1887.] A. Mukhopadhyay — Memoir on Plane AnahjHc Geometry. 329 



i|ie acute angle between every pair of the tliree axes is •- ; if, througli 



the vertices of the given triangle, diameters of the parabolas be drawn, 

 they intersect in a fixed point which may be determined geometrically, 

 viz., if equilateral triangles BDO, CEA, AFB be described externally on 

 the sides, the lines AD, BE, CF are diameters of the enveloping para- 



bolas and meet in a point, the acute angle between each pair being -. 



o 



§§. 26 — 27. JReciprocal Folars. 



§. 26. Reciprocal of Central Conic. — It is well-known that the 



first focal pedal of a conic, being the locus of the foot of the perpendicular 

 dropped from a focus on any tangent, is, in the case of central conies, 

 the circle described on the axis-major as diameter ; hence, as the reciprocal 

 of any curve is the inverse of its pedal, it is clear that the inverse of 

 pedal of the first focal pedal of any central conic is the reciprocal polar 

 of a circle, which reciprocal is known to be a conic ; hence it follows 

 that the second pedal of a conic with respect to a focus is the inverse of 

 a conic whose position and magnitude may be determined geometrically. 

 For we know that the recij^rocal of a circle of radius a, with respect to 

 a circle of radius k, is a conic which is an ellipse if the origin of recipro- 

 cation lies within the given circle, the focus of the conic is at the origin 



. h'^ . . . c 



of reciprocation, the semi-latus-rectum is — , the eccentricity is -, where 



CL CL 



c is the distance between the centres of the given circle and the circle of 

 reciprocation, and the directrix is a line at right angles to the central 



line drawn at a distance — from the origin of reciprocation. "Now, in 



the question under consideration, we have to find the reciprocal of the 

 circle described on the major axis as diameter, with a focus as origin of 

 reciprocation ; hence the conic is an ellipse, a focus of which is the 



focus of the given conic, the semi-latus-rectum is — , the eccentricity is 



equal to the eccentricity of the given conic, and the directrix is a line at 



right angles to the axis-major of the given conic, at a distance — from 



the given focus. 



These results are easily verified analytically, for the given conic 

 being 



f! a. ^-1 



remove the origin to the focus, say the negative one ; then the conic is 



42 



