Elements of Algebra

CHAP. VIII. ADDITIONS. 581 transformed equation $(\mu^2 - A)q^2 + 2\mu qr + r^2 = 0$; we shall have an equation, whose root will be $\frac{r}{s}$; then taking the approximate, deficient value of this root, we shall have the value of $\mu''$. Here again we shall substitute $\mu''r+s$, instead of $r$, &c. Let us now suppose, for example, that $t$ is the last remainder, which must be nothing, then $s$ will be the last but one, which must be $=1$; wherefore, if the formula $p^2 - Aq^2$, when transformed into terms of $s$ and $t$, is $Ps^2 + Qst + Rt^2$, by making $t=0$, and $s=1$, it must become $=1$, in order that the given equation, $p^2 - Aq^2 = 1$, may take place; and therefore $P$ must be $=1$. Thus, we shall only have to continue the above operations and transformations, until we arrive at a transformed formula, in which the coefficient of the first term is equal to unity; then, in that formula, we shall make the first of the two indeterminates, as $r$, equal to $1$, and the second, as $s$, equal to $0$; and, by going back, we shall have the corresponding values of $p$ and $q$. We might likewise work with the equation $p^2 - Aq^2 = 1$ itself, only taking care to abstract from the term $1$, which is known, and consequently from the other known terms, likewise, that may result from this, in the determination of the approximate values $\mu, \mu', \mu''$, &c. of $\frac{p}{q}$, $\frac{q}{r}$, $\frac{r}{s}$, &c. In this case, we shall try at each new transformation, whether the indeterminate equation can subsist, by making one of the two indeterminates $=1$, and the other $=0$; when we have arrived at such a transformation, the operation will be finished; and we shall have only to go back through the several steps, in order to have the required values of $p$ and $q$. Here, therefore, we are brought to the method of Chap. VII. To examine this method in itself, and independently of the principles from which we have just deduced it, it must appear indifferent whether we take the approximate values of $\mu, \mu', \mu''$, &c. less, or greater than the real values; since, in whatever way we take these values, those of $r, s, t$, &c. must go on decreasing to $0$. (Art. 6.) Wallis also expressly says, that we may employ the limits for $\mu, \mu', \mu''$, &c. either in plus, or in minus, at pleasure; and he even proposes this, as the proper means often of abridging the calculation. This is likewise remarked by Euler, Art. 102, et seq. of the chapter just now quoted. However, the following example will shew, that by setting about it in this