TWENTY-SIXTH ANNUAL MEETING. 17 



2. If the "central" polar circle of a point P with respect to an oval pass 

 through the point Q, then will the "central " polar circle of Q pass through P. 

 [By "central" polar circle is meant the circle passing through the origin and 

 the points of contact of " central " tangent circles from the point to the oval.] 



1. The locus of the point of intersection of two tangents to an ellipse, which are 

 at right angles to one another, is the director circle of radius equal to (a' + b')^ 

 when a and b are the semi-axes respectively. 



2. The locus of the point of intersection of two "central " tangent circles to 



an oval which cut one another orthogonally is a circle of radius when a 



{a- + b')h 

 and b are the semi-axes of the oval respectively. 



1. The equation of the locus of the foot of the perpendicular from the center 



of an ellipse on a tangent is r^ = a^ cos"- k + b'^ sin'^ k, the equation of the ellipse 



r"^ cos'^ k _, r'^ sin^ k 



being ^ = ;; ~r ; . 



a- 0- 



2. The equation of the locus of the extremity of the diameter through the origin 



of a "central " tangent circle to the oval given by the equation r^ = a^ eos''^ k -\- 



&^ si7i'' k is an ellipse whose equation is i = + — . 



a^ b- 



1. The sum of the reciprocals of the squares of any two diameters of an ellipse 

 which are at right angles to one another is constant. 



2. The sum of the squares of any two diameters of an oval which are at right 

 angles to one another is constant. 



1. The line joining the extremities of any two diameters of an ellipse which 

 are at right angles to one another will always touch a fixed circle. 



2. The "central" circle joining the extremities of any two diameters of an 

 oval which are at right angles to one another will always touch a fixed circle. 



1. The tangent at a point P of an ellipse meets the tangent at A, one ex- 

 tremity of the axis AC A', in the point Y; then is CY parallel to A'P, C be- 

 ing the center of the curve. 



2. The "central" tangent circle at a point Pof an oval cuts the "central" 

 tangent circle at A, one extremity of the axis AC A', in the point Y; then is C 

 F tangent at C to the " central " circle through P and A', C being the center of 

 the curve. 



1. If three of the sides of a quadrilateral inscribed in an ellipse are parallel 

 respectively to three given straight lines, then will the fourth side also be parallel 

 to a fixed straight line. 



2. Four points are taken on an oval: if three of the four "central" circles 

 through the consecutive pairs of these points have their "central diametei-s" 

 perpendicular to three given lines respectively, then will the "central diameter " 

 of the fourth "central" circle be perpendicular to a given line. [By "central 

 diameter" of a "central" circle is meant that diameter which passes through 

 the origin.] 



1. A parallelogram circumscribes a circle, and two of the angular points are 

 on fixed straight lines parallel to one another and equidistant from the center; 

 then are the other two on an ellipse of which the circle is the minor auxiliary 

 circle. 



2, Four "central" circles are inscribed in a circle so that the two opposite 

 ones have a common "central diameter"; two of the four points of intersection 



—2 



