Page:Scientific Memoirs, Vol. 3 (1843).djvu/694

following^[1] very simple example, in which we are to multiply $\scriptstyle {(a+bx^{1})}$ by $\scriptstyle {(A+B\cos ^{1}x)}$. We shall begin by writing $\scriptstyle {x^{0}}$, $\scriptstyle {x^{1}}$, $\scriptstyle {\cos ^{0}x}$, $\scriptstyle {\cos ^{1}x}$, above the columns $\scriptstyle {\mathbf {V} _{0}}$, $\scriptstyle {\mathbf {V} _{1}}$, $\scriptstyle {\mathbf {V} _{2}}$, $\scriptstyle {\mathbf {V} _{3}}$; then, since from the form of the two functions to be combined, the terms which are to compose the products will be of the following nature, $\scriptstyle {x^{0}.\cos ^{0}x}$, $\scriptstyle {x^{0}.\cos ^{1}x}$, $\scriptstyle {x^{1}.\cos ^{0}x}$, $\scriptstyle {x^{1}.\cos ^{1}x}$; these will be inscribed above the columns $\scriptstyle {\mathbf {V} _{4}}$, $\scriptstyle {\mathbf {V} _{5}}$, $\scriptstyle {\mathbf {V} _{6}}$, $\scriptstyle {\mathbf {V} _{7}}$. The coefficients of $\scriptstyle {x^{0}}$, $\scriptstyle {x^{1}}$, $\scriptstyle {\cos ^{0}x}$, $\scriptstyle {\cos ^{1}x}$ being given, they will, by means of the mill, be passed to the columns $\scriptstyle {\mathbf {V} _{0}}$, $\scriptstyle {\mathbf {V} _{1}}$, $\scriptstyle {\mathbf {V} _{2}}$ and $\scriptstyle {\mathbf {V} _{3}}$. Such are the primitive data of the problem. It is now the business of the machine to work out its solution, that is to find the coefficients which are to be inscribed on $\scriptstyle {\mathbf {V} _{4}}$, $\scriptstyle {\mathbf {V} _{5}}$, $\scriptstyle {\mathbf {V} _{6}}$, $\scriptstyle {\mathbf {V} _{7}}$. To attain this object, the law of formation of these same coefficients being known, the machine will act through the intervention of the cards, in the manner indicated by the following table:—

[2] Columns above which are written the functions of the variable.	Coeffi­cients.		Cards of the operations.		Cards of the variables.
	Given.	To be formed.	Number of the operations.	Nature of the operation.	Columns on which operations are to be performed.	Columns on which are to be inscribed the results of the operations.	Indication of change of value on any column submitted to an operation.	Results of the operation.
$\scriptstyle {x^{0}\ldots \ldots \ldots ^{1}\mathbf {V} _{0}}$	$\scriptstyle {a}$	"	"	"	"	"	"	""
$\scriptstyle {x^{1}\ldots \ldots \ldots ^{1}\mathbf {V} _{1}}$	$\scriptstyle {b}$	"	"	"	"	"	"	""
$\scriptstyle {\mathrm {Cos} ^{0}x\ldots ^{1}\mathbf {V} _{2}}$	A	"	"	"	"	"	"	""
$\scriptstyle {\mathrm {Cos} ^{1}x\ldots ^{1}\mathbf {V} _{3}}$	B	"	"	"	"	"	"	""
$\scriptstyle {x^{0}\cos ^{0}x..^{0}\mathbf {V} _{4}}$	…	$\scriptstyle {a}$A	1	$\scriptstyle {\times }$	$\scriptstyle {^{1}\mathbf {V} _{0}\times ^{1}\mathbf {V} _{2}=}$	$\scriptstyle {^{1}\mathbf {V} _{4}\ldots \ldots }$	$\scriptstyle {\left\{{\begin{aligned}&\scriptstyle {^{1}\mathbf {V} _{0}=^{1}\mathbf {V} _{0}}\\&\scriptstyle {^{1}\mathbf {V} _{2}=^{1}\mathbf {V} _{2}}\end{aligned}}\right\}}$	$\scriptstyle {^{1}\mathbf {V} _{4}=a}$A coefficients of $\scriptstyle {x}$
$\scriptstyle {x^{0}\cos ^{1}x..^{0}\mathbf {V} _{5}}$	…	$\scriptstyle {a}$B	2	$\scriptstyle {\times }$	$\scriptstyle {^{1}\mathbf {V} _{0}\times ^{1}\mathbf {V} _{3}=}$	$\scriptstyle {^{1}\mathbf {V} _{5}\ldots \ldots }$	$\scriptstyle {\left\{{\begin{aligned}&\scriptstyle {^{1}\mathbf {V} _{0}=^{0}\mathbf {V} _{0}}\\&\scriptstyle {^{1}\mathbf {V} _{3}=^{1}\mathbf {V} _{3}}\end{aligned}}\right\}}$	$\scriptstyle {^{1}\mathbf {V} _{5}=a}$B … … $\scriptstyle {x}$
$\scriptstyle {x^{1}\cos ^{0}x..^{0}\mathbf {V} _{6}}$	…	$\scriptstyle {b}$A	3	$\scriptstyle {\times }$	$\scriptstyle {^{1}\mathbf {V} _{1}\times ^{1}\mathbf {V} _{2}=}$	$\scriptstyle {^{1}\mathbf {V} _{6}\ldots \ldots }$	$\scriptstyle {\left\{{\begin{aligned}&\scriptstyle {^{1}\mathbf {V} _{1}=^{1}\mathbf {V} _{1}}\\&\scriptstyle {^{1}\mathbf {V} _{2}=^{0}\mathbf {V} _{2}}\end{aligned}}\right\}}$	$\scriptstyle {^{1}\mathbf {V} _{6}=b}$A … … $\scriptstyle {x}$
$\scriptstyle {x^{1}\cos ^{1}x..^{0}\mathbf {V} _{7}}$	…	$\scriptstyle {b}$B	4	$\scriptstyle {\times }$	$\scriptstyle {^{1}\mathbf {V} _{1}\times ^{1}\mathbf {V} _{3}=}$	$\scriptstyle {^{1}\mathbf {V} _{7}\ldots \ldots }$	$\scriptstyle {\left\{{\begin{aligned}&\scriptstyle {^{1}\mathbf {V} _{1}=^{0}\mathbf {V} _{1}}\\&\scriptstyle {^{1}\mathbf {V} _{3}=^{0}\mathbf {V} _{3}}\end{aligned}}\right\}}$	$\scriptstyle {^{1}\mathbf {V} _{7}=b}$B … … $\scriptstyle {x}$

It will now be perceived that a general application may be made of the principle developed in the preceding example, to every species of process which it may be proposed to effect on series submitted to calculation. It is sufficient that the law of formation of the coefficients be known, and that this law be inscribed on the cards of the machine, which will then of itself execute all

1. See Note E.
2. For an explanation of the upper left-hand indices attached to the $\scriptstyle {\mathbf {V} }$'s in this and in the preceding Table, we must refer the reader to Note D, amongst those appended to the memoir.—Note by Translator.