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Published online by Cambridge University Press: 24 October 2008
The problem of discriminating between the different real forms of the rational twisted quartic was first attempted by Adler in 1882. His catalogue is incomplete and some of his results are inaccurate, as has been pointed out by Richmond †. The work was however superseded by a paper in 1891 by Rohn‡ who considered the curve in relation to its allied Steiner quartic surface; his results may be compared with those given in the final section of this paper. Substantially the same results were obtained by Vietoris§ who mapped the curve on a conic by a somewhat different method from that used by Adler.
* Wiener Sitzungsberichte, 86 (1883), 919, 1201, 1212.Google Scholar
† Trans. Camb. Phil. Soc., 19 (1900), 132.Google Scholar
‡ Leipzig Berichte, 43 (1891), 1.Google Scholar
§ Wiener Sitzungsberichte, 125 (1916), 259.Google Scholar
∥ Proc. Camb. Phil. Soc., 29 (1933), 195CrossRefGoogle Scholar. This paper is referred to as T.
* Cf. Grace, and Young, , Algebra of Invariants, p. 198Google Scholar, or we may argue thus The polar prime of F with respect to the primal J is the polar prime of Z with respect to I. If this prime meets the trisecant at Z′, then Z and Z′ are conjugate with respect to I, while F and F′ which are conjugate with respect to I are the two points having the same Hessian Z. Hence since Z′ is the linear polar of F with respect to the points J i, the point Z is the linear polar of F′ with respect to the points K i.