14 



KANSAS ACADEMY OF SCIENCE. 



THE INVERSE OF CONICS AND C0NIC0ID3 FROM THE 



CENTER. 



By M. E. Rice, University of Kansas, Lawrence. 



For the purposes of this paper it is convenient first to discuss briefly the 

 locus of a point which moves so that the sum, or difference, of its distances 



from two fixed points bears a constant ratio 

 to its distance from a third point midway 

 between these two. 



In Fig. 1 let P be the moving point, F', 

 i^' and O the three fixed points. Take O 

 for origin, and the line FF" for the axis of 

 abscissae; then the condition is 



PF ±PF' 



= a constant. 



J^CJ. t. 



I 



[[x+OFf + U'f 



P 



This gives the equation, 



[{OF— xf +.2/2f = k (a;2 + y"^)^ 



(1) 



(2) 



For convenience let O F =■ O F' z=—, and k =^—\ equation (2) becomes, after 



clearing of radicals: 



a2 ^2 _|. y2 



(.^■2 ^ ylf 



(3) 



1 — e2 



It is evident from the form of this equation that the curve is symmetrical 

 about either axis; and that as e varies the shape of the curve will change ac- 

 cordingly. Hence e may conveniently be called the eccentricity of the curve; 

 and the fixed points i^and F', foci. 



Denote the numerical value of 



by 6-, whence e- = 



±h"- — cfi 



(4) 



(5) 



1 — t2 '•'-'■--- ' -f.62 



Equation (3) may now be written in the symmetrical form, 

 ce- .C2 ±Wy^ = (a;2 + y^f 

 according as e < 1 or e > i. 



Consider first the case when e <^ 1, and the equation is, 



a^x^ + Wy^^{x^ + y'-)- (6) 



The curve cuts the axes of coordinates at the points (=b «, o), {k, ± 6); and as 

 it is a closed curve, a may be called the minor, and h the major semi-axis respect- 

 ively. 



The distance from center to either focus is — . Since the equation contains no 



e 

 terms of a lower degree than the second, the origin is a conjugate point. Also 



the equation of imaginary tangents at the origin is 



a2 x^ _^ 52 y2 ^ Q_ (7) 



The circular points at infinity are imaginary double points for, making the 

 given equation homogeneous by means of the line at infinity. Ox -f 0,y -|- c = (3, 

 gives 



(.t2 + .?/-)-=0, ' _ (8, 



which is the equation of the lines through the origin and the points common to 

 the curve and the line at infinity. But this breaks up into 



{X 4- iyf {X - iyf = 0; (9) 



