It may be remarked that a slight discrepancy exists between the formulæ
given in the Memoir as the data for calculation, and the results of the calculation as developed in the last division of the table which accompanies it. To agree perfectly with this latter, the data should have been given as
The following is a more complicated example of the manner in which the engine would compute a trigonometrical function containing variables. To multiply
Let the resulting products be represented under the general form
This trigonometrical series is not only in itself very appropriate for illustrating the processes of the engine, but is likewise of much practical interest from its frequent use in astronomical computations. Before proceeding further with it, we shall point out that there are three very distinct classes of ways in which it may be desired to deduce numerical values from any analytical formula.
First. We may wish to find the collective, numerical value of the whole formula, without any reference to the quantities of which that formula is a function, or to the particular mode of their combination and distribution, of which the formula is the result and representative. Values of this kind are of a strictly arithmetical nature in the most limited sense of the term, and retain no trace whatever of the processes through which they have been deduced. In fact, any one such numerical value may have been attained from an infinite variety of data, or of problems. The values for