Elements of Algebra

CHAP. XIV. OF ALGEBRA. 425 $z = \frac{4(v+by) \times vy \times (v+ay)}{q^2}$, or $z = \frac{4vy(v+ay) \times (v+by)}{q^2}$. By these means we may take any numbers for $v$ and $y$, and assuming for $q^2$ the greatest square contained in the numerator, we shall easily determine the value of $z$; after which, we may return to the equations $m = az$, $n = bz$, and $p = v^2 - aby^2$, and shall obtain the formulæ required. 1. $p^2 + azq^2 = (v^2 - aby^2)^2 + 4avy(v+ay) \times (v+by)$, which is a square, whose root is $r = -v^2 - 2avy - aby^2$. 2. The second formula becomes $p^2 + bzq^2 = (v^2 - aby^2)^2 + 4bvy(v+ay) \times (v+by)$, which is also a square, whose root is $s = -v^2 - 2bvy - aby^2$, and the values both of $r$ and $s$ may be taken positive. It may be proper to analyse these results in some examples. 226. *Example 1.* Let $a = -1$, and $b = +1$, and let us endeavour to seek such a number for $z$, that the two formulæ $p^2 - zq^2$, and $p^2 + zq^2$, may become squares; namely, the first $r^2$, and the second $s^2$. We have therefore $p = v^2 + y^2$; and, in order to find $z$, we have only to consider the formula $z = \frac{4vy(v-y) \times (v+y)}{q^2}$; and, by giving different values to $v$ and $y$, we shall see those that result for $z$. \begin{tabular}{|c||c|c|c|c|c|c|} \hline & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline \hline $v$ & 2 & 3 & 4 & 5 & 16 & 8 \\ $y$ & 1 & 2 & 1 & 4 & 9 & 1 \\ $v-y$ & 1 & 1 & 3 & 1 & 7 & 7 \\ $v+y$ & 3 & 5 & 5 & 9 & 25 & 9 \\ $zq^4$ & $4 \times 6$ & $4 \times 30$ & $16 \times 15$ & $9 \times 16 \times 5$ & $36 \times 25 \times 16 \times 7$ & $16 \times 9 \times 14$ \\ \hline $q^2$ & 4 & 4 & 16 & $9 \times 16$ & $36 \times 25 \times 16$ & $16 \times 9$ \\ $z$ & 6 & 30 & 15 & 5 & 7 & 14 \\ $p$ & 5 & 13 & 17 & 41 & 337 & 65 \\ \hline \end{tabular} And by means of these values, we may resolve the following formulæ, and make squares of them: 1. We may transform into squares the formulæ $p^2 - 6q^2$, and $p^2 + bq^2$; which is done by supposing $p=5$, and $q=2$; for the first becomes $25 - 24 = 1$, and the second $25 + 24 = 49$. 2. Likewise, the two formulæ $p^2 - 30q^2$, and $p^2 + 30q^2$;