The Circular Locus. 15 



characterizes each of the required coinitants, and for tliis 



purpose the comitant equations may be employed. Thus, 



in the case of an ^--comitant, the equation may be written 



ti ( X' ~}~ ■?/"' — v' ) 

 21=^^-^^ — TT"^ ' ^ formula which directly suggests the 



following geometric process: Draw by the usual methods 

 the polar of the given point with respect to the circle of 

 radius r, and from the point in which this i)olar meets the 

 line joining the given point to the origin draw a parallel 

 to the real axis, then the x)erpendicular distance from this 

 line to the given point is equal in absolute magnitude to 

 double the quantity I, answering to the /^ of the preceding 

 constructions. Having found this quantity, the comitant 

 is constructed as before. 



It is to be observed that when I and r, are found by tliis 

 process, the coordinates of the given point are fully known. 

 hence this construction solves also the problem: To deter- 

 mine geometrically the coordinates of a given point of a 

 plane, when regarded as a point of a circular locus of given 

 radius, having the origin as center. 



Having now sufficiently considered the form. and posi- 

 tion of the comitant curves which make up the circular 

 locus, the next step will be to examine the most important 

 properties of the locus as they are discussed in the elemen- 

 tary analytical geometry, and observe what new light is 

 shed by the present construction upon the familiar pro- 

 cesses and results. 



"A right line," it is commonly said, intersects the circle 

 in two points, which may be real and discrete, coincident, 

 or imaginary. Just as the "circle," here used to mean the 

 geometric equivalent of the equation, is in the present in- 

 terpretation replaced by a double system of curves, together 

 forming the circular locus, so the right line gives place to 

 a double system of lines. These intersect the comitants 

 of the circular locus in every part of the plane, but since 

 these indefinitely numerous intersections of the separate 

 comitants are not liable to be confused with the two special 

 points mentioned in the theorem, the word "intersect" 



