Page:Scientific Memoirs, Vol. 3 (1843).djvu/728

or in successive powers of $\scriptstyle {n}$ itself, the index of the function we are ultimately to obtain, in which case the general form would be

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

and $\scriptstyle {x}$ would only enter in the coefficients. Again, other functions of $\scriptstyle {x}$ or of $\scriptstyle {n}$ instead of powers, might be selected. It might be in addition proposed, that the coefficients themselves should be arranged according to given functions of a certain quantity. Another mode would be to make equations arbitrarily amongst the coefficients only, in which case the several functions, according to either of which it might be possible to develop the $\scriptstyle {n}$th function of (5.), would have to be determined from the combined consideration of these equations and of (5.) itself.

The algebraical nature of the engine (so strongly insisted on in a previous part of this Note) would enable it to follow out any of these various modes indifferently; just as we recently showed that it can distribute and separate the numerical results of any one prescribed series of processes, in a perfectly arbitrary manner. Were it otherwise, the engine could merely compute the arithmetical $\scriptstyle {n}$th function, a result which, like any other purely arithmetical results, would be simply a collective number, bearing no traces of the data or the processes which had led to it.

Secondly, the law of development for the $\scriptstyle {n}$th function being selected, the next step would obviously be to develope (5.) itself, according to this law. This result would be the first function, and would be obtained by a determinate series of processes. These in most cases would include amongst them one or more cycles of operations.

The third step (which would consist of the various processes necessary for effecting the actual substitution of the series constituting the first function, for the variable itself) might proceed in either of two ways. It might make the substitution either wherever $\scriptstyle {x}$ occurs in the original (5.), or it might similarly make it wherever $\scriptstyle {x}$ occurs in the first function itself which is the equivalent of (5.). In some cases the former mode might be best, and in others the latter.

Whichever is adopted, it must be understood that the result is to appear arranged in a series following the law originally prescribed for the development of the $\scriptstyle {n}$th function. This result constitutes the second function; with which we are to proceed exactly as we did with the first function, in order to obtain the third function; and so on, $\scriptstyle {n-1}$ times, to obtain the $\scriptstyle {n}$th function. We easily perceive that since every successive function is arranged in a series following the same law, there would (after the first function is obtained) be a cycle, of a cycle, of a cycle, &c. of operations^[1], one, two, three, up to $\scriptstyle {n-1}$ times, in order to get the $\scriptstyle {n}$th function. We say, after the first function is obtained, because (for reasons on which we cannot here enter) the first

1. A cycle that includes $\scriptstyle {n}$ other cycles, successively contained one within another, is called a cycle of the $\scriptstyle {n+1}$th order. A cycle may simply include many other cycles, and yet only be of the second order. If a series follows a certain law for a certain number of terms, and then another law for another number of terms, there will be a cycle of operations for every new law; but these cycles will not be contained one within another,—they merely follow each other. Therefore their number may be infinite without influencing the order of a cycle that includes a repetition of such a series.