Page:Scientific Memoirs, Vol. 3 (1843).djvu/736

tion. We remarked in Note B., that any set of columns on which numbers are inscribed, represents merely a general function of the several quantities, until the special function have been impressed by means of the Operation and Variable-cards. Consequently, if instead of requiring the value of the function, we require that of its integral, or of its differential coefficient, we have merely to order whatever particular combination of the ingredient quantities may constitute that integral or that coefficient. In $\scriptstyle {ax^{n}}$, for instance, instead of the quantities

$\scriptstyle {\mathbf {V} _{0}}$	$\scriptstyle {\mathbf {V} _{1}}$	$\scriptstyle {\mathbf {V} _{2}}$	$\scriptstyle {\mathbf {V} _{3}}$
$\scriptstyle {a}$	$\scriptstyle {n}$	$\scriptstyle {x}$	$\scriptstyle {ax^{n}}$
$\scriptstyle {\underbrace {\quad \quad \quad \quad \quad } }$
$\scriptstyle {ax^{n}}$

being ordered to appear on $\scriptstyle {\mathbf {V} _{3}}$ in the combination $\scriptstyle {ax^{n}}$, they would be ordered to appear in that of

$\scriptstyle {anx^{n-1}}$.

They would then stand thus:—

$\scriptstyle {\mathbf {V} _{0}}$	$\scriptstyle {\mathbf {V} _{1}}$	$\scriptstyle {\mathbf {V} _{2}}$	$\scriptstyle {\mathbf {V} _{3}}$
$\scriptstyle {a}$	$\scriptstyle {n}$	$\scriptstyle {x}$	$\scriptstyle {anx^{n-1}}$
$\scriptstyle {\underbrace {\quad \quad \quad \quad \quad } }$
$\scriptstyle {anx^{n-1}}$

Similarly, we might have $\scriptstyle {{\frac {a}{n}}x^{(n+1)}}$, the integral of $\scriptstyle {ax_{n}}$.

An interesting example for following out the processes of the engine would be such a form as

$\scriptstyle {\int {\frac {x^{n}\;dx}{\sqrt {a^{2}-x^{2}}}}}$,

or any other cases of integration by successive reductions, where an integral which contains an operation repeated $\scriptstyle {n}$ times can be made to depend upon another which contains the same $\scriptstyle {n-1}$ or $\scriptstyle {n-2}$ times, and so on until by continued reduction we arrive at a certain ultimate form, whose value has then to be determined.

The methods in Arbogast's Calcul des Dérivations are peculiarly fitted for the notation and the processes of the engine. Likewise the whole of the Combinatorial Analysis, which consists first in a purely numerical calculation of indices, and secondly in the distribution and combination of the quantities according to laws prescribed by these indices.

We will terminate these Notes by following up in detail the steps through which the engine could compute the Numbers of Bernoulli, this being (in the form in which we shall deduce it) a rather complicated example of its powers. The simplest manner of computing these numbers would be from the direct expansion of (1.)

$\scriptstyle {{\frac {x}{e^{x}-1}}={\frac {1}{1+{\frac {x}{2}}+{\frac {x^{2}}{2.3}}+{\frac {x^{3}}{2.3.4}}+\&{\text{c.}}}}}$