1865.] T. A. Hirst on the Quadric Inversion of Plane Curves. 99 



In a similar manner the line ~R.,, which connects C and one of the inter- 

 sections y of AB and (P), has for its inverse a line R x touching, at B, a branch 

 of (P'). All these cases are included in the following theorems : 



i. The tangents at a principal point to one of two inverse curves are 

 respectively inverse to the riaht lines which connect the intersections of 

 the other curve and the principal line polar to that point, with the oppo- 

 site principal point. 



ii. If a non-principal line have r-pointic contact at a principal point 

 with any branch of one of two inverse curves, the inverse line ivill have 

 (r])-pointic contact with the other curve on t he principal line polar to 

 that principal' point. 



The second of these theorems, as will be shown in the next article, is 

 slightly modified when the line of ?'-pointic contact is a principal one ; 

 the first theorem, though still true, becomes susceptible of the following 

 simpler enunciation : 



iii. If a branch of one of two inverse curves touch a principal line at a 

 principal point, the other curve will have a branch touched by the polar of 

 this point at the pole of that line. 



The following examples will serve as illustrations of these three theorems : 

 Ex. 1. The primitive being a right line intersecting the principal lines in 

 a, ft, y, respectively (see figure), Aa will be the tangent at A to its inverse 

 conic ; and if B/J, Cy intersect in p, the inverse point p' will be the pole of 

 BC relative to the inverse conic. This pole is always real ; it may, more- 

 over, be easily constructed, even when B and C are imaginary, on observing 

 that p is also the intersection of the polar of a, relative to the fundamental 

 conic, with the harmonic conjugate of aA, relative to BC and the primitive 

 line. 



Ex. 2. The primitive being a conic passing through the origin and in- 

 tersecting the fundamental lines in ft and y, its inverse will be a cubic 

 passing through B and C, and having a double point at A (Art. 6). This 

 latter point will be a node, if the primitive conic cut BC in real points 

 a,, a 2 ; and Aa L , Aa 3 will be the tangents thereat. It will be a conjugate or 

 isolated point, however, when the primitive conic does not actually cut BC. 

 The tangents to the cubic at B and C will, as before, intersect in the 

 point p', inverse to the intersection p of B/3, Cy. If the primitive conic 

 touch the latter lines in ft and y, in which case it is manifest that it cannot 

 cut BC, then the cubic will have real points of inflection at B and C, 

 and, necessarily, a conjugate point at A*. The line Ap is obviously the 

 polar, relative to the primitive, as well as to the fundamental conic, of the 

 intersection a of the lines BC, /3y ; hence aA touches the primitive conic 

 at A, and, by i., a is a point on the cubic ; it is, in fact, the third point of in- 

 flection on this curve. 



Ex. 3. The primitive being a conic touching the fundamental lines in 



* The truth of a well-known theorem in cubics, ' Higher Plane Curves' (Art. 183), is 

 here rendered visible. 



