The Circular Locus. 25 



in each circle, situated at the center. Accor(linp:ly, in ex- 

 amining the intersection of X cos <p-\-Y sin <f=j) with the 

 locus, no indeterminateness was found to attach to the 

 position of the interior point of intersection, when p was 

 increased indefinitely; on the contrary, the limiting posi- 

 tion of the intersection was found to be the center of the 

 circle. Through this point every comitant jjasses, and 

 each is there touched by the corresponding comitant of 

 the asymptote; while this point of contact, in the case of 

 a given comitant — say the rr-comitant /i — has for its coor- 

 dinates x= oc -|- in, Y= i'x. It is to be observed that although 

 in this way each single comitant exhibits the position of 

 the point at infinity, the entire series of comitants can 

 illustrate only one form of infinite value which might be 

 assigned to the variable x. For beside the value cc + {//, 

 where /j- is finite, X might have the infinite value (x:{x-\-u) 

 with a finite ratio between x and c, or indeed it might be 

 x-\-icc, where x, the real part, is finite. But in these cases 

 the point of contact would fall on a comitant infinity. 



This remark applies with still greater significance to the 

 consideration of the second asymptote, x— iY=0. This 

 locus has nothing of the unique quality which makes the 

 former asymptote a correlative to the line infinity. Its 

 a;-comitant lies on the real axis and its ^/-comitant on 

 the imaginary axis, while any comitant // is at a distance 

 2a from the comitant zero. On the a;-comitant p. the point 

 whose coordinate x has the value cc + i/j. is at an infinite 

 distance from the origin; and similarly for the ?/-comitants. 

 So long, then, as ,'j- is finite, the point of intersection with 

 the line infinity is represented by the two points infinitely 

 distant on the two axes. Accordingly, every (finite) comi- 

 tant of the circular locus has for its individual asymptote 

 the corresponding comitant of the locus x— i y=0. But the 

 second circular point at infinity must not on this ground 

 be asserted to be represented only by the infinitely dis- 

 tant points of the axes, for if we take into account the 

 comitants infinity, the direction of the point in which the 

 asymptote locus meets the line infinity becomes indetermi- 

 nate; and this agrees with the fact that the exterior inter- 



