1865.] T. A. Hirst on the Quadric Inversion of Plane Curves. 103 



relative to either of these triangles will transform the quartic to a quintic 

 (Art. 6) with three tacnodes, the varieties of which will correspond to 

 those of the quartic. If, for instance, the primitive conic were inscribed 

 in the original principal triangle, then the quartic would have three 

 ceratoid cusps, and the quintic would be the remarkable one which Pliicker 

 has signalized (31. (. p. 222, fig. 35) as possessing three ramphoid cusps. 



(c) With respect to singularities of a higher order on the primitive 

 curve, and on a principal line, little more need be added. To a tacnode 

 at a would correspond, on the inverse curve, two branches touching the line 

 Aa, and having three-pointic contact with each other at A (51. (. p. 165, 

 fig. 20). In fact, as a general rule, the contact of the branches at A is 

 one higher in order than that of the corresponding branches at a. If a 

 were a ramphoid cusp on the primitive, A would also be a ramphoid cusp, 

 of higher order, on the inverse curve (21. (. p. 170), and so on. It is 

 worth observing, lastly, that although, by (a), the inverse of a tacnode at 

 A is an ordinary node at a, on the polar of A, the latter will itself become 

 a tacnode when approaches to coincidence with B or C (Art. 11, iii.), 

 and similarly 



i. If one of two inverse curves have a ramphoid cusp at a principal 

 point, to which a principal line is the tangent, the other will also have a 

 ramphoid cusp at the pole of that line, the tangent to which will be the 

 polar of that point. 



This is manifest, in fact, from (b), on considering, with Professor 

 Cayley *, a ramphoid cusp at B, with tangent BC, to arise from the coin- 

 cidence of a ceratoid cusp at a with a node at B. 



Special cases of quadric inversion. 



15. The special cases of inversion which correspond to particular hypo- 

 theses relative to the position of the origin and to the nature of the 

 fundamental conic are very numerous. The choice of these elements will 

 depend of course upon the nature of the properties which are to be investi- 

 gated by the method of inversion. A few only of the more useful of such 

 special cases can be here noticed. 



(1) When the fundamental conic (F) is an hyperbola with its centre at 

 the origin A, its asymptotes constitute the fundamental lines, and their in- 

 tersections with the line at infinity the fundamental points (Art. 3). Every 

 right line parallel to one of these asymptotes has, for its inverse, a right line 

 parallel to the other ; and the two intersect on the fundamental hyperbola 

 (Art. 8, ii.). The inverse of every other line in the plane is an hyperbola 

 passing through the origin, and having its asymptotes parallel to those of 

 the fundamental hyperbola (Art. 8, i.) ; moreover the conic (I) inverse to 

 the line at infinity resolves itself into these asymptotes (Art. 9, i.). Every 

 hyperbola which does not pass through the origin, but has its asymptotes 

 parallel to those of the fundamental one, has, for its inverse, an hyperbola 

 * Quarterly Journal of Mathematics, vol. vi. p. 74. 



