Page:Scientific Memoirs, Vol. 3 (1843).djvu/727

analysis there is a recurring group of one or more cycles; that is, a cycle of a cycle, or a cycle of cycles. For instance: suppose we wish to divide a series by a series, (1.)

$\scriptstyle {\frac {a+bx+cx^{2}+.\ldots .}{a'+b'x+c'x^{2}+.\ldots .}}$,

it being required that the result shall be developed, like the dividend and the divisor, in successive powers of $\scriptstyle {x}$. A little consideration of (1.), and of the steps through which algebraical division is effected, will show that (if the denominator be supposed to consist of $\scriptstyle {p}$ terms) the first partial quotient will be completed by the following operations:— (2.)

$\scriptstyle {\left\{(\div ),p(\times ,-)\right\}\quad }$ or $\scriptstyle {\left\{(1),p(2,3)\right\}}$,

that the second partial quotient will be completed by an exactly similar set of operations, which acts on the remainder obtained by the first set, instead of on the original dividend. The whole of the processes therefore that have been gone through, by the time the second partial quotient has been obtained, will be,— (3.)

$\scriptstyle {2\left\{(\div ),p(\times ,-)\right\}\quad }$ or $\scriptstyle {2\left\{(1),p(2,3)\right\}}$,

which is a cycle that includes a cycle, or a cycle of the second order. The operations for the complete division, supposing we propose to obtain $\scriptstyle {n}$ terms of the series constituting the quotient, will be,— (4.)

$\scriptstyle {n\left\{(\div ),p(\times ,-)\right\}\quad }$ or $\scriptstyle {n\left\{(1),p(2,3)\right\}}$.

It is of course to be remembered that the process of algebraical division in reality continues ad infinitum, except in the few exceptional cases which admit of an exact quotient being obtained. The number $\scriptstyle {n}$ in the formula (4.), is always that of the number of terms we propose to ourselves to obtain; and the $\scriptstyle {n}$th partial quotient is the coefficient of the $\scriptstyle {(n-1)}$th power of $\scriptstyle {x}$. There are some cases which entail cycles of cycles of cycles, to an indefinite extent. Such cases are usually very complicated, and they are of extreme interest when considered with reference to the engine. The algebraical development in a series, of the $\scriptstyle {n}$th function of any given function, is of this nature. Let it be proposed to obtain the $\scriptstyle {n}$th function of (5.)

$\scriptstyle {\phi (a,b,c\ldots \ldots x),~x}$ being the variable.

We should premise that we suppose the reader to understand what is meant by an $\scriptstyle {n}$th function. We suppose him likewise to comprehend distinctly the difference between developing an $\scriptstyle {n}$th function algebraically, and merely calculating an $\scriptstyle {n}$th function arithmetically. If he does not, the following will be by no means very intelligible; but we have not space to give any preliminary explanations. To proceed: the law, according to which the successive functions of (5.) are to be developed, must of course first be fixed on. This law may be of very various kinds. We may propose to obtain our results in successive powers of $\scriptstyle {x}$, in which case the general form would be

$\scriptstyle {C+C_{1}x+C_{2}x^{2}+}$ &c.,