Page:Scientific Memoirs, Vol. 3 (1843).djvu/715

either permanently remain on its column during the succeeding calculations, so that when all the functions had been computed, their values would simultaneously exist on $\scriptstyle {\mathbf {V} _{4}}$, $\scriptstyle {\mathbf {V} _{5}}$, $\scriptstyle {\mathbf {V} _{6}}$, &c.; or each result might (after being printed off, or used in any specified manner) be effaced, to make way for its successor. The square under $\scriptstyle {\mathbf {V} _{4}}$ ought, for the latter arrangement, to have the functions $\scriptstyle {ax^{n}}$, $\scriptstyle {x^{an}}$, $\scriptstyle {a~n~x}$, &c. successively inscribed in it.

Let us now suppose that we have two expressions whose values have been computed by the engine independently of each other (each having its own group of columns for data and results). Let them be $\scriptstyle {ax^{n}}$, $\scriptstyle {b.p.y}$. They would then stand as follows on the columns:—

$\scriptstyle {\mathbf {V} _{1}}$	$\scriptstyle {\mathbf {V} _{2}}$	$\scriptstyle {\mathbf {V} _{3}}$	$\scriptstyle {\mathbf {V} _{4}}$	$\scriptstyle {\mathbf {V} _{5}}$	$\scriptstyle {\mathbf {V} _{6}}$	$\scriptstyle {\mathbf {V} _{7}}$	$\scriptstyle {\mathbf {V} _{8}}$	$\scriptstyle {\mathbf {V} _{9}}$
+	+	+	+	+	+	+	+	+
0	0	0	0	0	0	0	0	0
0	0	0	0	0	0	0	0	0
0	0	0	0	0	0	0	0	0
0	0	0	0	0	0	0	0	0
$\scriptstyle {a}$	$\scriptstyle {n}$	$\scriptstyle {x}$	$\scriptstyle {ax^{n}}$	$\scriptstyle {b}$	$\scriptstyle {p}$	$\scriptstyle {y}$	$\scriptstyle {bpy}$	$\scriptstyle {\frac {ax^{n}}{bpy}}$
$\scriptstyle {\underbrace {\quad \quad \quad \quad } }$				$\scriptstyle {\underbrace {\quad \quad \quad \quad } }$

We may now desire to combine together these two results, in any manner we please; in which case it would only be necessary to have an additional card or cards, which should order the requisite operations to be performed with the numbers on the two result-columns, $\scriptstyle {\mathbf {V} _{4}}$ and $\scriptstyle {\mathbf {V} _{8}}$, and the result of these further operations to appear on a new column, $\scriptstyle {\mathbf {V} _{9}}$. Say that we wish to divide $\scriptstyle {ax^{n}}$ by $\scriptstyle {b.p.y}$. The numerical value of this division would then appear on the column $\scriptstyle {\mathbf {V} _{9}}$, beneath which we have inscribed $\scriptstyle {\frac {ax^{n}}{bpy}}$. The whole series of operations from the beginning would be as follows ($\scriptstyle {n}$ being $\scriptstyle {=7}$):—

$\scriptstyle {\left\{7(\times ),~2(\times ),~\div \right\}}$, or $\scriptstyle {\left\{9(\times ),~\div \right\}}$.

This example is introduced merely to show that we may, if we please, retain separately and permanently any intermediate results (like $\scriptstyle {ax^{n}}$, {{nowrap|$\scriptstyle {b.p.y}$.), which occur in the course of processes having an ulterior and more complicated result as their chief and final object $\scriptstyle {\left({\text{like }}{\frac {ax^{n}}{bpy}}\right)}$.

Any group of columns may be considered as representing a general function, until a special one has been implicitly impressed upon them through the introduction into the engine of the Operation and Variable-cards made out for a particular function. Thus, in the preceding example, $\scriptstyle {\mathbf {V} _{1}}$, $\scriptstyle {\mathbf {V} _{2}}$, $\scriptstyle {\mathbf {V} _{3}}$, $\scriptstyle {\mathbf {V} _{5}}$, $\scriptstyle {\mathbf {V} _{6}}$, $\scriptstyle {\mathbf {V} _{7}}$ represent the general function $\scriptstyle {\phi (a,n,b,p,x,y)}$ until the function $\scriptstyle {\frac {ax^{n}}{b.p.y}}$ has been determined on, and implicitly expressed by the placing of the right cards in the engine. The actual working of the mechanism, as regulated by these cards, then explicitly developes the value of the function. The inscription of a function under the brackets, and in the square under the result-column, in no way influences the processes or the results, and is merely a memorandum for the observer, to remind him of what is going on. It is the Operation and the Variable-cards only, which in reality determine the function. Indeed it should be distinctly kept in

3 a 3