Pagina:Werkecarlf02gausrich.djvu/37

$$1-y^{-1}{\frac {1-y^{-2m}}{1-y^{-2}}}+y^{-2}{\frac {(1-y^{-2m})(1-y^{-2m+2})}{(1-y^{-2})(1-y^{-4})}}-y^{-3}{\frac {(1-y^{-2m})(1-y^{-2m+2})(1-y^{-2m+2})}{(1-y^{-2})(1-y^{-4})(1-y^{-6})}}+{\text{etc.}}$$

usque ad terminum ${\textstyle m+1^{\text{tum }}}$

$$=(1-y^{-1})(1+y^{-2})(1-y^{-3})(1+y^{-4})\ldots (1\pm y^{-m})\dots \dots \dots [7]$$

Quodsi hic pro ${\textstyle y}$ accipitur radix propria aequationis ${\textstyle y^{n}-1=0}$, puta ${\textstyle r}$, atque simul statuitur ${\textstyle m=n-1}$, erit

$${\begin{aligned}&{\frac {1-y^{-2m}}{1-y^{-2}}}={\frac {1-r^{2}}{1-r^{-2}}}=-r^{2}\\&{\frac {1-y^{-2m+2}}{1-y^{-4}}}={\frac {1-r^{4}}{1-r^{-4}}}=-r^{4}\\&{\frac {1-y^{-2m+4}}{1-y^{-6}}}={\frac {1-r^{6}}{1-r^{-6}}}=-r^{6}{\text{ etc.}}\end{aligned}}$$

usque ad

$${\frac {1-y^{-2}}{1-y^{-2m}}}={\frac {1-r^{2n-2}}{1-r^{-2n+2}}}=-r^{2n-2}$$

ubi notandum, nullum denominatorum ${\textstyle 1-r^{-2}}$, ${\textstyle 1-r^{-4}}$ etc. fieri ${\textstyle =0}$. Hinc aequatio [7] hancce formam assumit

$$1+r+r^{4}+r^{9}+{\text{etc.}}+r^{(n-1)^{2}}=(1-r^{-1})(1+r^{-2})(1-r^{-3})\ldots (1+r^{-n+1})$$

Multiplicando in membro secundo huius aequationis terminum primum per ultimum, secundum per penultimum etc., habemus

$${\begin{aligned}&(1-r^{-1})(1+r^{-n+1})=r-r^{-1}\\&(1+r^{-2})(1-r^{-n+2})=r^{n-2}-r^{-n+2}\\&(1-r^{-3})(1+r^{-n+3})=r^{3}-r^{-3}\\&(1+r^{-4})(1-r^{-n+4})=r^{n-4}-r^{-n+4}{\text{ etc.}}\end{aligned}}$$

Ex his productis partialibus facile perspicietur conflari productum

$$(r-r^{-1})(r^{3}-r^{-3})(r^{5}-r^{-5})\ldots (r^{n-4}-r^{-n+4})(r^{n-2}-r^{-n+2})$$

quod itaque erit

$$=1+r+r^{4}+r^{9}+{\text{etc.}}+r^{(n-1)^{2}}=W$$

Haec aequatio identica est cum aequ. [5] in art. 12 e progressione prima derivata, ratiociniaque dein reliqua eodem modo adstruentur, ut in artt. 13 et 14.