Twenty- THiED Annual Meeting. 



85 



The polar equation of an ellipse, a 



real focus being a pole, is 



a(l — e2) 



r= . 



1 + e cos. e 



The locus of the foot of the perpen- 

 dicular from either real focus on a tan- 

 gent is the major auxiliary circle. 



The rectangle contained by the per- 

 pendiculars from the real foci on a tangent 

 is constant, and equal to 6-. 



The real focal radii to any point on 

 an ellipse make equal angles with the 

 tangent at that point. 



The major axis of an ellipse is divided 

 by a real focus into segments, such that 

 their product is equal to b^. 



The distance from the center to where 

 the normal cuts the axis of -y, or the sub- 

 normal, is e'-x^. 



A circle of radius a described from the 

 extremity of the conjugate axis cuts the 

 transverse axis in the real foci. 



The polar of either focus is a di- 

 rectrix. 



The distance from the center to either 



real directrix is—. 

 e 



The distance of any point on an ellipse 

 from a real focus is in a constant ratio to 

 its distance from the corresponding di- 

 rectrix; this ratio is equal to the real 

 eccentricity of the conic. 



The equation of the tangent at the 

 extremity of the real latus rectum is 

 y -\- ex^=a. 



The equation of the normal at the 



same point is y — —-{-ae^ =0. 

 e 

 The circle described on a real focal 

 radius as a diameter touches the auxiliary 

 circle. 



The polar equation of an ellipse, an 

 imaginary focus being the pole, is 

 ^_ b(l-g^) 

 l±e sin. 9 



The locus of the foot of the perpen- 

 dicular from either imaginary focus on a 

 tangent is the minor auxiliary circle. 



The rectangle contained by the per- 

 pendiculars from the imaginary foci on a 

 tangent is constant, and equal to a^. 



The imaginary focal radii to any point 

 on an ellipse make equal angles with the 

 tangent at that point. 



The minor axis of an ellipse is divided 

 by an imaginary focus into segments, 

 whose product is equal to a^. 



The distance from the center to where 

 the normal cuts the axis of y, or the con- 

 jugate subnormal, is e'-y'^. 



A circle of radius b described from the 

 extremity of the transverse axis cuts the 

 conjugate axis in the imaginary foci. 



The polar of an imaginary focus is an 

 imaginary directrix. 



The distance from the center to either 



imaginary directrix is—. 

 e 



The distance of any point on an ellipse 

 from an imaginary focus is in a constant 

 ratio to its distance from the correspond- 

 ing directrix; this ratio is equal to the 

 imaginary eccentricity of the conic. 



The equation of the tangent at the 

 extremity of the imaginary latus rectum 

 is X + ey ^= b. 



The equation of the normal at the 



same point is x 



e 

 The circle described on an imaginary 

 focal radius as a diameter touches the 

 minor auxiliary circle. 



Other theorems in respect to the ellipse might be added ; but enough have been 

 given to illustrate the theory. The above list contains the most of the important 

 propositions. 



If the conic be a hyperbola, two of the foci are still real, and two imaginary ; 

 but the imaginary foci coincide in position with the real foci of the conjugate hy- 

 perbola. The theorems concerning the imaginary focal properties of the hyper- 

 bola are almost all identical with the theorems concerning the real focal properties of 

 the conjugate hyperbola. For example, the locus of the foot of a perpendicular, 

 from an imaginary focus on a tangent, is the minor auxiliary circle ; and the locus 

 of the foot of a perpendicular, from a real focus of the conjugate hyperbola on a 



