TRANSACTIONS OF THE SECTIONS. 3 



On tlie Geometrical Transformation of Plane Curves. B>j Professor Crejioxa, 

 of Bologna. Communimted by T. A. Hirst, F.E.S. 



In a note on the geometrical transformation of plane curves, published in the 

 ' Giomale di Matematiche,' vol. i. p. ?>Qo, several remarkable properties possessed by 

 a certain system of curves of the rt-th order, situated in the same plane, were con- 

 sidered. The important one which forms the subject of this note has been more 

 recently detected, and has reference to the Jacobian of such a system, that is to 

 say, to the locus of a point whose polar lines, relative to all curve.s of the system, 

 are concurrent. 



The ciu'ves in question form, in fact, a reseau ; in other words, they satisfy, in com- 

 mon, — i__Ji:— 2 conditions in such a manner that through any two assumed points 



only one curve passes. Thoy have, moreover, so many fixed (fundamcniar) points 

 in common that no two curves intersect iu more than one variable point. In short, 

 if, in general, .r^. denote the number of fundamental points which are multiple 

 points of the r-th order on every curve of the reseau, the following two equations 

 are satisfied : — 



x^-^-^x^+Gx^ 



n(n-\)^ Z»(»+3) 



':> 



_o 



ari-l-4r,-l-9.r3 + (H-l)=.r„_, = «^-l. 



This being premised, the property alluded to is, that the Jacobian of every such 

 reseau resolves itself into y^ right lines, y.^ conies, 7/3 cubics, &c., and y„_i curves 

 of the order w — 1; where the integers y^, y.^, &c. also satisfy the above equations, 

 and constitute a conjugate solution to x-^, x.^, &c., being connected therewith by the 

 relation 



0-1 -l-x, +-r„_i=yi+2/» +yn-i- 



On a Generalization of the Method of Geometrical Inversion. 

 By T. A. HiEST, F.E.8. 



It is well known that Steiner, by assuming, instead of a conic, any fimdamental 

 curve wiiatever, succeeded in generalizing Poncelet's theory of reciprocal polars. 

 The ordinary method of inversion is susceptible of a generalization of the same 

 character, and may then be appropriately termed Quantic Inversim. A fixed origin 

 o being taken, the radius vector from it to any point p in the plane is, of course, 

 cut in {m—r) points p' by the ?--th polar oi p, relative to any fixed fundamental 



curve of the «i-th order. If p describe a 2>rimitive curve of the M-th order P , it 



can readily be shown that its corresponding points p' will generate a curve P of 



the order mn (independent of ?•) which, amongst other singularities, always possesses 

 a multiple point at the origin of the order nr. The properties of the series of 

 (wi— 1) inverse curves corresponding to any primitive w-ic, and relative to the 

 same origin and fundamental in-\c, formed the subject of the communication. 



"V\Tien m=2, the fimdamental curve is a conic which is intersected in two, real 

 or imaginarj', points Oj and o^ by the polar of the origin o. In this ease the first 

 and sole quadnc inverse of a given H-ic which passes a times through the origin o, 

 «i times through the point o„ and a.^ times through the point o,, is (if the sides of the 

 principal triangle 00^ o^ be excluded) a cm-ve of the order (2n—a—a^ — a,J, which 

 passes (n — a^ — a,) times through o,(n—a — a^) thj-ough o^, and (w — a — a.-^) through o^. 

 As a simple illustration of the utility of this special case of quantic inversion, when 

 employed as a method of transformation, it may bo rcmarlied that the number of 

 (double tangents to the quadric inverse curve, as determined by Pliicker's foTmiflfg', 

 is equal to the number of conies which can be drawn through three fixed points 

 SO as to have double contact with the primitive cm-ve. The results of quadric in- 

 version are identical with those obtained by the somewhat more general, but less 

 easily manipulated, transformations of Steiner and of Mag-nus. "VMien the funda- 

 mental curve is a circle around the origin, the fundamental poinds Oj, o^ coincide with 



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