xii PEEFACE.

Such an equation a;2— Ay* = l has long borne the name Pellian equation, after John Pell, due to a confusion on the part of Euler; it would have been more appropriately named after tfermat, who stated in 1657 that it has an infinitude of integral solutions if A is any positive integer not a square, and who stated in 1659 that he possessed a proof by indefinite descent. He proposed it as a challenge problem to the English mathematicians Lord Brouncker and John Wallis, who finally succeeded in discovering a tentative method of solution, without giving a proof of the existence of an infinitude of solutions. This theorem is really only the simplest and first known case of Dirichlet's elegant and very general theorem on the existence of units hi any algebraic field or domain. The former theorem is also of great importance in the theory of binary quadratic forms. Moreover, the problem to find all the rational solutions of the most general equation of the second degree in two unknowns reduces readily to that for x*—Ay*=B, all of whose solutions follow from one solution and the solutions

In 1765 Euler exhibited the method of solving a Pellian equation due to Brouncker and Wallis in a more convenient form by use of the continued fraction for VZ and found various important facts, but gave no proof that the process leads always to a solution in positive integers. This fundamental fact of the existence of solutions was first proved by Lagrange a year or two later; while in 1769 and 1770 he brought out his classic memoirs which give a direct method to find all integral solutions of x2—Ayz=B, as well as of an equation of degree n, by developing its real roots into continued fractions.

Of the further extensive literature on the Pellian equation, the most notable papers are those by Legendre, Gauss, Dirichlet, Jacobi, and Perott; limits for the least positive solution were obtained by Tchebychef in 1851 and by Remak, Perron, Schmitz, and Schur in 1913-18. Useful tables have been given by Euler, Legendre, Degen, Tenner, Koenig, Arndt, Cayley, Stern, Seeling, Roberts, Bickmore, Cunningham, and Whitf ord.

Chapter XIII treats of further single equations of the second degree, including axy+bx+cy+d=Q, xz—y2 = g, ax2-t~bxy+cyz = dz2 or d, the most general equation of the second degree in x} y, and its homogeneous form aX*+bY2+cZ2+dXY+eXZ+fYZ=Q. Criteria for integral solutions of the latter were stated by H. J. S. Smith (p. 431) and proved by Meyer for the case of an odd determinant, while its complete solution was given by Desboves (p. 432) when one solution is known. Lagrange's method for $2— Ay2 =#, cited above, was employed by Legendre in 1785 to prove the important theorem that, if no two of the integers a, 6, c have a common factor and if each is neither zero nor divisible by a square, then oz2+67/2-f C22=0 has integral solutions not all zero if, and only if, — 6c, — oc, — db are quadratic residues of a, 6, c, respectively, and a, 6, c are not all of the same sign. Gauss gave a proof by means of ternary quadratic forms, while a generalization was made by Dirichlet (p. 423) and Goldscheider (p. 426). Meyer gave criteria (pp. 432-3) for integral solutions of /=0;