Note on the Theory of the Pellian Equation and of Binary Quadratic Forms of a Positive Determinant*

Henry J. Stephen Smith
1875 Proceedings of the London Mathematical Society  
Art.l. Let 0 O = Mo~\ ' 1-••• beany continued fraction, of which A*i hi * The following summary of the contents of this Note may be of use to the reader :-Art. 1.-The relation, in a continued fraction, between the quantities « and 0. Art. 2.-The theorem that T + U^/D is equal to the product of the complete quotients in the development of ,/D. Art. 3.-The samo theorem for the period of complete quotients in the development of any quadratic surd. Art. 4.-Theorems as to the number of different
more » ... er of different periods of complete quotients; viz., equations (1) -(4). Art. 5.-Theorems as to the number of non-equivalent classes of quadratic forms; viz., equations (5) and (6). Art. 6.-Equations arising from a comparison. of the formulae (6) and (6) with those of Dirichlet. Arts. 7-13.-Discussion of the nature of the periods in the more important special cases. Art. 14.-On the symmetry of any periodic series. Art. 16.-On the arithmetical conditions under which [the various special cases present themselves.. . [It would bo difficult to say that anything in the Addition (Arts. 7-15) is new: the discussion there attempted has never been given completely (see Art. 7) ; but this may have been because no one has thought it worth giving.]
doi:10.1112/plms/s1-7.1.196 fatcat:sizz5i56trdl5jwc2x2s2g64jm