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.
(Received April 12 1923)
(Revised February 06 1923)
* 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.