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



the fundamental points, the inverse curve will be another conic possessing 

 the same properties (Art. 6). The fundamental conic is not the only one 

 of such a series which coincides with its own inverse (Art. 7) ; for there is 

 obviously a second one which cuts every line through the origin in a pair of 

 inverse points. 



Singularities of inverse Curves. 



1 2. If two of the intersections a, a L , a 2 &c. . . of the primitive curve (P) 

 with the principal line BC coincide ; in other words, if (P) touch BC at ; 

 then two of the branches of (P) will unite to form a cusp at A, at which 

 A<t will be the tangent. Similarly, if (P) touch a fundamental line, say 

 AC at j3, then on one of the branches of (P') there will be a cusp at C, at 

 which the line inverse to B/3 will be the tangent. In short 



i. If one of two inverse curves touch a principal line at a non-principal 

 point, the inverse of the connector of the point of contact with the opposite 

 principal point will touch, at the pole of that line, a cusped branch of the 

 other curve. 



The more general theorem is obviously this : 



ii. If one of two inverse curves have r-pointic contact with a principal line 

 at a non-principal point, r branches of the other curve will coalesce so as to 

 form a branch on which there will be a multiple point of the rth order at the 

 pole of that line ; and the sole tangent to this branch will be the inverse of 

 the connector of the point of contact of the first curve with the opposite 

 principal point. 



It may be added that, in general, the singularity at the principal point on 

 this branch will be invisible or cusp-like, according as r is odd or even. 

 Thus: 



Ex. 1. If the primitive conic considered in Art. 1 1, Ex. 2, not only pass 

 through the origin, but touch the principal line BC in a, its inverse cubic 

 will, besides passing through B and C, have a cusp at A, the tangent at 

 which will be Aa. 



Ex. 2. If the primitive curve be a cubic with a node ; then, the latter being 

 taken as origin, the tangents thereat as fundamental lines, and the real sta- 

 tionary tangent of the cubic as the third principal line ; the inverse curve will 

 be a quartic which touches the latter line at the fundamental points B and 

 C (Arts. 6 and 1 1, iii.), and has moreover a triple point at A, at which the sole 

 tangent passes through the point of inflection on the primitive cubic. This 

 tangent meets the quartic in four points coincident with A (51. (. p. 190)*. 

 13. If the r-pointic contact in the preceding theorem occur at a prin- 

 cipal point, we may conceive it to have arisen from the approach thereto of 

 a point on the principal line, where the contact was (r l)-pointic ; the 

 unique tangent to the branch of the inverse curve upon which there is a 

 multiple point of the order (r 1) will also, by this approach, have become 

 coincident with a principal line f. Hence 



* See also Salmon's ' Higher Plane Curves,' Art. 217. 



t The case corresponding to r=2 has already been considered in Art. 11, iii. 



