Pagina:Werkecarlf02gausrich.djvu/157

$${\begin{aligned}\varphi (b-a,b+a)&=\varphi (b-a,2b)-{\frac {1}{8}}((b-a)^{2}-1)\\&={\frac {1}{8}}(3bb-2ab-aa-4b+1)-\varphi (2b,b-a)\end{aligned}}$$

nec non

$$\varphi (2b,b-a)=\varphi (2b,a)-2\varphi (b,a)+{\frac {1}{4}}b(b-1-a)$$

adeoque

$$\varphi (b-a,b+a)=2\varphi (b,a)-\varphi (2b,a)+{\frac {1}{8}}(bb-aa-2b+1)$$

tandemque

$$g=2\varphi (2b,a)-2\varphi (b,a)+{\frac {1}{8}}(aa-2ab+bb+4b-1)$$

Evictum est itaque, eandem formulam pro ${\textstyle g}$ valere, sive sit ${\textstyle a-b}$ positivus sive negativus, dummodo ${\textstyle a}$, ${\textstyle b}$ sint positivi.

75.

Ut reductionem ulteriorem assequamur, statuemus

$${\begin{aligned}&L=\left[{\frac {a}{2b}}\right]+\left[{\frac {2a}{2b}}\right]+\left[{\frac {3a}{2b}}\right]+{\text{etc.}}+\left[{\frac {{\frac {1}{2}}ba}{2b}}\right]\\&M=\left[{\frac {({\frac {1}{2}}b+1)a}{2b}}\right]+\left[{\frac {({\frac {1}{2}}b+2)a}{2b}}\right]+\left[{\frac {({\frac {1}{2}}b+3)a}{2b}}\right]+{\text{etc.}}+\left[{\frac {ba}{2b}}\right]\\&N=\left[{\frac {a+b}{2b}}\right]+\left[{\frac {2a+b}{2b}}\right]+\left[{\frac {3a+b}{2b}}\right]+{\text{etc.}}+\left[{\frac {{\frac {1}{2}}ba+b}{2b}}\right]\end{aligned}}$$

Quum facile perspiciatur, haberi generaliter ${\textstyle [u]+\left[u+{\frac {1}{2}}\right]=[2u]}$, quamcunque quantitatem realem denotet ${\textstyle u}$, fit ${\textstyle L+N=\varphi (b,a)}$, et quum manifesto sit ${\textstyle L+M=\varphi (2b,a)}$, erit

$$\varphi (2b,a)-\varphi (b,a)=M-N$$

Porro autem obvium est, aggregatum termini primi seriei ${\textstyle N}$ cum penultimo termino seriei ${\textstyle M}$, puta ${\textstyle \left[{\frac {a+b}{2b}}\right]+\left[{\frac {(b-1)a}{2b}}\right]}$ fieri ${\textstyle ={\frac {1}{2}}(a-1)}$, atque eandem summam effici e termino secundo seriei ${\textstyle N}$ cum antepenultimo seriei ${\textstyle M}$, et sic porro: quare quum etiam terminus ultimus seriei ${\textstyle M}$ fiat ${\textstyle ={\frac {1}{2}}(a-1)}$, ultimus vero terminus seriei ${\textstyle N}$ sit ${\textstyle =\left[{\frac {a+2}{4}}\right]={\frac {1}{4}}(a\mp 1)}$, valente signo superiori vel inferiori, prout ${\textstyle a}$ est formae ${\textstyle 4n+1}$ vel ${\textstyle 4n-1}$: erit

$$M+N={\frac {1}{4}}(a-1)b+{\frac {1}{4}}(a\mp 1)$$

et proin

$$\varphi (2b,a)-\varphi (b,a)={\frac {1}{4}}(a-1)b+{\frac {1}{4}}(a\mp 1)-2N$$