1865.] T. A. Hirst on the Quadric Inversion of Plane Curves. 101 



i. If a branch of one of two inverse curves have r-pointic contact with a 

 principal line at a principal point, the other curve will have, at the pole of that 

 line, a multiple point of the order r 1 on a branch the sole tangent to which 

 is the polar of that principal point. 



Ex. The primitive curve being a cubic which has the fundamental 

 points B, C for points of inflection, and the fundamental lines for stationary 

 tangents, and consequently another point of inflection a on BC, the inverse 

 curve will be a quartic touching Aa at the origin, and passing twice through 

 each of the fundamental points (Arts. 6 and 10). From the present theorem 

 we conclude* further, that the latter points will be cusps on the quartic, and 

 that the fundamental lines will be the tangents thereat (St. (. p. 192, ix.). 



14. From preceding articles the following properties may also be de- 

 duced : 



i. If one of two inverse curves have a multiple point on a principal line, but 

 not at a principal point, the other will, in general, have a corresponding 

 number of branches touching each other at the pole of that line ; and at this 

 pole the common tangent to these branches will be inverse to the connector of 

 the first multiple point and the principal point opposite to the principal line 

 on which it is situated. 



To obtain a clear conception of the modifications which may present 

 themselves, it will suffice to consider the case where the primitive curve (P) 

 has a double point at a, on the polar BC of the origin A. 



(a) The inverse curve will, in general, have a tacnode at A (21. (. 

 p. 164, figs. 17, 18) in other words, two branches which there touch each 

 other, the common tangent being Aa ; these branches will, moreover, have 

 three-pointic contact, at A, with the conies inverse to the two tangents 

 at a to the primitive curve*. If one of the tangents at a coincide with 

 aA, one of the branches of the inverse curve will be inflected at A (21. (. 

 fig. 21) (Art. 11, iv.). If one of the branches at a touch the primitive 

 line BC, then one of the branches at A will be cusped, Aa being still the 

 common tangent to the cusped and to the ordinary branch (21. (. fig. 28). 

 If both these singularities occur on the primitive at the same time, the 

 inverse curve will present at A a triple point, with a single tangent, formed 

 by an inflected and a cusped branch (21. (. fig. 30). 



(b) If the tangents at a coincide, so that the branches of the primitive 

 curve there form an ordinary or ceratoid cusp (21. (L fig. 16, Spitze erster 

 Art), the conies of three-pointic contact at A with the corresponding 

 branches of the inverse curve will also coincide, and the latter will possess 

 a ramphoid cusp f (21. (. fig. 1 9, Spitze zweiter Art), at which A* will still 

 be the sole tangent, meeting the cusp in four coincident points. In the 

 special case where the tangent at the cusp a passes through the origin, the 



* By Art. 11, i., the conies inverse to any right line whatever through will have 

 two-pointic contact, at A, with each branch of the tacnode. 



f Prof. Cayley's term cusp-node is more appropriate (Quart. Journ. vol. vi. p. 74) ; the 

 singularity in question may also be regarded as a stationary point on a stationary 

 tangent, for the curve lies entirely on one side of the latter. 



