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XIII. — The General Form of the Involutive 1-1 Quadric Transformation in a 

 Plane. By Charles Tweedie, M.A., B.Sc. 



(Read 15th July 1901.) 



§ 1. In a communication read before the Society, 3rd December 1900, Dr Mum dis- 

 cusses the generalisation, for more than two pairs of variables, of the proposition that : If 



then 



i=(f-x)/(l-xy); r,={x*-y)l(l-xy). 



If we interpret (x, y) and (£, >?) as points in a plane, it is manifest that the transfor- 

 mation thereby obtained is a Cremona transformation. It has the special property of 

 being reciprocal or involutive in character; i.e., if the point P is transformed into Q, 

 then the repetition of the same transformation on Q transforms Q into P. Symbolically, 

 if the transformation is denoted by T, T(P) = Q, and T(Q) = T 2 (P) = P ; so that T 2 = 1, 

 and T = T -1 . Moreover, if the locus of P (x, y) is a straight line, the locus of Q (£, >;) is 

 in general a conic. 



§2. The object of this note is to find the most general bilinear transformation con- 

 necting two points (x,y), (£, v) of the form 



L 2 £+M 2 >7 + N 2 = 0j ' 



(Lj, etc., being linear functions of x and y) which possesses this property; i.e., the 

 most general 1-1 transformation which is involutive in character and in which to a 

 straight line corresponds a conic. 



This problem has already been discussed by Czuber (Monatshefte fur Mathematik 

 und PhysiJc, 1894), but unfortunately his discussion is not free from error, and one 

 of the best-known transformations of the kind — the so-called Hirst transformation — 

 entirely escapes his observation. Moreover, he describes the above transformation (I.) 

 as the most general 1-1 point transformation, which is by no means the case, as it is 

 not difficult to frame 1-1 point transformations in which to a straight line corre- 

 sponds a curve of higher degree than the second (v. Salmon's Higher Plane Curves). 

 In his paper, however, he discusses a very large variety of degenerate cases, and this 

 enables me to dispense with these entirely and to discuss only the leading case in 

 which to a straight line corresponds a conic. 



^3. The transformation (I.) would appear to be the most general 1-1 quadric 

 transformation. On solving for £ and >j we deduce 



f- 



L X M 2 - L 2 M 1 



_ N 1 L _-N,L 1 



VOL. XL. PART II. (NO. 13). 2 P 



