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



may readily be ascertained. For in the first place, since (P) has n points 

 on each principal line, its inverse must pass n times through each prin- 

 cipal point (Art. 4) ; and secondly, since a line B drawn through the 

 origin A intersects (P) in n points, none of which are, in general, situated 

 on the principal line BC, polar to Aj the same line R will intersect the 

 inverse curve, not only in the n points coincident with the origin, but also 

 in n points distinct therefrom. Hence 



The complete quadric inverse of any curve of the order n is a curve of the 

 order 2n, which has multiple points, of the order n, at each of the three 

 principal points. 



6. The term complete is here used because, under certain circumstances, 

 the inverse curve will break up into one or more of the principal lines, 

 each taken once or oftener, and aproper inverse curve (P') of lower order, and 

 which passes less frequently through the principal points. This, by Art. 4, 

 will be the case whenever the primitive curve (P) passes through a principal 

 point; and it is obvious that the order of (P') will then be less than 2n 

 by the total number of such passages. Further, the multiplicity of any 

 principal point on (P') will be less than n by the number of times the two 

 principal lines, which there intersect, enter into the complete inverse, in 

 other words by the number of passages of the primitive curve (P) through 

 the poles of those principal lines. 



Hence, if ajjtt, c denote, respectively, the multiplicities of the principal 

 points A, B, C on the curve (P), and if a', b', c' have the same significations 

 relative to the inverse curve (P') of the order n', we shall have 



n'=2n a b c, 



a'= nb c, 



b'= n-a-b, 



c'= nac. 



These equations, by transformation, may readily be made identical with 

 those which result therefrom by simply interchanging the accented and 

 like non-accented letters; this shows, of course, that between proper inverse 

 curves the same mutual relation exists as between inverse points. It is in 

 virtue of this mutual relation that the theorems to be given hereafter are 

 all conversely true ; the enunciations of the converse theorems may there- 

 fore, in all cases, be omitted. 



7. The above equations also furnish the following relations : 



n t a'=na, n'b'=nc, n'c'=nb, 

 from which we learn that 



i. The difference between the orders of two inverse curves is numerically 

 the same as that between the order of either, and the total number of its 

 passages through the three principal points. This latter difference, however, 

 has opposite signs for the two curves. 



ii. For a curve to be of the same order as its inverse, and to pass the 



