The Circular Locus. 17 



hence the branches of its comitants which are here met by 

 the corresponding linear comitants are both ovals, while at 

 the other intersection of the loci, the conchoidal branches 

 alone are present. If j) is made indefinitely great, the 

 latter intersection recedes to infinity, while the former 

 approaches the origin as its limiting position. In this 

 way is indicated the geometric representation of the "cir- 

 cular points at infinity," only one of which, it appears, is 

 situated at an infinite distance on the plane, and this one 

 must be regarded as in an indeterminate direction from 

 the finite points of the locus. But the further examination 

 of these important points may be deferred a little until the 

 subject of the asymptotes is reached. 



The geometric construction of the finite imaginary 

 points in which the circular locus may be intersected 

 by a real linear locus needs little further statement. The 

 real parts of the loci — the circle and non-secant line — are 

 supposed given. A perpendicular is to be drawn to the 

 line from the center of the circle (the origin), and from -its 

 foot a tangent to the circumference. The length of this 

 tangent, laid off on the above-mentioned perpendicular in 

 both directions from its intersection with the given line, 

 marks out the points required. 



By the aid of the circle may also be solved a problem, 

 which may often be required, viz: To draw other comitants 

 of a real linear locus, when the comitants zero (?'. e., the 

 real line) are given. The distance between two comitants 

 of one system, say the ic-comitant /^ and the ic-comitant y, 

 for a given difPerence between ,'j. and v, depends on the 

 direction of the real line, and is independent of the latter's 

 distance from the origin. Hence if the given line happens 

 to be a secant, a parallel may be drawn to it which will lie 

 outside the circumference, and then the points of intersec- 

 tion of the loci found as above. Selecting one of these 

 l^oints — for instance, the exterior one— its distance from 

 the given line proves to be to the coefiicients of i in its co- 

 ordinates X and Y, as unity to sin <p and —cos <p respectively. 

 The corresponding distance of the two comitants m will be 

 to // in the same ratio, hence these distances are /^ cosec <p 

 and —A* sec <p. 



