16 Colorado College Studies. 



may be retained, and the proposition will read, ''A circular 

 and a linear locus intersect at two points/' By a point of 

 intersection of two loci must of course be meant one whose 

 coordinates, when it is regarded as a point of the one locus, 

 are the same as when it is taken to belong to the other. 

 If such a point, then, lies on the ^-comitant I of one locus, 

 it is on the .r-comitant ^ of the other (the c having the 

 same value for the two), and a similar statement is true of 

 the other system of comitants. Thus a point of intersec- 

 tion of two loci may be defined as a point in ichicli two 

 comifanis {viz., one of each series) belonging to one locus 

 are met hijflie corresponding comitants of the other locus. 

 The theorem of analytical geometry, above quoted, speci- 

 fies not only the number of intersections of the two loci, 

 but their kinds. This, however, is done under the tacit 

 assumption that both loci are real. The full statement 

 intended, therefore, is as follows: "A circular and a linear 

 locus, each of which has a real branch, intersect at two 

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

 or conjugate imaginary." The locus of a linear equation 

 with real coefficients (called, for brevity, a real linear locus) 

 has the two comitants zero coinciding in a single right 

 line, while each of the series of comitants consists of 

 lines parallel to this. Real intersections, whether in dis- 

 crete or consecutive points, are formed by the real parts 

 of the two loci, just as in the ordinary constructions of 

 analytical geometry. The situation of the imaginary inter- 

 sections may be easily studied by combining the two equa- 

 tions x' + Y- = ?-- and X cos cr + Y sin c'=j9, whil e assu ming 

 p)>r. Elimination gives x=j) cos <s±i sin <fs/pi~ — r, and 

 Y=p sin (f^i cos (f^^p'—r'^, whence 



x-|-tY = (j)±'v/jr— r") (cos f-f / sin c). 

 It is at once apparent that the two intersections lie 

 on the perpendicular drawn from the origin to the real 

 part of the linear locus, at equal distances on opposite 

 sides of the latter, but always on the same side of 

 the origin. The intersection nearest the origin is wit hin 

 the real part of the circular locus, since p—'v^j/— >*■<?•, 



