Page:Scientific Memoirs, Vol. 3 (1843).djvu/734

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Diagram for the computation by the Engine of the Numbers of Bernoulli. See Note G. (page 722 *et seq.*)
Number of Operation.		Nature of Operation.		Variables acted upon.	Variables receiving results.	Indication of change in the value on any Variable.	Statement of Results.	Data.			Working Variables.										Result Variables.
								$\scriptstyle {^{1}\mathbf {V} _{1}}$	$\scriptstyle {^{1}\mathbf {V} _{2}}$	$\scriptstyle {^{1}\mathbf {V} _{3}}$	$\scriptstyle {^{0}\mathbf {V} _{4}}$	$\scriptstyle {^{0}\mathbf {V} _{5}}$	$\scriptstyle {^{0}\mathbf {V} _{6}}$	$\scriptstyle {^{0}\mathbf {V} _{7}}$	$\scriptstyle {^{0}\mathbf {V} _{8}}$	$\scriptstyle {^{0}\mathbf {V} _{9}}$	$\scriptstyle {^{0}\mathbf {V} _{10}}$	$\scriptstyle {^{0}\mathbf {V} _{11}}$	$\scriptstyle {^{0}\mathbf {V} _{12}}$	$\scriptstyle {^{0}\mathbf {V} _{13}}$	$\scriptstyle {^{1}\mathbf {V} _{21}}$	$\scriptstyle {^{1}\mathbf {V} _{22}}$	$\scriptstyle {^{1}\mathbf {V} _{23}}$	$\scriptstyle {^{1}\mathbf {V} _{24}}$
								$\scriptstyle {\bigcirc }$	$\scriptstyle {\bigcirc }$	$\scriptstyle {\bigcirc }$	$\scriptstyle {\bigcirc }$	$\scriptstyle {\bigcirc }$	$\scriptstyle {\bigcirc }$	$\scriptstyle {\bigcirc }$	$\scriptstyle {\bigcirc }$	$\scriptstyle {\bigcirc }$	$\scriptstyle {\bigcirc }$	$\scriptstyle {\bigcirc }$	$\scriptstyle {\bigcirc }$	$\scriptstyle {\bigcirc }$	$\scriptstyle {\bigcirc }$	$\scriptstyle {\bigcirc }$	$\scriptstyle {\bigcirc }$	$\scriptstyle {\bigcirc }$
								0	0	0	0	0	0	0	0	0	0	0	0	0	$\scriptstyle {B_{1}}$ in a decimal fraction.	$\scriptstyle {B_{3}}$ in a decimal fraction.	$\scriptstyle {B_{5}}$ in a decimal fraction.	0
								0	0	0	0	0	0	0	0	0	0	0	0	0				0
								1	2	4	0	0	0	0	0	0	0	0	0	0				0
								1	2	$\scriptstyle {n}$											$\scriptstyle {B_{1}}$	$\scriptstyle {B_{3}}$	$\scriptstyle {B_{5}}$	$\scriptstyle {B_{7}}$
1		$\scriptstyle {\times }$		$\scriptstyle {^{1}\mathbf {V} _{2}\times ^{1}\mathbf {V} _{3}}$	$\scriptstyle {^{1}\mathbf {V} _{4},^{1}\mathbf {V} _{5},^{1}\mathbf {V} _{6}}$	$\scriptstyle {\left\{{\begin{array}{c}\scriptstyle {^{1}\mathbf {V} _{2}=^{1}\mathbf {V} _{2}}\\\scriptstyle {^{1}\mathbf {V} _{3}=^{1}\mathbf {V} _{3}}\end{array}}\right\}}$	$\scriptstyle {=2n}$	…	2	$\scriptstyle {n}$	$\scriptstyle {2n}$	$\scriptstyle {2n}$	$\scriptstyle {2n}$
2		$\scriptstyle {-}$		$\scriptstyle {^{1}\mathbf {V} _{4}-^{1}\mathbf {V} _{1}}$	$\scriptstyle {^{1}\mathbf {V} _{4}}$	$\scriptstyle {\left\{{\begin{array}{c}\scriptstyle {^{1}\mathbf {V} _{4}=^{2}\mathbf {V} _{4}}\\\scriptstyle {^{1}\mathbf {V} _{1}=^{1}\mathbf {V} _{1}}\end{array}}\right\}}$	$\scriptstyle {=2n-1}$	1	…	…	$\scriptstyle {2n-1}$
3		$\scriptstyle {+}$		$\scriptstyle {^{1}\mathbf {V} _{5}+^{1}\mathbf {V} _{1}}$	$\scriptstyle {^{2}\mathbf {V} _{5}}$	$\scriptstyle {\left\{{\begin{array}{c}\scriptstyle {^{1}\mathbf {V} _{5}=^{2}\mathbf {V} _{5}}\\\scriptstyle {^{1}\mathbf {V} _{1}=^{1}\mathbf {V} _{1}}\end{array}}\right\}}$	$\scriptstyle {=2n+1}$	1	…	…	…	$\scriptstyle {2n+1}$
4		$\scriptstyle {\div }$		$\scriptstyle {^{2}\mathbf {V} _{4}\div ^{2}\mathbf {V} _{5}}$	$\scriptstyle {^{1}\mathbf {V} _{11}}$	$\scriptstyle {\left\{{\begin{array}{c}\scriptstyle {^{3}\mathbf {V} _{5}=^{0}\mathbf {V} _{5}}\\\scriptstyle {^{3}\mathbf {V} _{4}=^{0}\mathbf {V} _{4}}\end{array}}\right\}}$	$\scriptstyle {={\frac {2n-1}{2n+1}}}$	…	…	…	0	0	…	…	…	…	…	$\scriptstyle {={\frac {2n-1}{2n+1}}}$
5		$\scriptstyle {\div }$		$\scriptstyle {^{1}\mathbf {V} _{11}\div ^{1}\mathbf {V} _{2}}$	$\scriptstyle {^{2}\mathbf {V} _{11}}$	$\scriptstyle {\left\{{\begin{array}{c}\scriptstyle {^{1}\mathbf {V} _{11}=^{2}\mathbf {V} _{11}}\\\scriptstyle {^{1}\mathbf {V} _{2}=^{1}\mathbf {V} _{2}}\end{array}}\right\}}$	$\scriptstyle {={\frac {1}{2}}\cdot {\frac {2n-1}{2n+1}}}$	…	2	…	…	…	…	…	…	…	…	$\scriptstyle {={\frac {1}{2}}\cdot {\frac {2n-1}{2n+1}}}$
6		$\scriptstyle {-}$		$\scriptstyle {^{0}\mathbf {V} _{13}-^{2}\mathbf {V} _{11}}$	$\scriptstyle {^{1}\mathbf {V} _{13}}$	$\scriptstyle {\left\{{\begin{array}{c}\scriptstyle {^{2}\mathbf {V} _{11}=^{0}\mathbf {V} _{11}}\\\scriptstyle {^{0}\mathbf {V} _{13}=^{1}\mathbf {V} _{13}}\end{array}}\right\}}$	$\scriptstyle {=-{\frac {1}{2}}\cdot {\frac {2n-1}{2n+1}}}$	…	…	…	…	…	…	…	…	…	…	0		$\scriptstyle {=-{\frac {1}{2}}\cdot {\frac {2n-1}{2n+1}}=A_{0}}$
7		$\scriptstyle {-}$		$\scriptstyle {^{1}\mathbf {V} _{3}-^{1}\mathbf {V} _{1}}$	$\scriptstyle {^{1}\mathbf {V} _{10}}$	$\scriptstyle {\left\{{\begin{array}{c}\scriptstyle {^{1}\mathbf {V} _{2}=^{1}\mathbf {V} _{3}}\\\scriptstyle {^{1}\mathbf {V} _{3}=^{2}\mathbf {V} _{2}}\end{array}}\right\}}$	$\scriptstyle {=n-1(=3)}$	1	…	$\scriptstyle {n}$	…	…	…	…	…	…	$\scriptstyle {n}$
8		$\scriptstyle {+}$		$\scriptstyle {^{1}\mathbf {V} _{2}+^{0}\mathbf {V} _{7}}$	$\scriptstyle {^{1}\mathbf {V} _{7}}$	$\scriptstyle {\left\{{\begin{array}{c}\scriptstyle {^{1}\mathbf {V} _{2}=^{1}\mathbf {V} _{2}}\\\scriptstyle {^{0}\mathbf {V} _{7}=^{1}\mathbf {V} _{7}}\end{array}}\right\}}$	$\scriptstyle {=2+U=2}$	…	2	…	…	…	…	2
9		$\scriptstyle {\div }$		$\scriptstyle {^{1}\mathbf {V} _{6}\div ^{1}\mathbf {V} _{7}}$	$\scriptstyle {^{3}\mathbf {V} _{11}}$	$\scriptstyle {\left\{{\begin{array}{c}\scriptstyle {^{1}\mathbf {V} _{6}=^{1}\mathbf {V} _{6}}\\\scriptstyle {^{0}\mathbf {V} _{11}=^{3}\mathbf {V} _{11}}\end{array}}\right\}}$	$\scriptstyle {={\frac {2n}{2}}=A_{1}}$	…	…	…	…	…	$\scriptstyle {2n}$	2	…	…	…	$\scriptstyle {{\frac {2n}{2}}=A_{1}}$
10		$\scriptstyle {\times }$		$\scriptstyle {^{1}\mathbf {V} _{12}\times ^{3}\mathbf {V} _{11}}$	$\scriptstyle {^{1}\mathbf {V} _{12}}$	$\scriptstyle {\left\{{\begin{array}{c}\scriptstyle {^{1}\mathbf {V} _{21}=^{1}\mathbf {V} _{21}}\\\scriptstyle {^{3}\mathbf {V} _{11}=^{3}\mathbf {V} _{11}}\end{array}}\right\}}$	$\scriptstyle {=B_{1}\cdot {\frac {2n}{2}}=B_{1}A_{1}}$	…	…	…	…	…	…	…	…	…	…	$\scriptstyle {{\frac {2n}{2}}=A_{1}}$	$\scriptstyle {B_{1}\cdot {\frac {2n}{2}}=B_{1}A_{1}}$	……	$\scriptstyle {B_{1}}$
11		$\scriptstyle {+}$		$\scriptstyle {^{1}\mathbf {V} _{12}+^{1}\mathbf {V} _{13}}$	$\scriptstyle {^{2}\mathbf {V} _{12}}$	$\scriptstyle {\left\{{\begin{array}{c}\scriptstyle {^{1}\mathbf {V} _{12}=^{0}\mathbf {V} _{12}}\\\scriptstyle {^{1}\mathbf {V} _{13}=^{2}\mathbf {V} _{13}}\end{array}}\right\}}$	$\scriptstyle {=-{\frac {1}{2}}\cdot {\frac {2n-1}{2n+1}}+B_{1}\cdot {\frac {2n}{2}}}$	…	…	…	…	…	…	…	…	…	…	……	0	$\scriptstyle {\left\{-{\frac {1}{2}}\cdot {\frac {2n-1}{2n+1}}+B_{1}\cdot {\frac {2n}{2}}\right\}}$
12		$\scriptstyle {-}$		$\scriptstyle {^{1}\mathbf {V} _{10}-^{1}\mathbf {V} _{1}}$	$\scriptstyle {^{2}\mathbf {V} _{10}}$	$\scriptstyle {\left\{{\begin{array}{c}\scriptstyle {^{1}\mathbf {V} _{10}=^{2}\mathbf {V} _{10}}\\\scriptstyle {^{1}\mathbf {V} _{1}=^{1}\mathbf {V} _{1}}\end{array}}\right\}}$	$\scriptstyle {=n-2=2}$	1	…	…	…	…	…	…	…	…	$\scriptstyle {n-2}$
13	$\scriptstyle {\left\{{\begin{matrix}\ \\\\\ \\\ \\\ \\\ \\\ \\\ \\\ \\\ \\\ \\\ \\\ \\\ \\\ \\\ \\\ \\\ \\\ \\\ \\\ \\\ \ \end{matrix}}\right.}$	$\scriptstyle {\left\{{\begin{matrix}\ \\\\\ \\\ \\\ \\\ \\\ \\\ \ \end{matrix}}\right.}$	$\scriptstyle {-}$	$\scriptstyle {^{1}\mathbf {V} _{6}-^{1}\mathbf {V} _{1}}$	$\scriptstyle {^{2}\mathbf {V} _{6}}$	$\scriptstyle {\left\{{\begin{array}{c}\scriptstyle {^{1}\mathbf {V} _{6}=^{2}\mathbf {V} _{6}}\\\scriptstyle {^{1}\mathbf {V} _{1}=^{1}\mathbf {V} _{1}}\end{array}}\right\}}$	$\scriptstyle {=2n-1}$	1	…	…	…	…	$\scriptstyle {2n-1}$
14			$\scriptstyle {+}$	$\scriptstyle {^{1}\mathbf {V} _{1}-^{1}\mathbf {V} _{7}}$	$\scriptstyle {^{2}\mathbf {V} _{7}}$	$\scriptstyle {\left\{{\begin{array}{c}\scriptstyle {^{1}\mathbf {V} _{1}=^{1}\mathbf {V} _{1}}\\\scriptstyle {^{1}\mathbf {V} _{7}=^{2}\mathbf {V} _{7}}\end{array}}\right\}}$	$\scriptstyle {=2+1=3}$	1	…	…	…	…	…	3
15			$\scriptstyle {\div }$	$\scriptstyle {^{2}\mathbf {V} _{6}\div ^{2}\mathbf {V} _{7}}$	$\scriptstyle {^{1}\mathbf {V} _{8}}$	$\scriptstyle {\left\{{\begin{array}{c}\scriptstyle {^{2}\mathbf {V} _{6}=^{2}\mathbf {V} _{6}}\\\scriptstyle {^{2}\mathbf {V} _{7}=^{2}\mathbf {V} _{7}}\end{array}}\right\}}$	$\scriptstyle {={\frac {2n-1}{3}}}$	…	…	…	…	…	$\scriptstyle {2n-1}$	3	$\scriptstyle {\frac {2n-1}{3}}$
16			$\scriptstyle {\times }$	$\scriptstyle {^{1}\mathbf {V} _{8}\times ^{3}\mathbf {V} _{11}}$	$\scriptstyle {^{4}\mathbf {V} _{11}}$	$\scriptstyle {\left\{{\begin{array}{c}\scriptstyle {^{1}\mathbf {V} _{8}=^{0}\mathbf {V} _{8}}\\\scriptstyle {^{3}\mathbf {V} _{11}=^{4}\mathbf {V} _{11}}\end{array}}\right\}}$	$\scriptstyle {={\frac {2n}{2}}\cdot {\frac {2n-1}{3}}}$	…	…	…	…	…	…	3		0	…	…	$\scriptstyle {{\frac {2n}{2}}\cdot {\frac {2n-1}{3}}}$
17		$\scriptstyle {\left\{{\begin{matrix}\ \\\\\ \\\ \\\ \\\ \\\ \\\ \ \end{matrix}}\right.}$	$\scriptstyle {-}$	$\scriptstyle {^{2}\mathbf {V} _{6}-^{1}\mathbf {V} _{1}}$	$\scriptstyle {^{3}\mathbf {V} _{6}}$	$\scriptstyle {\left\{{\begin{array}{c}\scriptstyle {^{2}\mathbf {V} _{6}=^{3}\mathbf {V} _{6}}\\\scriptstyle {^{1}\mathbf {V} _{1}=^{1}\mathbf {V} _{1}}\end{array}}\right\}}$	$\scriptstyle {=2n-2}$	1	…	…	…	…	$\scriptstyle {2n-2}$
18			$\scriptstyle {+}$	$\scriptstyle {^{1}\mathbf {V} _{1}+^{2}\mathbf {V} _{7}}$	$\scriptstyle {^{3}\mathbf {V} _{7}}$	$\scriptstyle {\left\{{\begin{array}{c}\scriptstyle {^{2}\mathbf {V} _{7}=^{3}\mathbf {V} _{7}}\\\scriptstyle {^{1}\mathbf {V} _{1}=^{1}\mathbf {V} _{1}}\end{array}}\right\}}$	$\scriptstyle {=3+1=4}$	1	…	…	…	…	…	4
19			$\scriptstyle {\div }$	$\scriptstyle {^{3}\mathbf {V} _{6}\div ^{3}\mathbf {V} _{7}}$	$\scriptstyle {^{1}\mathbf {V} _{9}}$	$\scriptstyle {\left\{{\begin{array}{c}\scriptstyle {^{3}\mathbf {V} _{6}=^{3}\mathbf {V} _{6}}\\\scriptstyle {^{3}\mathbf {V} _{7}=^{3}\mathbf {V} _{7}}\end{array}}\right\}}$	$\scriptstyle {={\frac {2n-2}{4}}}$	…	…	…	…	…	$\scriptstyle {2n-2}$	4	…	$\scriptstyle {\frac {2n-2}{4}}$	…	$\scriptstyle {\left\{{\begin{array}{c}\scriptstyle {{\frac {2n}{2}}\cdot {\frac {2n-1}{3}}\cdot {\frac {2n-2}{3}}}\\\scriptstyle {=A_{2}}\end{array}}\right\}}$
20			$\scriptstyle {\times }$	$\scriptstyle {^{1}\mathbf {V} _{9}\times ^{4}\mathbf {V} _{11}}$	$\scriptstyle {^{5}\mathbf {V} _{11}}$	$\scriptstyle {\left\{{\begin{array}{c}\scriptstyle {^{1}\mathbf {V} _{9}=^{0}\mathbf {V} _{9}}\\\scriptstyle {^{4}\mathbf {V} _{11}=^{5}\mathbf {V} _{11}}\end{array}}\right\}}$	$\scriptstyle {={\frac {2n}{2}}\cdot {\frac {2n-1}{3}}\cdot {\frac {2n-2}{4}}=A_{3}}$	…	…	…	…	…	…	…	…	0	…
21			$\scriptstyle {\times }$	$\scriptstyle {^{1}\mathbf {V} _{22}\times ^{5}\mathbf {V} _{11}}$	$\scriptstyle {^{6}\mathbf {V} _{11}}$	$\scriptstyle {\left\{{\begin{array}{c}\scriptstyle {^{1}\mathbf {V} _{22}=^{1}\mathbf {V} _{22}}\\\scriptstyle {^{0}\mathbf {V} _{12}=^{0}\mathbf {V} _{12}}\end{array}}\right\}}$	$\scriptstyle {=B_{3}\cdot {\frac {2n}{2}}\cdot {\frac {2n-1}{3}}\cdot {\frac {2n-2}{3}}=B_{3}A_{3}}$	…	…	…	…	…	…	…	…	…	…	0	$\scriptstyle {B_{3}A_{3}}$	…	…	$\scriptstyle {B_{3}}$
22			$\scriptstyle {+}$	$\scriptstyle {^{3}\mathbf {V} _{12}+^{2}\mathbf {V} _{13}}$	$\scriptstyle {^{3}\mathbf {V} _{13}}$	$\scriptstyle {\left\{{\begin{array}{c}\scriptstyle {^{3}\mathbf {V} _{12}=^{0}\mathbf {V} _{12}}\\\scriptstyle {^{2}\mathbf {V} _{13}=^{3}\mathbf {V} _{13}}\end{array}}\right\}}$	$\scriptstyle {=A_{0}+B_{1}A_{2}+B_{3}A_{3}}$	…	…	…	…	…	…	…	…	…	…	……	0	$\scriptstyle {\left\{A_{3}+B_{2}A_{2}+B_{3}A_{3}\right\}}$
23			$\scriptstyle {-}$	$\scriptstyle {^{2}\mathbf {V} _{10}+^{1}\mathbf {V} _{1}}$	$\scriptstyle {^{3}\mathbf {V} _{10}}$	$\scriptstyle {\left\{{\begin{array}{c}\scriptstyle {^{3}\mathbf {V} _{10}=^{3}\mathbf {V} _{10}}\\\scriptstyle {^{1}\mathbf {V} _{1}=^{1}\mathbf {V} _{1}}\end{array}}\right\}}$	$\scriptstyle {=n-3=1}$	1	…	…	…	…	…	…	…	…	$\scriptstyle {n-3}$
Here follows a repetition of Operations thirteen to twenty-three.
24			$\scriptstyle {+}$	$\scriptstyle {^{4}\mathbf {V} _{12}+^{0}\mathbf {V} _{24}}$	$\scriptstyle {^{1}\mathbf {V} _{24}}$	$\scriptstyle {\left\{{\begin{array}{c}\scriptstyle {^{4}\mathbf {V} _{12}=^{0}\mathbf {V} _{12}}\\\scriptstyle {^{0}\mathbf {V} _{24}=^{1}\mathbf {V} _{24}}\end{array}}\right\}}$	$\scriptstyle {=B_{7}}$	…	…	…	…	…	…	…	…	…	…	……	……	……	…	…	…	$\scriptstyle {B_{7}}$
25			$\scriptstyle {+}$	$\scriptstyle {^{1}\mathbf {V} _{1}+^{1}\mathbf {V} _{3}}$	$\scriptstyle {^{1}\mathbf {V} _{3}}$	$\scriptstyle {\left\{{\begin{array}{c}\scriptstyle {\left.{\begin{array}{c}\scriptstyle {^{1}\mathbf {V} _{1}=^{1}\mathbf {V} _{1}}\\\scriptstyle {^{1}\mathbf {V} _{3}=^{1}\mathbf {V} _{3}}\end{array}}\right\}}\\{\begin{array}{c}\scriptstyle {^{4}\mathbf {V} _{6}=^{0}\mathbf {V} _{6}}\\\scriptstyle {^{1}\mathbf {V} _{7}=^{0}\mathbf {V} _{7}}\end{array}}\end{array}}\right.}$	$\scriptstyle {=n+1=4+1=5}$ by a Variable-card. by a Variable-card.