1865.] T. A. Hirst on the Quadric Inversion of Plane Curves. 97 



conies. Now, by Arts. 4 and 6, the inverse of a conic which has double 

 contact with (I) at the inverse circular points is, in general, a quartic curve 

 having double points at A, B, C, and touching, at the circular points, the 

 line at infinity. This curve, therefore, is also the envelope of similar conies 

 circumscribed to the triangle ABC *. The point h clearly belongs to the 

 above series of conies having double contact with (I), and H must be its 

 polar relative to (I) -f; so that we may resume as follows*: 



iv. The right lines whose inverse conies are equilateral hyperbolas, all 

 pass through the point h which is inverse to the intersection of the three 

 perpendiculars of the principal triangle; the circle circumscribed to this 

 triangle is the inverse of the polar H of the point h, relative to the conic (I) 

 which is inverse to the line at infinity ; the imaginary intersections of H and 

 (I) are the inverse circular points, and all lines which envelope a conic (S) 

 having double contact at these points with the conic (I) are inverse to conies 

 which are similar to each other. 



Tangents to inverse curves at inverse points. 



10. To a pencil of right lines L, whose centre is p, corresponds, by 

 quadric inversion, a pencil of conies (L') passing through the three prin- 

 cipal points (Art. 8, i.) and the inverse point p'. To each element of the 

 one pencil corresponds manifestly but one element of the other ; so that 

 the lines L through p, and the tangents L', at p', to their respective inverse 

 conies (L') constitute two homographic pencils]; ; and since two corresponding 

 rays of the latter coincide with pp', every other pair must intersect on a 

 fixed line D (see figure). Since, moreover, to the rays p B, p C correspond, 

 respectively, the rays p' C, p' B (Art. 8, ii.), it is obvious that the line D is 

 simply one of the three diagonals of the complete quadrilateral p Bp'C, the 

 other two diagonals being pp 1 and BC. Hence if a be the intersec- 

 tion of the latter, the former D will, in virtue of a well-known property of 

 the quadrilateral, pass through the harmonic conjugates d' and of a, rela- 

 tive respectively to pp' and BC. Now a is at once recognized to be the 

 pole, relative to the fundamental conic, of pp' or R ; so that d, the inverse 



* From this a new definition may be readily deduced of the interesting curve, of the 

 fourth order and third class (with three cusps, and the circular points for points of contact 

 of an infinitely distant double tangent), which presented itself to Steiner as the envelope 

 of the line passing through the feet of the perpendiculars let fall from any point of a 

 circle upon the sides of an inscribed triangle, and of which he has enunciated (merely) 

 so many remarkable properties in a paper published in vol. liii. of Crelle's Journal. The 

 curve is, in fact, the envelope of an hyperbola circumscribed to an equilateral triangle, and 

 having its asymptotes inclined to each other as are any two sides of that triangle. The 

 curve may also be generated as a hypocycloid, and appears to be identical with the one 

 whose equation is given at p. 214 of Dr. Salmon's ' Higher Plane Curves.' 



t Chasles's Sections Coniques, Art. 47-1. 



J Chasles, "Principe de correspondancc cntre deux ofy'efs variables," &c., Comptes 

 Eendus, Dec. 24, 1855, and Sections Coniques, Art. 325. See also Cremona's ' Tcoria 

 geometrica delle curve piane,' p. 7. 



