THE CIRCULAR LOCUS 



Geometrically Constructed to Show Imaginary Values of the 

 Variables. 



By FRANK H. LOUD. 



In the paper ou "A Geometrical Construction for the 

 Imaginary Points and Branches of Plane Curves," which 

 I read before this society on November 18, 1892, the gen- 

 eral method of construction therein outlined was illus- 

 trated by the simple example of the equation x'" + y''= ?*'"'; 

 and it was shown by a diagram how the full significance of 

 this equation, for imaginary as well as real values of the 

 variables, may be geometrically interpreted by the aid of 

 additional curves, accompanying the circle described 

 around the origin. A more full and methodical treatment 

 of the theory of this plan of construction, in its general 

 application, has since been contributed to the Annals of 

 Mdthematics, but at the date of the present reading has 

 not appeared in print. It is the object of the present 

 paper to continue somewhat further the treatment of the 

 circular locus just mentioned, as a special case of the gen- 

 eral theory, borrowing, from the two papers above named, 

 as much as may be necessary to render this intelligible to 

 a reader who has seen neither of the others. 



The principle of construction employed is of course 

 based upon the well-known geometric interpretation of im- 

 aginaries, in which a+ia is represented by a line drawn 

 from the origin to a point a units to the right of a vertical 

 axis and « units above a horizontal axis, called the imagi- 

 nary and the real axis respectively. 



The apparatus postulated in this usual representation 

 of imaginaries— a pair of rectangular axes in a plane — 

 coincides precisely with that of the still more familiar 

 Cartesian method of translating algebraic equations into 



