92 T. A. Hirst on the Quadric Inversion of Plane CuTves. [Mar. 2, 



whatever is taken, as a fundamental curve, and the origin is placed any- 

 where in its plane. In this manner many descriptive relations which in 

 the ordinary, theory are masked, regain the generality and prominence to 

 which they are entitled. Having long ago convinced myself of the utility 

 of this generalized method of inversion, I deem it desirahle to establish, for 

 the sake of future reference, its chief general principles. With the view of 

 securing the greatest possible familiarity with the effects of inversion, I 

 employ 'purely geometrical considerations, and everywhere give preference 

 to a direct and immediate contempktion of the several geometrical forms 

 which present themselves. The examples occasionally introduced, are 

 given for the sake of illustration merely ; they do not exhibit the full 

 power of the method. Moreover, to prevent, as much as possible, the 

 extension of a paper intended for publication in the Proceedings of the 

 Royal Society, no attempt has been made to subject such special cases to 

 exhaustive treatment. The figures are, for the most part, simple ; the 

 fundamental one being given, the rest may readily be drawn or imagined ; 

 when treating of the effects of inversion on the higher singularities of 

 curves, however, I have thought it desirable to refer by the initials 

 (91. (L) to articles and figures in Plucker's elaborate work on the Theorie 

 der Algebraischen Curven. The relation which the present method bears 

 to the still more general one of quadric transformation, as developed in 

 1832 by Steiner in his Geometrische Gestalten, and by Magnus in the 

 eighth volume of Crelle's Journal, offers several points of interest to which 

 I propose to return on a future occasion*. 



Definitions. 



2. Two points, p and^', conjugate to each other with respect to a fixed 

 fundamental conic (F), and likewise collinear with any fixed origin A in 

 the plane of the latter, are said to be inverse to each other, relative to that 

 conic and origin. In other words, the inverse of a point is the intersection 

 of its polar, relative to (F), and the line which connects it with the origin 

 A. From this the following property is at once deduced : 



i. The several pairs of inverse points p, p', on any line R through the 

 origin A, form an involution, the foci of which are the intersections, real 

 or imaginary, of that line and the fundamental conic (F). 



* I have recently been interested to find that the method of quadric inversion was 

 distinctly suggested, though never developed, by Prof. Bellavitis of Padua no less 

 than twenty-seven years ago. Considering the date of its appearance, the memoir, in 

 the last paragraph of which this suggestion was made, is in many respects a remarkable 

 one. It is entitled Saggio di Geomctria Derivata, and will be found in the fourth vo- 

 lume of the Nuovi Saggi delP I. ft. Accad. di Science, Leffcre cd Arti di Padova. Two 

 years previously, that is to say in 1836, the same geometer had developed, very fully, 

 the principles of the ordinary method of cyclic inversion ; which latter, after a lapse of 

 .seven years, appears to have been first proposed in England by Mr-. J. W. Stubbs, E.A., 

 Fellow of Trinity College, Dublin, in a paper "On the Application of a new Method 

 to the' Geometry of Curves nnd Curve Surfaces," published in the Philosophical Maga- 

 zine, vol. xxiii. p. .3/W. 



