The Circular Locus. 7 



of its capacity of representing at least one such set. To 

 render visible this effect of the equation, it is necessary to 

 indicate, for points in different parts of the plane, by what 

 coordinates they may, consistently with the equation, be 

 represented; and this is effected, as already mentioned, by 

 indicating the imaginary part of each coordinate. If a 

 line is drawn through all these points of the plane for 

 which the imaginary part of x has a certain constant value, 

 say 1/, another through all those points for which its value 

 is 2/, etc., then this system of lines will indicate to the eye 

 the value of the imaginary part of x for all points of 

 the plane. Another series of lines will indicate in the 

 same way the manner in which the imaginary part of y 

 varies over the plane; and when these two things are known 

 for any iDoint, the real parts of the coordinates become also 

 known, and the geometric significance of the equation is 

 fully set forth. It is true that unless a point be situated 

 exactly on one of the lines of each series, its coordinates 

 are only approximately indicated; but, as will be seen, it 

 will always be possible to draw a line of each set through 

 any assigned point of the plane, and ascertain the value of 

 the coefficient of i which remains constant upon that line; 

 thus the determination of the coordinates may be made 

 precise. 



To these lines I have ventured to give the name of 

 comitants, designating the two series as ic-comitants and 

 2/-comitants respectively, and affixing to either name the 

 number or symbol denoting the constant value of the co- 

 efficient of I in the expression for the coordinate of any 

 point upon it. Thus the ic-comitant //. is defined to mean 

 a line (or curve) drawn through every point for which the 

 imaginary part of the value of x has the value i'j-. 



I will also adopt in the remainder of this paper, the 

 rule of denoting by small capitals a, b, etc., quantities 

 which are entirely unrestricted in value, and hence are, in 

 general, complex; while for their real parts I use italic let- 

 ters, and Greek letters for the coefficients of the imaginary 



