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Published online by Cambridge University Press: 20 January 2009
The first part of the following investigation was begun before the discovery that Mr E. Kasner had already touched upon the apolarity theory of double binary forms in an important work on the Inversion Group (Transactions of the American Mathematical Society, Vol. I (1900), pp. 471–473). The theory is carried further in what follows, with special reference to the (2, 2) form. The second part answers questions raised by Professor A. R. Forsyth in the Quarterly Journal, 1910, p. 113. It appears that the general (2, 2) form admits of three independent automorphic transformations, but the general (n, n) form admits of none, if n exceeds two.
Previous discussions by the author are (1) Proc. Roy. Soc. Edin., XLIII. (1922–3), pp. 43–50; (2) Proc. Edin. Math. Soc, XLI. (1922–3), pp. 116–127, and (3) Proc. Roy. Soc. Edin., XLIV. (1923–4) pp. 23–50.
* Loc. cit.
* An expressive notation, apparently due to Dr R. Weitzenböck, meaning that the preceding statements are true for ail values of ξ.
* Cf. Proc. Roy. Soc. Edin, XLIV. p. 23.
* Cf. Proc. Soy. Soc. Edin., XLIV. p. 30.
† Cf. Kasner, loc. cit. Also Proc. R. S. E., XLIV. p. 25.
* Cf. Proc. R. S. E., XLIV. p. 29.
* Homographic Transformations. Quarterly Journal (1910), p. 113. Also, in the same volume, a note by Steinthal on p. 221.
* Cf. Forsyth, loc. cit., § 12, which agrees with this result, as the (2, 2) curve there treated is restricted by three conditionr.
† Cf. Proc. Roy. Soc. Edin., XLIV., p. 36. This geometrical application of double binary algebra was suggested by Mr J. H. Grace, to whom my thanks are due.
* Cf. FORSYTH, loc. cit., where the problem is suggested but unanswered. The case when 
vanishes, i.e. a third degree invariant vanishes, is solved. An algebraic solution of the general problem is given by Steinthal, loc. cit. but the method used throws no light on the geometrical theory.
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