24 Colorado College Studies. 



sections of the circular and linear locus is not practically 

 appropriate, since it invokes the aid of higlier curves to 

 treat a problem which is in fact amenable to the ruler and 

 compasses. A similar objection must hold in other cases. 



The problems thus far considered embrace the con- 

 struction of linear loci as secants, tangents, or polars, when 

 imaginary quantities enter the problem in any way what- 

 ever; and are hence adaptable to the treatment of any 

 specific elementary question which involves only one circle. 

 But there is one j)articular case of tangents which, on ac- 

 count of its importance as well as of the special peculiari- 

 ties it exhibits, requires a separate investigation. This 

 case is that of the asymptotes. 



The equations of the asymptotes of the locus X'4-y'=?-'' 



Y Y . . 



are x-I-/y=0, or -=i, and x— ?'y=0, or-= —i. It is atonce 

 X X , 



apparent that the equation x4-?'y=0 for any (finite) values 

 of X and y can be satisfied only at the origin ; though from the 



form-=i, it appears that infinite values of x and y might 



differ numerically by any finite amount. Hence the finite 

 comitants of the locus all pass through the origin, while any 

 line whatever of the i^lane may be taken as a comitant in- 

 finity. Now a real "line at infinity'" is characterized by 

 opposite properties, i. e., its comitants p., wdien ,'j- is any 

 finite quantity, lie altogether in the infinitely distant region 

 of the plane, unless the comitant zero be regarded as 

 parallel to an axis, when any other parallel to the same, 

 though at a finite distance therefrom, becomes a finite 

 comitant. But by assuming //. infinite we may identify its 

 comitants with any lines in the finite region, irrespective of 

 direction. Hence the first "circular point at infinity" re- 

 garded as the intersection of these two loci, is at any finite 

 point; and the statement that all circles pass through this 

 one "circular point," is only in this sense true, that one 

 analytical expression will serve to designate for all circles 

 the infinite coordinates which will satisfy their equations; 

 while, as the geometric equivalent of this analytical ex- 

 pression is indeterminate, the point in question is in fact 

 geometrically different for different circles. It is, actually. 



