Mr. Stubbs's New Theorems relative to the Conic Sections. 209 



cyclic plane. Now taking the curve the intersection of these 

 cones with a sphere the centre of which is at the common vertex, 

 we derive the following theorem : — Given two great circles and 

 a point on a sphere: if through the given point two arcs of great 

 circles be drawn to cut one of the given arcs, so that the di- 

 stance between the points of section shall be constant, they 

 will form with the other given arc a triangle; the spherical 

 conic circumscribing this triangle, and having the first arc for 

 one of its cyclic arcs, constantly touches a fixed spherical conic 

 having the same arc as a cyclic arc ; and as we change the 

 length of the constant intercept the pole of its cyclic arc lies 

 in a fixed arc of a great circle. Now taking the correspond- 

 ing properties in the reciprocal cone or curve : if two points 

 on a sphere and an arc of a great circle be given, and from 

 one of the points arcs be drawn, including a given angle, to 

 meet the given arc ; if the points of intersection be joined with 

 the other given point, and a spherical conic be inscribed in 

 the triangle so formed, having its focus at the first given point, 

 it will constantly touch a fixed conic section with the same 

 focus, and whose directrix, as the given angle changes in mag- 

 nitude, constantly passes through a given point : by making 

 the radius of the sphere infinite, a similar property of the plane 

 conic sections may be deduced. 



In a similar manner the following theorems give as results 

 the annexed properties. 



II. A chord of a circle passes through a fixed point, the 

 tangents at its extremities intersect in a right line by inversion 

 (the pole at the given point); a chord of a circle passes through 

 a given point in its segments, circles are described touching 

 the given circle at the extremities of the chord, the locus of 

 the point of intersection is a circle ; hence, 



" If from any point of a fixed line in the plane of a conic 

 section two tangents be drawn to it, and two conic sections 

 confocal with the given one be described so as to touch each 

 the fixed line and one of the tangents at its point of contact, 

 a tangent drawn to them in common will owelope a fourth 

 conic section with the same focus." 



III. The angle in the same segment of a circle is constant; 

 taking any point in the circle as pole, and inverting, the fol- 

 lowing property is derived : Given a line and two points in it, 

 and a fixed point in its plane, two circles passing each through 

 the latter point and one of the other two, and intersecting on 

 the given line, cut at a given angle ; hence, 



" If through the vertex of an angle of a fixed triangle any 

 line be drawn to meet the opposite side, and if in the triangles 

 so formed conic sections be inscribed, having a fixed point for 



Phil. Mag. S. 3. Vol. 25. No. 165. Sept. 1844. P 



