1865.] T. A. Hirst on the Quadric Inversion of Plane Curves. 95 



same number of times through each of the three principal points , it is neces- 

 sary and sufficient, first, that its order be equal to the total number of its 

 passages through the three principal points ; and secondly, that it pass as 

 frequently through one fundamental point as through the other. 



It would be easy, by the second theorem, to determine the number and 



nature of the several curves of a given order which have inverse curves of 



the same order and like properties, relative to the principal points. This 



determination, however, as well as the solution of the allied question 



Under what conditions will a curve of given order coincide with its own 



inverse ? 



will more appropriately form the subject of a separate paper. It will be 

 sufficient to note here that a right line through the origin, and the 

 fundamental conic itself, regarded as a primitive curve, are the simplest in- 

 stances of the kind under consideration. Another example is also alluded 

 to in Art. 11. Ex.3. 



Conies inverse to Right Lines, 



8. From Arts. 2 and 6, as well as directly from elementary geometrical 

 principles *, it follows that 



i. The inverse of a right line is, in general, a conic passing through the 

 three principal points and the two intersections of the fundamental conic and 

 the primitive line, as well as through the pole of the latter, relative to the 

 former. 



It is only when the primitive line passes through a principal point that 

 the inverse conic breaks up into the fixed principal line, polar to that 

 point, and another right line (the proper inverse) through the principal 

 point opposite to that fixed principal line (Art. 4). Thus : 



ii. The proper inverse of a right line passing through one of the two 

 fundamental points is a right line passing through the other, and these inverse 

 right lines always intersect on the fundamental conic. 



The following is an immediate consequence of this and the theorem i. 

 of Art. 4 : 



iii. If one of two inverse curves have r-pointic contact, not on a principal 

 line, with a right line passing through a fundamental point, the other will 

 have r-pointic contact with the inverse line, through the other fundamental 

 point, and the points of contact, being inverse to each other, will be collinear 

 with the origin. 



The modification which this theorem suffers when one of the points of 

 contact is on a principal line, will shortly be fully considered. 



9. The line at infinity has also its inverse conic (I), which is of import- 

 ance in many inquiries connected with inversion. On observing (Art. 2) 

 that the inverse of the infinitely distant point of any line R through 



* An elegant demonstration of this theorem, identical with tlio elementary one 

 alluded to, has been inserted by M. Chasles in Art. 209 of his excellent TraiU des Sec- 

 tions Conigucs, the first part of which has just reached me. 



