1865.] T. A. Hirst on the Quadric Inversion of Plane Curves. 93 



Two curves are said to be inverse to each other, of course, when the se- 

 veral points of one are inverse to those of the other. The latter is some- 

 times referred to as the primitive, and the former as its inverse ; the relation 

 hetween the two curves, however, is a mutual one, and the distinction is 

 merely introduced for convenience. In order to obtain clear conceptions 

 of the various relations which exist between inverse curves, it will be found 

 convenient to place the origin A outside the fundamental conic (F). The 

 modifications to be introduced when the origin is placed elsewhere are quite 

 obvious, and, except in a few instances, need not be specially alluded to. 



3. Adopting terms introduced by Magnus, the origin A and the two 

 points of contact B and C of the tangents from A to the fundamental conic 

 (F) are called the principal points ; the triangle, of which they form the 

 corners, is termed the principal triangle, and its sides BC, CA, AB, respect- 

 ively polar to A, C, B, are the principal lines. Occasionally it will be con- 

 venient to refer to B and C as distinct from A, which is always real ; the 

 two former will then be called the fundamental points, and the principal 

 lines AB, AC, which there touch (F), will in like manner be called the fun- 

 damental lines polar to B and C. 



4. This premised, it is manifest from Art. 2 that, in general, a pointy 

 has but one inverse pointy'. The only exceptions, in fact, are the three 

 principal points, each of which is obviously inverse to every point in the 

 principal line which constitutes its polar. It is further evident that each 

 point of the fundamental conic (F) coincides with its own inverse, and that 

 the several points of (F) are the only ones in the plane which possess" this 

 property. 



Hence may be inferred, without difficulty, the following theorem : 



i. If any two curves have r-pointic contact with each other at a point p, 



not on a principal line, their inverse curves will also have r-pointic contact 



with each other at the inverse point p'. 



Relative orders of inverse Curves. 



5. The order of the curve inverse to a given curve (P) of the order n 



i 2 



