Elements of Algebra

476 ADDITIONS. CHAP. I. $A = \alpha$ $B = \beta A + 1$ $C = \gamma B + A$ $D = \delta C + B$ $E = \epsilon D + C$ &c. $A' = 1$ $B' = \beta$ $C' = \gamma B' + A'$ $D' = \delta C' + B'$ $E' = \epsilon D' + C'$ &c. we shall have this series of fractions converging towards the quantity a, $\frac{A}{A'}$, $\frac{B}{B'}$, $\frac{C}{C'}$, $\frac{D}{D'}$, $\frac{E}{E'}$, $\frac{F}{F'}$, &c. If the quantity a be rational, and represented by any fraction $\frac{v}{v'}$, it is evident that this fraction will always be the last in the preceding series; since then the continued fraction will be terminated, and the last fraction of the above series must always be equal to the whole continued fraction. But if the quantity a be irrational, or transcendental, then the continued fraction necessarily going on ad infinitum, we may also continue ad infinitum the series of converging fractions. 11. Let us now examine the nature of these fractions. 1st, It is evident that the numbers A, B, C, &c. must continually increase, as well as the numbers A', B', C', &c. for 1st, if the numbers $\alpha$, $\beta$, $\gamma$, &c. are all positive, the numbers A, B, C, &c. A', B', C', &c. will also be positive, and we shall evidently have $B > A$, $C > B$, $D > C$, &c. and $B' = \beta$, or $> A'$, $C' > B'$, $D' > C'$, &c. 2dly, If the numbers $\alpha$, $\beta$, $\gamma$, &c. are all, or partly negative, then amongst the numbers, A, B, C, &c. and, A', B', C', there will be some positive, and some negative; but in that case we must consider that we have, by the preceding formulæ, $\frac{B}{A} = \beta + \frac{1}{\alpha}$, $\frac{C}{B} = \gamma + \frac{A}{B}$, $\frac{D}{C} = \delta + \frac{B}{C}$, &c. whence we immediately see, that, if the numbers $\alpha$, $\beta$, $\gamma$, &c. are different from unity, whatever their signs be, we shall necessarily have, neglecting the signs, $\frac{B}{A} > 1$; and therefore $\frac{A}{B} < 1$; consequently, $\frac{C}{B} > 1$, and so on: therefore $B > A$, $C > B$, &c. There is no exception to this but when some of the numbers $\alpha$, $\beta$, $\gamma$, &c. are equal to unity. Suppose, for example, that the number $\gamma$ is the first which is equal to $\pm 1$; we