Page:Scientific Memoirs, Vol. 3 (1843).djvu/722

Note E.—Page 684.

This example has evidently been chosen on account of its brevity and simplicity, with a view merely to explain the manner in which the engine would proceed in the case of an analytical calculation containing variables, rather than to illustrate the extent of its powers to solve cases of a difficult and complex nature. The equations of page 679 are in fact a more complicated problem than the present one.

We have not subjoined any diagram of its development for this new example, as we did for the former one, because this is unnecessary after the full application already made of those diagrams to the illustration of M. Menabrea's excellent tables.

It may be remarked that a slight discrepancy exists between the formulæ

${\begin{aligned}&\scriptstyle {(a+bx')}\\&\scriptstyle {(A+B\cos 'x)}\end{aligned}}$

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

${\begin{aligned}&\scriptstyle {(ax^{0}+bx')}\\&\scriptstyle {(A\cos ^{0}x+B\cos 'x).}\end{aligned}}$

The following is a more complicated example of the manner in which the engine would compute a trigonometrical function containing variables. To multiply by

${\begin{aligned}&\scriptstyle {A+A_{1}\cos \theta +A_{2}\cos 2\theta +A_{3}\cos 3\theta +\ldots }\\&\scriptstyle {B+B_{1}\cos \theta }\end{aligned}}$

Let the resulting products be represented under the general form (1.)

$\scriptstyle {C_{0}+C_{1}\cos \theta +C_{2}\cos 2\theta +C_{3}\cos 3\theta +~.~.~.~.~.}$

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 $\scriptstyle {x}$ and $\scriptstyle {y}$ in the two equations (see Note D.), come under this class of numerical results.

Secondly. We may propose to compute the collective numerical value of each term of a formula, or of a series, and to keep these results