Page:Scientific Memoirs, Vol. 3 (1843).djvu/737

which is in fact a particular case of the development of

$\scriptstyle {\frac {a+bx+cx^{2}+\&{\text{c.}}}{a'+b'x+c'x^{2}+\&{\text{c.}}}}$

mentioned in Note E. Or again, we might compute them from the well-known form (2.)

$\scriptstyle {B_{2n-1}=2.{\frac {1.2.3\ldots 2n}{(2\pi )^{2n}}}\cdot \left\{1+{\frac {1}{2^{2n}}}+{\frac {1}{3^{2n}}}+\ldots \right\}}$

or from the form (3.)

$\scriptstyle {B_{2n-1}={\frac {\pm 2n}{(2^{2n}-1)2^{n-1}}}\left\{{\begin{aligned}&\scriptstyle {{\frac {1}{2}}n^{2n-1}}\\\scriptstyle {-(n-1)}&\scriptstyle {^{2n-1}\left\{1+{\frac {1}{2}}\cdot {\frac {2n}{1}}\right\}}\\\scriptstyle {+(n-2)}&\scriptstyle {^{2n-1}\left\{1+{\frac {2n}{1}}+{\frac {1}{2}}\cdot {\frac {2n.(2n-1}{1.2}}\right\}}\\\scriptstyle {-(n-3)}&\scriptstyle {^{2n-1}\left\{{\begin{matrix}&\scriptstyle {1+{\frac {2n}{1}}+{\frac {2n.2n-1}{1.2}}+}\\&\scriptstyle {+{\frac {1}{2}}\cdot {\frac {2n.(2n-1).(2n-2)}{1.2.3}}}\end{matrix}}\right\}}\\\scriptstyle {+\cdots }&\scriptstyle {\qquad \cdots \qquad \cdots \qquad \cdots }\end{aligned}}\right\}}$

or from many others. As however our object is not simplicity or facility of computation, but the illustration of the powers of the engine, we prefer selecting the formula below, marked (8.). This is derived in the following manner:— If in the equation (4.)

$\scriptstyle {{\frac {x}{e^{x}-1}}=1-{\frac {x}{2}}+B_{1}{\frac {x^{2}}{2}}+B_{3}{\frac {x^{4}}{2.3.4}}+B_{5}{\frac {x^{6}}{2.3.4.5.6}}+\ldots }$

(in which $\scriptstyle {B_{1}}$, $\scriptstyle {B_{3}}$…, &c. are the Numbers of Bernoulli), we expand the denominator of the first side in powers of $\scriptstyle {x}$, and then divide both numerator and denominator by $\scriptstyle {x}$, we shall derive (5.)

$\scriptstyle {1=\left(1-{\frac {x}{2}}+B_{1}{\frac {x^{2}}{2}}+B_{3}{\frac {x^{4}}{2.3.4}}+\ldots \right)\left(1+{\frac {x}{2}}+{\frac {x^{2}}{2.3}}+{\frac {x^{3}}{2.3.4}}\ldots \right)}$

If this latter multiplication be actually performed, we shall have a series of the general form (6.)

$\scriptstyle {1+D_{1}x+D_{2}x^{2}+D_{3}x^{3}\ldots \ldots \ldots \ldots }$

in which we see, first, that all the coefficients of the powers of $\scriptstyle {x}$ are severally equal to zero; and secondly, that the general form for $\scriptstyle {D_{2n}}$ the co-efficient of the $\scriptstyle {2n+1}$th term, (that is of $\scriptstyle {x^{2n}}$ any even power of $\scriptstyle {x}$), is the following:—(7.)

$\scriptstyle {\left.{\begin{aligned}\scriptstyle {\frac {1}{2.3\ldots 2n+1}}&\scriptstyle {-{\frac {1}{2}}\cdot {\frac {1}{2.3.\ldots 2n}}+{\frac {B_{1}}{2}}\cdot {\frac {1}{2.3\ldots 2n-1}}+{\frac {B_{3}}{2.3.4}}\cdot {\frac {1}{2.3\ldots 2n-3}}+}\\&\scriptstyle {+{\frac {B_{5}}{2.3.4.5.6}}\cdot {\frac {1}{2.3\ldots 2n-5}}+\cdots +{\frac {B_{2n-1}}{2.3\ldots 2n}}\cdot 1=0}\end{aligned}}\right\}}$