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



of d ', must be the pole of D or ad 1 , as well as the harmonic conjugate of A 

 relative to^ 1 (Art. 2, i.). Consequently, from the fact that when L touches, 

 at p, a primitive curve (P), the conic (L'), and hence its tangent L', 

 must touch, aty, the inverse curve (P f ) (Art. 4, i.), we at once deduce the 

 following theorem, by means of which the tangent at any point of a curve 

 inverse to a given one may, for all positions of the origin, be readily con- 

 structed : 



i. The tangents, at two inverse points, to two inverse curves intersect 

 on the polar, relative to the fundamental conic, of the harmonic conjugate 

 of the origin with respect to their points of contact. 

 Hence we may also deduce the following property : 

 ii. To a multiple point on one of two inverse curves, but not on a 

 principal line, corresponds, on the other, a multiple point of the same 

 order of multiplicity, and the tangents to corresponding (inverse) branches 

 all intersect on the polar, relative to the fundamental conic, of the har- 

 monic conjugate of the origin, relative to the two multiple points. 



The reality, taction, and general distribution of the several branches will 

 be the same at two such multiple points ; the latter, in fact, will merely 

 differ in certain secondary properties. For instance, a branch inflected at 

 one of these points would not, in general, correspond to an inflected branch 

 at the other ; the latter branch, however, would have three-pointic contact, at 

 this multiple point, with the conic inverse to the tangent of the inflected 

 branch at the first multiple point. 



1 1 . The tangents at the principal points to two inverse curves may be 

 thus investigated. 



Exclusive of the principal points JB and C, let the primitive curve (P) 

 intersect the principal line BC in the points a, a 1} a 2 , &c. . ., and conceive 

 a right line R to rotate around the origin A. Exclusive of A this line R 

 will intersect (P) in n a points^?, respectively inverse to the n' a' points 

 p' in which it intersects the inverse curve (P') (Art. 7). It is clear, how- 

 ever, that whenever, by the rotation of R, p approaches one of the points 

 a, p' will approach to coincidence with A, so that R will there touch a 

 branch of (P') ; and more generally, that if R should have (? l)~pointic 

 contact at a with (P) it would, at the same time, have r-pointic contact at A 

 with one of the branches of (P'). 



Similarly, if (P) intersect the fundamental line AC in the points ft, /3 t , 

 /3 2 , &c. ., A and C being excluded, and a right line R I} turning around 15, 

 intersect (P) in nb points^*, their n' c' inverse points^' (Art. 7) will be 

 the intersections of (P') and the line R 2 , inverse to R x and passing through 

 C (Art. 8, ii.). Each pair of points^?, p', moreover, will be colliuear with 

 A (Art. 2). Hence it follows that whenever, by the rotation of R, two 

 points p and ft approach each other, the inverse point p' will approach C, 

 so that the line R 2 will there touch a branch of (P) ; and, as before, if R t 

 have (r l)-pointic contact with(P) at a point ft, R 2 will have r-pointic 

 contact at C with a branch of (P'). 



