4 Colorado College Studies. 



rather, to reduce tbe problem of determining it to a finite 

 number of solutions. Let four of these be supplied by 

 assigning, in the finite region, lines which the curve must 

 touch, and a fifth by requiring that the double tangent to 

 the curve meet the line infinity in a specified point, K. 

 Then a set of curves is determined, to each one of which one 

 tangent is drawn at / and one at J, meeting at a definite 

 point. Now let K move along the line infinity. We may 

 regard each curve of the set as undergoing a continuous dis- 

 jjlacement and distortion; and if the attention be fixed upon 

 one such varying curve, we shall see its focus describing a 

 locus, defined by the intersection of a tangent at / with a 

 conjugate tangent at J. This separation of the tangents at 

 / and J belonging to one curve from those belonging to any 

 other affords an indication that the whole pencil of tangents 

 through I is resolvable into partial pencils, in each of which 

 a ray through I is met by a single ray through J^projectively 

 corresponding to it. If this be the case, the locus of each 

 focus is a circle. But when the point K, in its motion 

 along the line infinity, reaches the point at which that line 

 is cut by one of the four given tangents, the curve has four 

 contacts with lines through K, and being of third cla^ss only, 

 it must break up into the point K and a curve of second 

 class cutting the line at infinity at I and J, and touching the 

 three remaing given lines. Such a curve, of course, is a circle, 

 and its center is the focus of the disintegrating third-class 

 curve. We have thus the theorem that, when five lines are 

 given, the centers of the circles which touch three of these 

 lines lie by fours upon circles which have a common point; 

 that is, if four given lines be a, b, c, d, the center of a circle 

 touching a, b, c and those of one touching a, b, d, one touch- 

 ing a, c. d and one touching 6, c, d lie on a single circle; 

 moreover, if a fifth line be added, and the proper single circle 

 selected for each of the five sets of four lines formed by 

 dropping one line from the given five, these five circles pass 

 through a point. 



