Page:Scientific Memoirs, Vol. 3 (1843).djvu/729

function might in many cases be developed through a set of processes peculiar to itself, and not recurring for the remaining functions.

We have given but a very slight sketch, of the principal general steps which would be requisite for obtaining an $\scriptstyle {n}$th function of such a formula as (5.). The question is so exceedingly complicated, that perhaps few persons can be expected to follow, to their own satisfaction, so brief and general a statement as we are here restricted to on this subject. Still it is a very important case as regards the engine, and suggests ideas peculiar to itself, which we should regret to pass wholly without allusion. Nothing could be more interesting than to follow out, in every detail, the solution by the engine of such a case as the above; but the time, space and labour this would necessitate, could only suit a very extensive work.

To return to the subject of cycles of operations: some of the notation of the integral calculus lends itself very aptly to express them: (2.) might be thus written:— (6.)

$\scriptstyle {(\div ),\Sigma (+1)^{p}(\times ,-)}$ or $\scriptstyle {(1),\Sigma (+1)^{p}(2,3)}$,

where $\scriptstyle {p}$ stands for the variable; $\scriptstyle {(+1)^{p}}$ for the function of the variable, that is, for $\scriptstyle {\phi p}$; and the limits are from 1 to $\scriptstyle {p}$, or from 0 to $\scriptstyle {p-1}$, each increment being equal to unity. Similarly, (4.) would be,— (7.)

$\scriptstyle {\Sigma (+1)^{n}\left\{(\div ),\Sigma (+1)^{p}(\times ,-)\right\}}$

the limits of $\scriptstyle {n}$ being from 1 to $\scriptstyle {n}$, or from 0 to $\scriptstyle {n-1}$, (8.)

or $\scriptstyle {\Sigma (+1)^{n}\left\{(1),\Sigma (+1)^{p}(2,3)\right\}}$.

Perhaps it may be thought that this notation is merely a circuitous way of expressing what was more simply and as effectually expressed before; and, in the above example, there may be some truth in this. But there is another description of cycles which can only effectually be expressed, in a condensed form, by the preceding notation. We shall call them varying cycles. They are of frequent occurrence, and include successive cycles of operations of the following nature:— (9.)

$\scriptstyle {p(1,2,..m),{\overline {p-1}}(1,2\ldots m),{\overline {p-2}}(1,2..m)\ldots {\overline {p-n}}(1,2..m)}$,

where each cycle contains the same group of operations, but in which the number of repetitions of the group varies according to a fixed rate, with every cycle. (9.) can be well expressed as follows:— (10.)

$\scriptstyle {\Sigma p(1,2,..m)}$, the limits of $\scriptstyle {p}$ being from $\scriptstyle {p-n}$ to $\scriptstyle {p}$.

Independent of the intrinsic advantages which we thus perceive to result in certain cases from this use of the notation of the integral calculus, there are likewise considerations which make it interesting, from the connections and relations involved in this new application. It has been observed in some of the former Notes, that the processes used in analysis form a logical system of much higher generality than the applications to number merely. Thus, when we read over any algebraical formula, considering it exclusively with reference to the processes of the engine, and putting aside for the moment its abstract signification as to the relations of quantity, the symbols $\scriptstyle {+}$, $\scriptstyle {\times }$, &c., in reality represent (as their immediate and proximate effect, when the formula is