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 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 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 occurs in the original (5.), or it might similarly make it wherever 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 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, times, to obtain the 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[27], one, two, three, up to times, in order to get the th function. We say, after the first function is obtained, because (for reasons on which we cannot here enter) the first function might in many cases be developed through a set [Pg 53] of processes peculiar to itself, and not recurring for the remaining functions.