Page:Scientific Memoirs, Vol. 3 (1843).djvu/741

values undergo alterations during a performance of the operations (13.…23), and consequently the columns present a new set of values for the next performance of (13….23) to work on.

At the termination of the repetition of operations (13…23) in computing $\scriptstyle {B_{7}}$, the alterations in the values on the Variables are, that

${\begin{aligned}\scriptstyle {\mathbf {V} _{6~}}&\scriptstyle {=2n-4{\text{ instead of }}2n-2.}\\\scriptstyle {\mathbf {V} _{7~}}&\scriptstyle {=6\ldots \ldots \ldots \ldots 4.}\\\scriptstyle {\mathbf {V} _{10}}&\scriptstyle {=0\ldots \ldots \ldots \ldots 1.}\\\scriptstyle {\mathbf {V} _{13}}&\scriptstyle {=A_{0}+A_{1}B_{1}+A_{3}B_{3}+A_{5}B_{5}{\text{ instead of }}A_{0}+A_{1}B_{1}+A_{3}B_{3}}\end{aligned}}$

In this state the only remaining processes are first: to transfer the value which is on $\scriptstyle {\mathbf {V} _{13}}$, to $\scriptstyle {\mathbf {V} _{24}}$; and secondly to reduce $\scriptstyle {\mathbf {V} _{6}}$, $\scriptstyle {\mathbf {V} _{7}}$, $\scriptstyle {\mathbf {V} _{13}}$ to zero, and to add^[1] one to $\scriptstyle {\mathbf {V} _{3}}$, in order that the engine may be ready to commence computing $\scriptstyle {B_{9}}$. Operations 24 and 25 accomplish these purposes. It may be thought anomalous that Operation 25 is represented as leaving the upper index of $\scriptstyle {\mathbf {V} _{3}}$ still = unity. But it must be remembered that these indices always begin anew for a separate calculation, and that Operation 25 places upon $\scriptstyle {\mathbf {V} _{3}}$ the first value for the new calculation.

It should be remarked, that when the group (13…23) is repeated, changes occur in some of the upper indices during the course of the repetition: for example, $\scriptstyle {^{3}\mathbf {V} _{6}}$ would become $\scriptstyle {^{4}\mathbf {V} _{6}}$ and $\scriptstyle {^{5}\mathbf {V} _{6}}$.

We thus see that when $\scriptstyle {n=1}$, nine Operation-cards are used; that when $\scriptstyle {n=2}$, fourteen Operation-cards are used; and that when $\scriptstyle {n>2}$, twenty-five Operation-cards are used; but that no more are needed, however great $\scriptstyle {n}$ may be; and not only this, but that these same twenty-five cards suffice for the successive computation of all the Numbers from $\scriptstyle {B_{1}}$ to $\scriptstyle {B_{2n-1}}$ inclusive. With respect to the number of Variable-cards, it will be remembered, from the explanations in previous Notes, that an average of three such cards to each operation (not however to each Operation-card) is the estimate. According to this the computation of $\scriptstyle {B_{1}}$ will require twenty-seven Variable-cards; $\scriptstyle {B_{3}}$ forty-two such cards; $\scriptstyle {B_{5}}$ seventy-five; and for every succeeding $\scriptstyle {B}$ after $\scriptstyle {B_{5}}$, there would be thirty-three additional Variable-cards (since each repetition of the group (13…23) adds eleven to the number of operations required for computing the previous $\scriptstyle {B}$). But we must now explain, that whenever there is a cycle of operations, and if these merely require to be supplied with numbers from the same pairs of columns and likewise each operation to place its result on the same column for every repetition of the whole group, the process then admits of a cycle of Variable-cards for effecting its purposes. There is obviously much more symmetry and simplicity in the arrangements, when cases do admit of repeating the Variable as well as the Operation-cards. Our present example is of this nature. The only exception to a perfect identity in all the processes and columns used, for every repetition of Operations (13…23) is, that Operation 21 always requires one of its factors from a new column, and Operation 24 always puts its result

1. It is interesting to observe, that so complicated a case as this calculation of the Bernoullian Numbers, nevertheless, presents a remarkable simplicity in one respect; viz., that during the processes for the computation of millions of these Numbers, no other arbitrary modification would be requisite in the arrangements, excepting the above simple and uniform provision for causing one of the data periodically to receive the finite increment unity.