Elements of Algebra

CHAP. VII. ADDITIONS. 555 Second Method. 70. Let the formula $cy^2 - 2nyz + Bz^2$ undergo such transformations as those we have already made (Art. 54), and we shall invariably be brought by the transformations, to an equation, such as $L\xi^2 - 2M\xi\psi + N\psi^2$, the numbers, L, M, N, being whole numbers, depending upon the given numbers c, B, n, so that we have $M^2 - LN = n^2 - CB = A$; and farther, that $2M$ may not be greater (abstracting from the signs) than the number L, nor the number N; the numbers $\xi$ and $\psi$ will likewise be integer, but depending on the indeterminate numbers $y$ and $z$. For example, let c be less than B, and let us put the formula in question into this form, $B'y^2 - 2nyy' + By'^2$, making $c = B'$, and $z = y'$; if $2n$ be not greater than $B'$, it is evident that this formula will already of itself have the requisite conditions; but if $2n$ be greater than $B'$, then we must suppose $y = my' + y''$; and, by substitution, we shall have the transformed formula, $B''y'^2 - 2n'y'y'' + B'y''^2$, where $n' = n - mB'$, and $B'' = m^2B' - 2mn + B = \frac{n'^2 - A}{B'}$. Now, as the number $m$ is indeterminate, we may, by supposing it an integer, take it such, that the number $n - mB'$ may not be greater than $\frac{1}{2}B'$, abstracting from the sign; then $2n'$ will not surpass $B'$. So that, if $2n'$ does not even exceed $B'$, the preceding transformed formula will already be in the case which we have seen; but if $2n'$ is greater than $B'$, we shall then continue to suppose $y' = m'y'' + y'''$, which will give this new transformation, $B'''y''^2 - 2n''y''y''' + B''y'''^2$, where $n'' = n' - m'B''$, and $B''' = m'^2B'' - 2m'n' + B' = \frac{n''^2 - A}{B''}$. We shall now determine the whole number $m'$, so that $n' - m'B''$ may not be greater than $\frac{B''}{2}$, by which means $2n''$ will not exceed $B''$; so that we shall have the required transformation, if $2n''$ does not even exceed $B''$; but if $2n''$ exceed $B''$, we shall again suppose $y'' = m''y''' + y^{iv}, \ldots, \ldots$