Kansas University Quarterly. 



Vol. I. OCTOBER, 1892. No. 2. 



Unicursal Curves by Method of Inversion. 



BY HENRY BYRON NEWSON. 



This paper contains a summary of the work done during the last 

 school year by my class in Modern Geometry. Since many of the 

 results were suggested or entirely wrought out by class-room discussion, 

 it becomes practically impossible to assign to each member of the 

 class his separate portion. Many of the results were contributed by 

 Messrs. M. E. Rice, A. L. Candy, H. C. Riggs, and Miss Annie L. 

 MacKinnon. 



The reader who is not familiar with the method of Geometric 

 Inversion should read Townsend's Modern Geometry, chapters IX. 

 and XXIV; or a recent monograph entitled, "Das Princep der 

 Reziproken Radien," by C. Wolff, of Erlangen. 



When a conic is inverted from a point on the curve, the inverse 

 curve is a nodal, circular cubic. 



This is shown analytically as follows: let the equation of the conic 



be written 



ax2-|-2hxy-|-by2-j-2gx^-2fy=o; 



which shows that the origin is a point on the curve. Substituting for 



X and y and ~ — — , we have as the equation of the 



x2-j-y2 x^-fy- 



inverse curve 



ax3 + 2hxy+by2 + 2(gx+fy)(x2+y2)=o. 

 The terms of the second degree show that the origin is a double 

 point on the cubic; and is a crunode, acnode, or cusp, according as 

 the conic is a hyperbola, ellipse, or parabola. The terms of the 

 third degree break up into three linear factors, viz: gx + fy, x-fiy, and 

 X— iy, which are the equations of the three lines joining the origin to 

 the three points where the line at infinity cuts the cubic; thus showing 

 that the cubic passes through the imaginary circular points at infinity. 



(47) KAN. UNIV. QUAR., VOL. I., NO. i, OCT., 1S92. 



