, tertius
; secundus vero fit
![{\displaystyle ={\frac {1}{4}}bb+\sum \left[{\frac {aa-2ax}{2b}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ed1813695625391112ed1317314af9d060194f31)
Sed fit, scribendo terminos inverso ordine,
![{\displaystyle \Sigma \left[{\frac {aa-2ax}{2b}}\right]=\left[{\frac {a}{2b}}\right]+\left[{\frac {3a}{2b}}\right]+\left[{\frac {5a}{2b}}\right]+\ldots +\left[{\frac {(b-1)a}{2b}}\right]=\varphi (2b,a)-\varphi (b,a)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/91fbef5fe97f5f6e1ecc0f605ae7ef1548e8cd22)
Formula itaque nostra sequentem induit formam:
![{\displaystyle g=\varphi (a-b,a+b)+\varphi (2b.a)-\varphi (a,b)-\varphi (b,a)+{\frac {1}{4}}bb}](https://wikimedia.org/api/rest_v1/media/math/render/svg/75456be932027dd1766423e862d50654bd2dc3a9)
Consideremus primo terminum
, qui protinus transmutatur in
sive in
![{\displaystyle \varphi (a-b,2b)+{\frac {1}{8}}((a-b)^{2}-1)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7504f296015925c4ec1166064321b993eb40592a)
Dein quum per theorema generale fiat
, dum
sunt integri positivi inter se primi, habemus
![{\displaystyle \varphi (a-b,2b)={\frac {1}{2}}b(a-b-1)-\varphi (2b,a-b)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7b58d735794603bbb1fe4b9a59584ff4cdd0e51d)
adeoque
![{\displaystyle \varphi (a-b,a+b)={\frac {1}{8}}(aa+2ab-3bb-4b-1)-\varphi (2b,a-b)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a775edac1e87d7be6f8eb62591945100dbd67937)
Disponamus partes ipsius
sequenti modo
![{\displaystyle {\begin{aligned}&{\left[{\frac {a-b}{2b}}\right]+\left[{\frac {3(a-b)}{2b}}\right]+\left[{\frac {5(a-b)}{2b}}\right]+{\text{etc.}}+\left[{\frac {(b-1)(a-b)}{2b}}\right]}\\+&{\left[{\frac {a-b}{b}}\right]+\left[{\frac {2(a-b)}{b}}\right]+\left[{\frac {3(a-b)}{b}}\right]+{\text{etc.}}+\left[{\frac {{\frac {1}{2}}b(a-b)}{b}}\right]}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/070c5322b25fac4fb89780fa0d280886d27d64cd)
Series secunda manifesto fit
![{\displaystyle =\varphi (b,a-b)=\varphi (b,a)-1-2-3-{\text{etc.}}-{\frac {1}{2}}b=\varphi (b,a)-{\frac {1}{8}}(bb+2b)d}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cab32ebc63d29e4121511ee1b043dbf1d7a93062)
seriem primam ordine terminorum inverso ita exhibemus:
![{\displaystyle \left[{\frac {1}{2}}(a+1-b)-{\frac {a}{2b}}\right]+\left[{\frac {1}{2}}(a+3-b)-{\frac {3a}{2b}}\right]+\left[{\frac {1}{2}}(a+5-b)-{\frac {5a}{2b}}\right]+{\text{etc.}}+\left[{\frac {1}{2}}(a-1)-{\frac {(b-1)a}{2b}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c28c0282b96344d1da7e885f65e15d83533ed1ab)
quae expressio, quum denotante
![{\textstyle t}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b2bc926f90178739fccd01a96c6fa778ab3535d6)
numerum integrum,
![{\textstyle u}](https://wikimedia.org/api/rest_v1/media/math/render/svg/24e12e26e505ed5b02c7648a89bbc6737038b2de)
fractum, generaliter sit
![{\textstyle [t-u]=t-1-[u]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8969fc455d607c4950a4aa50b4af508306f5e611)
, mutatur in sequentem
![{\displaystyle {\begin{aligned}&{\frac {1}{8}}b(2a-4-b)-\left[{\frac {a}{2b}}\right]-\left[{\frac {3a}{2b}}\right]-\left[{\frac {5a}{2b}}\right]-{\text{etc.}}-\left[{\frac {(b-1)a}{2b}}\right]\\=&{\frac {1}{8}}b(2a-4-b)-\varphi (2b,a)+\varphi (b,a)\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8eddd565020e121d3ed1efb05aa89fee1e62aba7)