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



the origin is the middle point of the segment which (F) intercepts on R, 

 and that (Art. 4) the infinitely distant points of (F) are necessarily also on 

 (I), it will be seen thp.t 



i. The conic (I), circumscribed to the principal triangle, which is inverse 

 to the line at infinity, is similar and similarly situated to the fundamental conic, 

 of which latter, in fact, it bisects all chords that converge to the origin. 



By means of this conic (1) the asymptotes to any inverse curve may be 

 readily constructed. Since the next article, however, will be devoted to 

 the construction of the tangent at any point whatever of an inverse curve, 

 it will be sufficient here to note the following obvious corollaries of the 

 above theorem : 



ii. The asymptotes of either of two inverse curves are respectively parallel 

 to the right lines connecting the origin with the intersections of the other 

 curve and the conic (I), which circumscribes the principal triangle and is, at 

 the same time, similar and similarly situated to the fundamental conic (F). 



iii. The conic inverse to a given right line will be a hyperbola, a parabola, 

 or an ellipse, according as that line cuts, touches, or does not meet the conic (I), 

 which circumscribes the primitive triangle and is similar and similarly placed 

 to the fundamental conic (F). 



The conic (F) and the circle circumscribed to the " principal triangle 

 ABC have, of course, conjugate to BC, a second common chord H, inverse to 

 that circle (Art. 8, i.), and this chord is clearly the only line in the plane 

 whose inverse is a circle. The imaginary intersections of H with (I) are 

 inverse to the circular points at infinity, and consequently lie, with the 

 latter, on a pair of imaginary lines intersecting in the origin A. 



Again, it is well known that all chords of (I) which subtend a right angle 

 at A pass through a fixed point h*. The conies inverse to such 

 chords are readily seen to be equilateral hyperbolas, and like their primitive 

 lines they all pass through a fixed point in fact, through the point h' 

 inverse to h ; this point h', moreover, is well known to be the intersection 

 of the three perpendiculars of the triangle ABC, about which all the equi- 

 lateral hyperbolas are circumscribed f. It further follows from a known 

 theorem, that the chords of (I) which subtend a constant angle at A envelope 

 a conic which has double contact with (I) at the inverse circular points, or, in' 

 other words, at the imaginary intersections of (I) and H ; and conversely that 

 all tangents to such a conic intercept on (I) an arc which subtends at A a 

 constant angle J. The conies inverse to such tangents are, when the latter 

 actually cut (I), hyperbolas whose asymptotes are inclined to each other at 

 a constant angle that is to say (in order to embrace all cases), similar 



* Salmon's ' Conic Sections,' 4th ed., Art. 181, Ex. 2. 



f Ibid. Art. 228, Ex.1. Two distinct theorems in conies are thus brought, by inversion, 

 into juxtaposition, and we have a simple example of the duality which this method, like 

 that of reciprocal polars, imparts to every theorem. 



t Chasles's Sections Coniques, Arts, 473, 474; also Salmon's ' Conic Sections,' Arts. 2?6, 

 277, 296. 



