Elements of Algebra

556 ADDITIONS. CHAP. VII. Now, it is evident, that these operations cannot go on to infinity; for since $2n$ is greater than $D'$, and $2n'$ is not, $n'$ will evidently be less than $n$; in the same manner, $2n'$ is greater than $D''$, and $2n''$ is not, wherefore $n''$ will be less than $n'$, and so on; so that the numbers $n, n', n'', \&c.$ will form a decreasing series of integers, which of course cannot go on to infinity. We shall therefore arrive at a formula, in which the coefficient of the middle term will not be greater than those of the two extreme terms, and which will likewise have the other properties already mentioned; as is evident from the nature of the transformations employed. In order to facilitate the transformation of the formula, $cy^2 - 2nyz + Bz^2$ into this, $L\xi^2 - 2M\xi\psi + N\psi^2$, let us denote by $D$ the greater of the two extreme coefficients $c$ and $B$, and the other coefficient by $D'$; and, vice versâ, let us denote by $\theta$ the variable quantity, whose square shall be found multiplied by $D'$, and the other variable quantity by $\theta'$; so that the given formula may take this form, $D'\theta^2 - 2n\theta\theta' + D\theta'^2$, where $D'$ is less than $D$; then we have only to make the following calculation: $m = \frac{n}{D'}$, $n' = n - mD'$, $D'' = \frac{n'^2 - A}{D'}$, $\theta = m\theta' + \theta''$, $m' = \frac{n'}{D''}$, $n'' = n' - m'D''$, $D''' = \frac{n''^2 - A}{D''}$, $\theta' = m'\theta'' + \theta'''$, $m'' = \frac{n''}{D'''}$, $n''' = n'' - m''D'''$, $D^{\text{iv}} = \frac{n'''^2 - A}{D'''}$, $\theta'' = m''\theta''' + \theta^{\text{iv}}$, $\&c. ~ \&c. ~ \&c.$ where it must be observed, that the sign $=$, which is put after the letters $m, m', m'', \&c.$ does not express a perfect equality, but only an equality as approximate as possible, so long as we understand only integer numbers by $m, m', m'', \&c.$ The sign $=$ being only employed for want of a better. These operations must be continued, until in the series $n, n', n'', \&c.$ we find a term, as $n_q$, which (abstracting from the sign) does not exceed the half of the corresponding term,