Pagina:Werkecarlf03gausrich.djvu/181

$$={\frac {\alpha }{n+1}}+{\frac {\beta }{n}}+{\frac {\gamma }{n-1}}+{\frac {\delta }{n-2}}+{\text{etc. }}$$

qua quantitate posita ${\textstyle =M^{(\mu )}R^{(\mu )},}$ erit integrale quaesitum

$$=\Delta (AR+A^{\prime }R^{\prime }+A^{\prime \prime }R^{\prime \prime }+A^{\prime \prime \prime }R^{\prime \prime \prime }+{\text{etc.}}+A^{(n)}R^{(n)})$$

Exempli caussa apponemus computum coëfficientis ${\textstyle R^{\prime \prime }}$ pro ${\textstyle n=5.}$ Fit hic

$${\begin{aligned}&T^{\prime \prime }=5^{5}t^{5}-13.5^{4}t^{4}+59.5^{3}t^{3}-107.5^{2}tt+60.5.t\\&M^{\prime \prime }=2\times 1\times (-1)\times (-2)\times (-3)=-12\end{aligned}}$$

Hinc ${\textstyle -12R^{\prime \prime }={\frac {3125}{6}}-1625+{\frac {7375}{4}}-{\frac {2675}{3}}+150=-{\frac {25}{12}},}$ adeoque ${\textstyle R^{\prime \prime }={\frac {25}{144}}.}$

Computus aliquanto brevior evadit, statuendo ${\textstyle 2t-1=u.}$ Tunc fit

$$T^{(\mu )}={\frac {(nu+n)(nu+n-2)(nu+n-4)\ldots (nu-n+4)(nu-n+2)(nu-n)}{2^{n}(nu+n-2\mu )}}$$

Ponamus

$${\frac {(nnuu-nn).(nnuu-(n-2)^{2}).(nnuu-(n-4)^{2}).(nnuu-(n-6)^{2})\ldots }{nnuu-(n-2\mu )^{2}}}=U^{(\mu )}$$

ubi numerator desinere debet in ${\textstyle \ldots (nnuu-9)(nnuu-1),}$ si ${\textstyle n}$ est impar, vel in ${\textstyle \ldots (nnuu-4)nu,}$ si ${\textstyle n}$ est par, eritque

$$T^{(\mu )}={\frac {(nu-n+2\mu )U^{(\mu )}}{2^{n}}}$$

Iam integrale ${\textstyle \int T^{(\mu )}\operatorname {d} t}$ a ${\textstyle t=0}$ usque ad ${\textstyle t=1}$ acceptum aequale est integrali

$$\int {\frac {1}{2}}T^{(\mu )}\operatorname {d} u=\int {\frac {nuU^{(\mu )}\operatorname {d} u}{2^{n+1}}}+\int {\frac {(2\mu -n)U^{(\mu )}\operatorname {d} u}{2^{n+1}}}$$

ab ${\textstyle u=-1}$ usque ad ${\textstyle u=+1.}$

Statuendo itaque

$$U^{(\mu )}=\alpha u^{n-1}+\beta u^{n-3}+\gamma u^{n-5}+\delta u^{n-7}+{\text{etc. }}$$

(sponte enim patet, potestates ${\textstyle u^{n-2},u^{n-4},u^{n-6}}$ etc. abesse), integralis pars ${\textstyle \int {\frac {nuU^{(n)}\operatorname {d} u}{2^{n+1}}}}$ evanescet pro valore impari ipsius ${\textstyle n,}$ pars altera ${\textstyle \int {\frac {(2\mu -n)U^{(\mu )}\operatorname {d} u}{2^{n+1}}}}$ vero pro valore pari, unde integrale ${\textstyle \int T^{(\mu )}\operatorname {d} t}$ fiet pro ${\textstyle n}$ pari

$$={\frac {n}{2^{n}}}({\frac {\alpha }{n+1}}+{\frac {\beta }{n-1}}+{\frac {\gamma }{n-3}}+{\frac {\delta }{n-5}}+{\text{etc.}})$$