# The Cayley-Spottiswoode coordinates of a conic in 3-space

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## Abstract

The Cayley-Spottiswoode coordinates of a conic in 3-space COMPOSITIO MATHEMATICA H. S. RUSE TheCayley-Spottiswoode coordinates of a conic in 3-space Compositio Mathematica, tome 2 (1935), p. 438-462. <http://www.numdam.org/item?id=CM_1935__2__438_0> © Foundation Compositio Mathematica, 1935, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (http: //http://www.compositio.nl/) implique l’accord avec les conditions gé- nérales d’utilisation (http://www.numdam.org/legal.php). Toute utilisa- tion commerciale ou impression systématique est constitutive d’une in- fraction pénale. Toute copie ou impression de ce fichier doit conte- nir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ The Cayley-Spottiswoode coordinates of a conic in 3-space. by H. S. Ruse Edinburgh If a conic in a three-dimensional projective space is defined by the quadratic complex of lines which meet it, the coefficients in the equation of the complex may be regarded as coordinates of the conic. The coordinates thus defined, analogous to the Plücker coordinates of a line, are due essentially to Cayley 1), but were defined independently and differently by Spottiswoode 2). They were employed recently by J. A. Todd 3) to represent the conics of 3-space by points of 19-space, for which purpose he introduced a symmetrical and concise notation which, with certain modifications, is used in the present paper. Each conic has twenty-one distinct homogeneous coordinates which satisfy certain identical relations; these are obtained below by a method which shows that the symbolic calculus 4) employed by Todd admits of a geometrical interpretation. A variety of other formulae are also established, expressing the condition that two conics should intersect, that they should be coplanar, and so on. Todd’s notation is extended so as to be brought into confor- mity with that of tensor, o

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