Page:Scientific Memoirs, Vol. 3 (1843).djvu/738

Multiplying every term by ($\scriptstyle {2.3\ldots 2n}$), we have (8.)

$\scriptstyle {\left.{\begin{aligned}\scriptstyle {0=-}&\scriptstyle {{\frac {1}{2}}\cdot {\frac {2n-1}{2n+1}}+B_{1}\left({\frac {2n}{2}}\right)+B_{3}\left({\frac {2n.2n-1.2n-2}{2.3.4}}\right)+}\\&\scriptstyle {+B_{5}\left({\frac {2n.2n-1\ldots \ldots \ldots 2n-4}{2.3.4.5.6}}\right)+\ldots +B_{2n-1}}\end{aligned}}\right\}}$

which it may be convenient to write under the general form:— (9.)

$\scriptstyle {0=A_{0}+A_{1}B_{1}+A_{3}B_{3}+A_{5}B_{5}+\ldots +B_{2n-1}\ldots \ldots \ldots }$

$\scriptstyle {A_{1}}$, $\scriptstyle {A_{3}}$, &c. being those functions of $\scriptstyle {n}$ which respectively belong to $\scriptstyle {B_{1}}$, $\scriptstyle {B_{3}}$, &c.

We might have derived a form nearly similar to (8.), from $\scriptstyle {D_{2n-1}}$ the coefficient of any odd power of $\scriptstyle {x}$ in (6.); but the general form is a little different for the coefficients of the odd powers, and not quite so convenient.

On examining (7.) and (8.), we perceive that, when these formulæ are isolated from (6.) whence they are derived, and considered in themselves separately and independently, $\scriptstyle {n}$ may be any whole number whatever; although when (7.) occurs as one of the $\scriptstyle {D}$'s in (6.), it is obvious that $\scriptstyle {n}$ is then not arbitrary, but is always a certain function of the distance of that $\scriptstyle {D}$ from the beginning. If that distance be $\scriptstyle {=d}$, then

$\scriptstyle {2n+1=d}$, and $\scriptstyle {n={\frac {d-1}{2}}}$ (for any even power of $\scriptstyle {x}$.

$\scriptstyle {2n=d}$, and $\scriptstyle {n={\frac {d}{2}}}$ (for any odd power of $\scriptstyle {x}$.)

It is with the independent formula (8.) that we have to do. Therefore it must be remembered that the conditions for the value of $\scriptstyle {n}$ are now modified, and that $\scriptstyle {n}$ is a perfectly arbitrary whole number. This circumstance, combined with the fact (which we may easily perceive) that whatever $\scriptstyle {n}$ is, every term of (8.) after the $\scriptstyle {(n+1)}$th is $\scriptstyle {=0}$, and that the ($\scriptstyle {n+1}$)th term itself is always $\scriptstyle {=B_{2n-1}\cdot {\frac {1}{1}}=B_{2n-1}}$, enables us to find the value (either numerical or algebraical) of any $\scriptstyle {n}$th Number of Bernoulli $\scriptstyle {B_{2n-1}}$, in terms of all the preceding ones, if we but know the values of $\scriptstyle {B_{1}}$, $\scriptstyle {B_{3}\ldots B_{2n-3}}$. We append to this Note a Diagram and Table, containing the details of the computation for $\scriptstyle {B_{7}}$, ($\scriptstyle {B_{1}}$, $\scriptstyle {B_{3}}$, $\scriptstyle {B_{5}}$ being supposed given).

On attentively considering (8.), we shall likewise perceive that we may derive from it the numerical value of every Number of Bernoulli in succession, from the very beginning, ad infinitum, by the following series of computations:—

1st Series.—Let $\scriptstyle {n=1}$, and calculate (8.) for this value of $\scriptstyle {n}$. The result is $\scriptstyle {B_{1}}$.

2nd Series.—Let $\scriptstyle {n=2}$. Calculate (8.) for this value of $\scriptstyle {n}$, substituting the value of $\scriptstyle {B_{1}}$ just obtained. The result is $\scriptstyle {B_{3}}$.

3rd Series.—Let $\scriptstyle {n=3}$. Calculate (8.) for this value of $\scriptstyle {n}$, substituting the values of $\scriptstyle {B_{1}}$, $\scriptstyle {B_{3}}$ before obtained. The result is $\scriptstyle {B_{5}}$. And so on, to any extent.

The diagram^[1] represents the columns of the engine when just prepared

1. See the diagram at the end of these Notes.