Elements of Algebra

xxx CONTENTS. Page Chap. XIV. Solution of some Questions that belong to this Part of Algebra - 413 XV. Solutions of some Questions in which Cubes are required - - - 449 ADDITIONS BY M. DE LA GRANGE. Advertisement - - - - 463 Chap. I. Of Continued Fractions - - - 465 II. Solution of some New and Curious Arithmetical Problems - - - - 495 III. Of the Resolution in Integer Numbers of Equations of the First Degree containing two Unknown Quantities - - - 530 IV. General Method for resolving in Integer Equations of two Unknown Quantities, one of which does not exceed the First Degree - 534 V. A direct and general Method for finding the values of $x$, that will render Quantities of the form $\sqrt{(a+bx+cx^2)}$ Rational, and for resolving, in Rational Numbers, the indeterminate Equations of the second Degree, which have two Unknown Quantities, when they admit of Solutions of this kind - - 537 Resolution of the Equation $Ap^2 + Bq^2 = z^2$ in Integer Numbers - - - 539 VI. Of Double and Triple Equalities - - 547 VII. A direct and general Method for finding all the values of $y$ expressed in Integer Numbers, by which we may render Quantities of the form $\sqrt{(Ay^2 + B)}$, rational; $A$ and $B$ being given Integer Numbers; and also for finding all the possible Solutions, in Integer Numbers, of indeterminate Quadratic Equations of two unknown Quantities - - - 550 Resolution of the Equation $cy^2 - 2nyz + nz^2 = 1$ in Integer Numbers - - - 552 First Method - - - - ib. Second Method - - - - 555 Of the Manner of finding all the possible Solutions of the Equations $cy^2 - 2nyz + nz^2 = 1$, when we know only one of them - - 559 Of the Manner of finding all the possible Solutions, in whole Numbers, of Indeterminate Quadratic Equations of two Unknown Quantities - - - - - 565 VIII. Remarks on Equations of the Form $p^2 = Aq^2 + 1$, and on the common Method of resolving them in whole Numbers - - - 578 IX. Of the Manner of finding Algebraic Functions of all Degrees, which, when multiplied together, may always produce similar Functions 583