1865.] T. A. Hirst on the Quadric Inversion of Plane Curves. 105 



centre of the inverse circle, but the inverse of the origin relative to the 

 latter circle. If the primitive curve had points of inflection at the circula 1 ' 

 points, the inverse f of the intersection / (a triple focus) of the stationary 

 tangents thereat would not only be a focus of the inverse curve, but its 

 corresponding directrix would bisect A/' perpendicularly. The focal rela- 

 tions of cyclic inverse curves, however, deserve closer examination ; and 

 I propose on another occasion to return to them. 



(3) When the fundamental conic consists of a pair of real, right lines 

 F lf F 2 intersecting at F, the fundamental points B, C coincide with F, and 

 the principal line BC with the harmonic conjugate of AF relative to F lt F a . 

 The conic (I) inverse to the line at infinity is now an hyperbola, of which 

 AF is a diameter, and the asymptotes of which are parallel to Fj and F 2 . 

 Harmonic conjugates relative to the latter lines now constitute pairs of inverse 

 lines, and the conic inverse to every other line, cutting BC say in a, is a 

 conic touched at A and F by Aa and BC, so that the conies inverse to all 

 lines parallel to BC are concentric, and have AF for a common diameter. 

 The conies inverse to all lines passing through a fixed point a of AF have 

 obviously three-pointic contact with each other at F, so that the conies 

 inverse to lines parallel to AF have, at F, three-pointic contact with the 

 hyperbola (I). All conies touching BC at F, but not passing through A, 

 are inverse to conies having the same properties, and all conies passing 

 through A and F, but not touching BC at the latter point, give by inversion 

 conies of a similar description. The tangents at inverse points p,p' to two 

 inverse curves now intersect on the harmonic conjugate of BC, relative to 

 Fp,F/. 



(3a) The fundamental conic may consist of a pair of right lines, per- 

 pendicular to each other. The results are then similar to, and somewhat 

 simpler than those just noticed. 



(4) When the fundamental conic consists of a pair of imaginary right 

 lines intersecting at a real point F, the effects of inversion are analogous 

 to those considered in (3). To secure real constructions, it is merely ne- 

 cessary to assume the point-ellipse F to be concentric with, similar, and simi- 

 larly placed to a given ellipse (E). The lines FA and BC will then be con- 

 jugate diameters of (E), and any two inverse points p, p 1 will also lie on con- 

 jugate diameters. When the imaginary lines F,,, F 2 pass through the cir- 

 cular points, we have the following species of inversion : 



(4fl) The fundamental conic consists of a point-circle F. The principal 

 line BC is now perpendicular to AF at F, arid the connector of every pair 

 of inverse points p,p' subtends a right angle at F. Inverse right lines 

 through F, are perpendicular to each other, and the inverse of any right 

 line in the plane is a conic, through A, to which AF is the normal at F. 

 The circle (I) on AF as diameter, is the inverse of the line at infinity, 

 and all right lines which touch one and the same circle, concentric with (I), 

 are inverse to conies which are similar to each other these conies being 

 ellipses, parabolas, or hyperbolas, according as the circle in question is 



