Pagina:Werkecarlf02gausrich.djvu/156

Hinc fit $$\varphi (2b,a-b)=2\varphi (b,a)-\varphi (2b,a)+{\frac {1}{4}}b(a-3-b)$$ et proin $$\varphi (a-b,a+b)=\varphi (2b,a)-2\varphi (b,a)+{\frac {1}{8}}(aa-bb+2b-1)$$ Substituendo hunc valorem in formula pro ${\textstyle g}$ supra tradita, insuperque ${\textstyle \varphi (a,b)+\varphi (b,a)={\frac {1}{4}}b(a-1)}$, obtinemus $$g=2\varphi (2b,a)-2\varphi (b,a)+{\frac {1}{8}}(aa-2ab+bb+4b-1)$$

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Per ratiocinia prorsus similia absolvitur casus is, ubi manentibus ${\textstyle a}$, ${\textstyle b}$ positivis ${\textstyle a-b}$ est negativus sive ${\textstyle b-a}$ positivus. Aequationes ${\textstyle p(a-2x)=2b\eta +a\theta }$, ${\textstyle p(b-a+2x)=2b\zeta +(b-a)\theta }$ docent, ${\textstyle {\frac {1}{2}}a-x}$ atque ${\textstyle x+{\frac {1}{2}}(b-a)}$ positivos, et proin ${\textstyle x}$ alicui numerorum ${\textstyle -{\frac {1}{2}}(b-a-1)}$, ${\textstyle -{\frac {1}{2}}(b-a-3)}$, ${\textstyle -{\frac {1}{2}}(b-a-5)\ldots +{\frac {1}{2}}(a-1)}$ aequalem esse debere. Porro ex aequatione ${\textstyle px+(b-a)\eta =a\zeta }$ sequitur, pro valoribus negativis ipsius ${\textstyle x}$ conditionem, ex qua ${\textstyle \eta }$ debet esse positivus, iam contineri sub conditione, ex qua ${\textstyle \zeta }$ debet esse positivus, contrarium vero evenire, quoties ipsi ${\textstyle x}$ valor positivus tribuatur. Hinc valores ipsius ${\textstyle y}$ pro valore determinato negativo ipsius ${\textstyle x}$ inter ${\textstyle {\frac {(a+b)x}{a-b}}}$ et ${\textstyle {\frac {p-2ax}{2b}}}$, contra pro valore positivo ipsius ${\textstyle x}$ inter ${\textstyle {\frac {bx}{a}}}$ et ${\textstyle {\frac {p-2ax}{2b}}}$ contenti esse debent: manifesto pro ${\textstyle x=0}$ hi limites sunt ${\textstyle 0}$ et ${\textstyle {\frac {p-2ax}{2b}}}$, valore ${\textstyle y=0}$ ipso excluso. Hinc colligitur $$g=-\sum \left[{\frac {(a+b)x}{a-b}}\right]+\sum \left[{\frac {1}{2}}b+{\frac {aa-2ax}{2b}}\right]-\sum \left[{\frac {bx}{a}}\right]$$ ubi in termino primo summatio extendenda est per omnes valores negativos ipsius ${\textstyle x}$ inde a ${\textstyle -1}$ usque ad ${\textstyle -{\frac {1}{2}}(b-a-1)}$; in secunda per omnes valores ipsius ${\textstyle x}$ inde a ${\textstyle -{\frac {1}{2}}(b-a-1)}$ usque ad ${\textstyle {\frac {1}{2}}(a-1)}$; in tertia per omnes valores positivos ipsius ${\textstyle x}$ inde a ${\textstyle +1}$ usque ad ${\textstyle {\frac {1}{2}}(a-1)}$: hoc pacto e summatione prima prodit ${\textstyle -\varphi (b-a,b+a)}$, e secunda perinde ut in art. praec. ${\textstyle {\frac {1}{4}}bb+\varphi (2b,a)-\varphi (b,a)}$, denique e tertia ${\textstyle -\varphi (a,b)}$, sive habetur $$g=-\varphi (b-a,b+a)+\varphi (2b,a)-\varphi (b,a)-\varphi (a,b)+{\frac {1}{4}}bb$$ Iam simili modo ut in art. praec. evolvitur