Page:Scientific Memoirs, Vol. 3 (1843).djvu/724

Variables for Results.
$\scriptstyle {\overbrace {\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } }$
$\scriptstyle {\mathbf {V} _{20}}$	$\scriptstyle {\mathbf {V} _{21}}$	$\scriptstyle {\mathbf {V} _{22}}$	$\scriptstyle {\mathbf {V} _{23}~\cdot ~\cdot }$		$\scriptstyle {\cdot ~\cdot ~\mathbf {V} _{31}}$	$\scriptstyle {\mathbf {V} _{32}}$	$\scriptstyle {\mathbf {V} _{33}}$	$\scriptstyle {\mathbf {V} _{34}}$
$\scriptstyle {BA}$	$\scriptstyle {BA_{1}}$	$\scriptstyle {BA_{2}}$	$\scriptstyle {BA_{3}}$		$\scriptstyle {B_{1}A}$	$\scriptstyle {B_{1}A_{1}}$	$\scriptstyle {B_{1}A_{2}}$	$\scriptstyle {B_{1}A_{3}}$
	$\scriptstyle {\cos \theta }$	$\scriptstyle {\cos 2\theta }$	$\scriptstyle {\cos 3\theta }$		$\scriptstyle {\cos \theta }$	$\scriptstyle {(\cos \theta .\cos \theta )}$	$\scriptstyle {\vdots }$	$\scriptstyle {(\cos 3\theta .\cos \theta )}$
							$\scriptstyle {(\cos 2\theta .\cos \theta )}$

The variable belonging to each coefficient is written below it, as we have done in the diagram, by way of memorandum. The only further reduction which is at first apparently possible in the preceding result, would be the addition of $\scriptstyle {\mathbf {V} _{21}}$ to $\scriptstyle {\mathbf {V} _{31}}$ (in which case $\scriptstyle {B_{1}A}$ should be effaced from $\scriptstyle {\mathbf {V} _{31}}$). The whole operations from the beginning would then be—

First Series of
Operations.

$\scriptstyle {^{1}\mathbf {V} _{10}\times ^{1}\mathbf {V} _{0}=^{1}\mathbf {V} _{20}}$
$\scriptstyle {^{1}\mathbf {V} _{10}\times ^{1}\mathbf {V} _{1}=^{1}\mathbf {V} _{21}}$
$\scriptstyle {^{1}\mathbf {V} _{10}\times ^{1}\mathbf {V} _{2}=^{1}\mathbf {V} _{22}}$
$\scriptstyle {^{1}\mathbf {V} _{10}\times ^{1}\mathbf {V} _{3}=^{1}\mathbf {V} _{23}}$

Second Series of
Operations.

$\scriptstyle {^{1}\mathbf {V} _{11}\times ^{1}\mathbf {V} _{0}=^{1}\mathbf {V} _{31}}$
$\scriptstyle {^{1}\mathbf {V} _{11}\times ^{1}\mathbf {V} _{1}=^{1}\mathbf {V} _{32}}$
$\scriptstyle {^{1}\mathbf {V} _{11}\times ^{1}\mathbf {V} _{2}=^{1}\mathbf {V} _{33}}$
$\scriptstyle {^{1}\mathbf {V} _{11}\times ^{1}\mathbf {V} _{3}=^{1}\mathbf {V} _{34}}$

Third Series, which contains
only one (final) operation.

$\scriptstyle {^{1}\mathbf {V} _{21}+^{1}\mathbf {V} _{31}=^{2}\mathbf {V} _{21}}$, and
$\scriptstyle {\mathbf {V} _{31}}$ becomes $\scriptstyle {=0}$.

We do not enter into the same detail of every step of the processes as in the examples of Notes D. and G., thinking it unnecessary and tedious to do so. The reader will remember the meaning and use of the upper and lower indices, &c., as before explained.

To proceed: we know that (3.)

$\scriptstyle {\cos n\theta .\cos \theta ={\frac {1}{2}}\cos {\overline {n+1}}\theta +{\frac {1}{2}}{\overline {n-1}}.\theta ~.~.~.}$

Consequently, a slight examination of the second line of (2.) will show that by making the proper substitutions, (2.) will become

$\scriptstyle {BA}$	$\scriptstyle {+BA_{1}\;\;.\cos \theta }$	$\scriptstyle {+BA_{2}\;\;.\cos 2\theta }$	$\scriptstyle {+BA_{3}.\cos 3\theta }$
	$\scriptstyle {+B_{1}A.\cos \theta }$
$\scriptstyle {+{\frac {1}{2}}B_{1}A_{1}}$		$\scriptstyle {+{\frac {1}{2}}B_{1}A_{1}.\cos 2\theta }$
	$\scriptstyle {+{\frac {1}{2}}B_{1}A_{2}.\cos \theta }$		$\scriptstyle {+{\frac {1}{2}}B_{1}A_{2}.\cos 3\theta }$
		$\scriptstyle {+{\frac {1}{2}}B_{1}A_{3}.\cos 2\theta }$		$\scriptstyle {+{\frac {1}{2}}B_{1}A_{3}.\cos 4\theta }$
$\scriptstyle {C_{0}}$	$\scriptstyle {C_{1}}$	$\scriptstyle {C_{2}}$	$\scriptstyle {C_{3}}$	$\scriptstyle {C_{4}}$
These coefficients should respectively appear on
$\scriptstyle {\mathbf {V} _{20}}$	$\scriptstyle {\mathbf {V} _{21}}$	$\scriptstyle {\mathbf {V} _{22}}$	$\scriptstyle {\mathbf {V} _{23}}$	$\scriptstyle {\mathbf {V} _{24}}$

We shall perceive, if we inspect the particular arrangement of the results in (2.) on the Result-columns as represented in the diagram, that, in order to effect this transformation, each successive coefficient upon $\scriptstyle {\mathbf {V} _{32}}$, $\scriptstyle {\mathbf {V} _{33}}$, &c. (beginning with $\scriptstyle {\mathbf {V} _{32}}$), must through means of proper cards be divided by two^[1]; and that one of the halves thus

1. This division would be managed by ordering the number two to appear on any separate new column which should be conveniently situated for the purpose, and then directing this column (which is in the strictest sense a Working-Variable) to divide itself successively with $\scriptstyle {\mathbf {V} _{32}}$, $\scriptstyle {\mathbf {V} _{33}}$, &c.