nous avons d’abord, entre z {\displaystyle z} et c , {\displaystyle c,} la relation
d’où
On peut aussi mettre la relation entre Z {\displaystyle \mathrm {Z} } et c {\displaystyle c} (art. 99), sous la forme
on a ensuite,
mais
il vient donc, en substituant et réduisant,
on a par suite,
on a aussi,
et 
la relation
et 
(art. 99), sous la forme
![{\displaystyle {\begin{aligned}c-{\frac {1}{c}}&={\frac {2z+2+2{\sqrt {\overset {}{z}}}{\sqrt {{\overset {}{z}}+1}}}{{\sqrt {\overset {}{z}}}+{\sqrt {{\overset {}{z}}+1}}}}2{\sqrt {\overset {}{z}}}={\frac {2[(z+1)+{\sqrt {\overset {}{z}}}{\sqrt {{\overset {}{z}}+1}}]}{{\sqrt {\overset {}{z}}}+{\sqrt {{\overset {}{z}}+1}}}}2{\sqrt {\overset {}{z}}}\\&=2{\sqrt {(z+1)}}2{\sqrt {\overset {}{z}}}=4\left(z^{2}+z\right)^{\frac {1}{2}},\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fbef9743aa3c4224b4731d33c9d6e4133553937e)
![{\displaystyle {}=4\left(z^{2}+z\right)^{\frac {1}{2}}\left[\left({\sqrt {\overset {}{z}}}+{\sqrt {{\overset {}{z}}+1}}\right)^{2}+{\frac {1}{({\sqrt {\overset {}{z}}}+{\sqrt {{\overset {}{z}}+1}})^{2}}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d257810bef10669ab29c56ac976ce7f0c3a87c2c)
![{\displaystyle {}=4\left(z^{2}+z\right)^{\frac {1}{2}}\left\{{\frac {\left[(2z+1)+2{\sqrt {\overset {}{z}}}{\sqrt {{\overset {}{z}}+1}}\right]^{2}+1}{\left({\sqrt {\overset {}{z}}}+{\sqrt {{\overset {}{z}}+1}}\right)^{2}}}\right\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7161df5c92e106c863d8d9a817a9456f8facda61)
