Draw OM, and produce it to meet 



the circle of radius r at N. Draw NQ, parallel to MC, 

 meeting the imaginary axis at Q. Finally, describe about 

 O, with radius OQ, an arc meeting at P and P' a parallel 

 to the real axis drawn through K. Then P and P' are 

 points of the conchoidal branch. 



For the oval, extend the tangent GB on the opposite 

 side of the axis of reals, and take BK' in that direction, of 

 any length not exceeding OD. Draw OK ' and CH, to in- 

 tersect at F ' , and then F ' B, meeting the imaginary axis 

 at S. Through S, a parallel to the real axis is to be drawn, 

 meeting the semi-circumference at T. Then an arc, with 

 radius OT, described about O, will meet the parallel to the 

 real axis drawn through K',in points belonging to the 

 oval. 



If a parallel ruler is used in drawing, this construction 

 may prove more convenient than the preceding, since the 

 cbmpasses will have to be adjusted only once in locating 

 each pair of new points. It should be mentioned that after 

 the distances OD and OE have been determined, we may, 

 if convenience require, replace the points B, G, H, K, K ' , 

 on a tangent to the circle, by points at equal heights on 

 any other parallel to the imaginary axis. 



In each of the foregoing constructions for a comitant //. 

 it has been assumed that the value of // is directly given. 

 An important modification of the problem may be stated 

 as follows: Through any given point of the plane to 

 draw the comitants of a circular locus of given radius, 

 whose center is at the origin. Here it is necessary to de- 

 termine at the outset the quantity (coefficient of i) which 



