104 T. A. Hirst on the Quadric Inversion of Plane Curves. [Mar. 2, 



possessing the same properties (Art. 6) ; and if the primitive have likewise 

 its centre at the origin, the latter will also be the centre of its inverse 

 (Art. 11, Ex. 3). The tangents to two inverse curves at two inverse 

 points p,p' now intersect on a line D bisecting pp', and parallel to the 

 diameter of the fundamental conic, which is conjugate to pp' (Art. 10). 



(la) When the fundamental conic is an equilateral hyperbola, the 

 origin being still at its centre, the method of inversion becomes identical 

 with the "hyperbolic transformation" investigated by Professor Schiaparelli, 

 of Milan, in his interesting memoir, " Sulla trasformazione geometrica 

 delle figure " *. The line D, upon which the tangents to two inverse curves 

 at inverse points p,p' intersect, not only bisects pp', but is now inclined at 

 the same angle as pp' to each of the fixed asymptotes of the fundamental 

 hyperbola. 



(2) The fundamental conic being an ellipse and the origin at its centre, 

 the inverse of every right line in the plane will be an ellipse passing 

 through the origin, and at the same time similar, as well as similarly 

 placed, to the fundamental ellipse. The ellipse (I) inverse to the line at 

 infinity now resolves itself to a point coincident with the origin. Every 

 ellipse not passing through the origin, but similar and similarly placed to 

 the fundamental one, has for its inverse an ellipse with the same proper- 

 ties ; and should the primitive be likewise concentric with the fundamental 

 ellipse, so also will be the inverse. The tangents at two inverse points p,p' 

 to inverse curves again intersect on a line D which bisects pp', and is par- 

 allel to the diameter of the fundamental ellipse conjugate to pp'. 



(2a) When the fundamental conic is a circle with its centre at the origin, 

 we have, as already stated, the case of cyclic inversion, and the imaginary 

 circular points at infinity are the fundamental points ; the line D becomes, 

 as is well known, the perpendicular to pp' through its middle point. From 

 the theorems ii. and iii. of Art. 8 we should now infer that 



(i) The cyclic inverse of a right line through one of the circular 

 points is a right line through the other, and the two intersect on the 

 fundamental circle. 



(ii) The cyclic inverse of a simple focus of any curve is always a focus 

 of the inverse curve unless the first focus should coincide with the origin, 

 in which case the inverse curve would have cusps at the circular points at 

 infinity (Art. 12, i.). 



It is important to notice that this theorem does not hold for a double 

 focus / of the primitive curve that is to say, for the intersection of the 

 tangents to this curve at the circular points. For in this case the con- 

 nectors of the inverse point /' with the circular points would merely inter- 

 sect the inverse curve on the fundamental lines (Art. 11, i.); the line 

 joining the points of intersection a common chord of the point-circle f 

 and the inverse curve would, however, bisect A/' perpendicularly. Thus 

 it is that the cyclic inverse of the centre / of a primitive circle is not a 

 * Memoria della Eeale Accademia delle Scicnse di Torino, Serie II. torn. xxi. 



