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



cusp at A on the inverse curve will also assume the ceratoid form, but it 

 will be of higher order than the primitive one, since the tangent at A will 

 meet it in five, instead of in three coincident points (21. d. p. 167, iv.). 

 If the principal line BC be itself the tangent at the ordinary cusp a, the 

 inverse curve will have a triple point at A, the sole tangent Aa at which wiil 

 meet the curve in Jive coincident points. To the eye, this singularity will 

 have the form of a point of inflection (21. GT. p. ] 74, fig. 29). 



The following examples will illustrate the production, by inversion, of 

 cusps of both kinds. 



Ex. 1. The primitive curve being a cusped cubic, and the origin A 

 being placed at its real point of inflection, let the stationary tangent 

 be chosen as a fundamental line, and any line whatever through the cusp a 

 as the polar of the origin. If the points B, C, in which the latter inter- 

 sects the cubic and the stationary tangent, be considered as the funda- 

 mental points, the inverse curve will (Art. 6) be a quartic curve passing 

 once through B, and twice through each of the points C and A. The 

 tangent to the quartic at B will be the inverse of the line joining C to the 

 third intersection y of the cubic with the fundamental line AB (Art. 11, i.). 

 The point C will be a ceratoid cusp on the quartic with CB for its tan- 

 gent (Art. 13, i.). Lastly, A a will be the tangent at A to a ramphoid 

 cusp on the inverse curve (b) . The latter, therefore, is identical with the very 

 remarkable quartic curve to which Prof. Sylvester's recent researches on 

 the roots of equations of the fifth degree has imparted so great an interest. 

 (Phil. Trans. 1865). It is termed by him the 'Bicorn,' and is the one 

 which, in Pliicker's classification, is numbered xvi. (21. (. p. 193). Since 

 the primitive cubic from which it has been derived may itself be regarded 

 as the inverse of a conic (Art. 12, Ex. 1), it is obvious that many pro- 

 perties of the Bicorn may be deduced from those of a conic, by double 

 inversion, relative to two sets of principal points. 



Ex. 2. The primitive being a conic, its inverse will, in general, be a 

 quartic curve passing twice through each principal point (Art. 6). 



All the ten varieties of such qnartics which have been described by 

 Pliicker (21. (. p. 195) correspond, in a very simple manner, to the dif- 

 ferent positions which the primitive conic may have*. Now itis well known | 

 that, in general, two triangles maybe inscribed in this conic, each of which 

 shall, at the same time, be circumscribed to the principal triangle ; whence 

 we infer that two triangles may be inscribed in the quartic, so that a double 

 point shall lie on each side of each triangle. A second inversion, therefore, 



* For instance, if the principal triangle be self-conjugate relative to the primitive 

 conic, the inverse quartic will have, at each principal point, both its branches inflected 

 (Art. 11, ii.)- I n this case i* i" further obvious that two principal points must neces- 

 sarily lie outside the primitive conic ; so that one principal point will necessarily be a 

 conjugate point on the quartic (Art. 11, Ex. 2). Inversion, in fact, renders visible 

 the many curious properties, signalized by Pliicker, which quartic curves present when- 

 ever their double points are also points of inflexion (3. 3E. p. 199). 



t Salmon's Conic Sections, 4th ed. p. 237; Chasles's Sections Coniques, Art. 246. 



