Page:Scientific Memoirs, Vol. 3 (1843).djvu/742

on a new column. But as these variations follow the same law at each repetition, (Operation 21 always requiring its factor from a column one in advance of that which it used the previous time, and Operation 24 always putting its result on the column one in advance of that which received the previous result), they are easily provided for in arranging the recurring group (or cycle) of Variable-cards.

We may here remark that the average estimate of three Variable-cards coming into use to each operation, is not to be taken as an absolutely and literally correct amount for all cases and circumstances. Many special circumstances, either in the nature of a problem, or in the arrangements of the engine under certain contingencies, influence and modify this average to a greater or less extent. But it is a very safe and correct general rule to go upon. In the preceding case it will give us seventy-five Variable-cards as the total number which will be necessary for computing any $\scriptstyle {B}$ after $\scriptstyle {B_{3}}$. This is very nearly the precise amount really used, but we cannot here enter into the minutiæ of the few particular circumstances which occur in this example (as indeed at some one stage or other of probably most computations) to modify slightly this number.

It will be obvious that the very same seventy-five Variable-cards may be repeated for the computation of every succeeding Number, just on the same principle as admits of the repetition of the thirty-three Variable-cards of Operations (13…23) in the computation of any one Number. Thus there will be a cycle of a cycle of Variable-cards.

If we now apply the notation for cycles, as explained in Note E, we may express the operations for computing the Numbers of Bernoulli in the following manner:—

(1…7), (24, 25)	gives $\scriptstyle {B_{1}}$	= 1st number	; ($\scriptstyle {n}$ being	=
(1…7), (8…12), (24, 25)	$\scriptstyle {B_{3}}$	= 2nd	; ($\scriptstyle {n}$	=
(1…7), (8…12), (13…23), (24, 25)	$\scriptstyle {B_{5}}$	= 3rd	; ($\scriptstyle {n}$	=
(1…7), (8…12), 2(13…23), (24, 25)	$\scriptstyle {B_{7}}$	= 4th	; ($\scriptstyle {n}$	=
(1…7),(8…12),$\scriptstyle {\Sigma (+1)^{n-2}}$(13…23),(24, 25)	$\scriptstyle {B_{2n-1}}$	= $\scriptstyle {n}$th	; ($\scriptstyle {n}$	=

Again,

$\scriptstyle {(1\ldots 7),(24,25),{\overset {\sum (+1)^{n}}{{\text{limits }}1{\text{ to }}n}}\left\{(1\ldots 7),(8\ldots 12),{\overset {\sum (n+2)(13\ldots 23),(24,25)}{{\text{limits }}0{\text{ to }}(n+2)}}\right\}}$

represents the total operations for computing every number in succession, from $\scriptstyle {B_{1}}$ to $\scriptstyle {B_{2n-1}}$ inclusive.

In this formula we see a varying cycle of the first order, and an ordinary cycle of the second order. The latter cycle in this case includes in it the varying cycle.

On inspecting the ten Working-Variables of the diagram, it will be perceived, that although the value on any one of them (excepting $\scriptstyle {\mathbf {V} _{4}}$ and $\scriptstyle {\mathbf {V} _{5}}$) goes through a series of changes, the office which each performs is in this calculation fixed and invariable. Thus $\scriptstyle {\mathbf {V} _{6}}$ always prepares the numerators of the factors of any $\scriptstyle {A}$; $\scriptstyle {\mathbf {V} _{7}}$ the denominators. $\scriptstyle {\mathbf {V} _{8}}$ always receives the $\scriptstyle {(2n-3)}$th factor of $\scriptstyle {A_{2n-1}}$, and $\scriptstyle {\mathbf {V} _{9}}$ the $\scriptstyle {(2n-1)}$th. $\scriptstyle {\mathbf {V} _{10}}$ always decides which of two courses the succeeding processes are to follow, by feeling for the value of $\scriptstyle {n}$ through