The Circular Locus. 



13 



oval determine the species, in the classification of Plticker 



as in that of Newton: so that if --= . the curve is of 



4 

 species 152, but, for other values of ^, of sjDecies 148. 



The equations of the comitants also sujj^gest a process 

 of geometrical construction entirely different from that 

 deduced from the original equation of the locus. (See Fig. 



II. ) If, as before, it is desired to construct the £r-comitant />, 

 let the axes and the circle of radius ?• and center O be again 

 drawn, and, in addition, a semicircle having for its diameter 

 that radius, OC, of the former circle which extends along the 

 imaginary axis in the direction indicated by the sign of ."■. 

 Draw also, through B, one of the points in which the circle 

 of radius r meets the real axis, a tangent to this circle. 

 All the foregoing may remain unerased during the entire 

 construction. On the tangent last drawn, take points G 

 and H, so that BG=//. and BII=2//. Draw OG, and pro- 

 duce it sufficiently, so that a distance equal to GB may be 

 laid off from G on the produced part, to E, while the same 

 distance from G toward O, fixes the point D. Then OE 

 will be the maximum distance of the conchoidal branch 

 and OD that of the oval branch, from the axis of reals, and 

 the extremities of these maximum ordinates are to be 

 located on the other axis, in opposite directions from the 



