The Circular Locus. 29 



A more important modification results from admitting 

 imaginary values of the radius. The equation X' -f y' == r' 

 (when R=r+//'), is at once seen to belong to a singly in- 

 finite series of different loci, which vary in form with the 

 magnitude of the ratio p : r. The case already considered 

 is that in which this ratio is zero; and the most closely 

 analogous case, as may be readily conjectured, is that in 

 which the ratio is infinite. Those two cases alone can be 

 represented by equations with real coefficients. 



The comitants of the locus X' + y''+,"""=0, like those of 

 X' + ¥-=?•", belonging to Newton's "defective hyperbolas 

 having a diameter," being symmetrically divided by one 

 of the axes. The j^-comitant is a straight line, co- 

 inciding with the real axis, and the successive comitants I, 

 when ^ is small, are conchoidal curves, convex toward this 

 axis, and all touching it at the origin. Each lies within 

 its predecessor, and exhibits a greater curvature, until the 

 ic-comitant /> is reached, when the curve, whose shape at 

 first might suggest a bowl, and later a purse with a narrow- 

 ing neck, has at length completely closed together, and has 

 a node on the imaginary axis, at a height of p above the 

 origin. The succeeding comitants have ovals, like the comi- 

 tants of the locus X" +¥"=?•'", but these lie on the same 

 side of the real axis as the conchoidal branches, and op- 

 posite the convexity instead of the concavity of the latter. 

 All the ovals, as in the former case, touch the real axis at 

 the origin. Thus it is seen that in a single series of comi- 

 tants occur successively representatives of Newton's species 

 45, 41, and 39 — the conchoidal hyperbola pur a, nodafa, 

 and cum ovali, the last having the oval on the convex side. 

 As ^ increases beyond p to infinity, the conchoidal branch 

 grows more straight, and the oval shrinks to a point at the 

 origin. 



Between this form and that of the locus X' + Y"=r-' there 

 is an unbroken gradation, corresponding to the positive 

 values of the ratio p : r; and similarly another for the nega- 

 tive values. In the intermediate forms, however, the comi- 

 tants are no longer symmetrical in respect to either axis, 



