362
NOTES DU TRADUCTEUR.
Cette relation est appelée relation de Nicolic, son inventeur.
En introduisant cette expression dans la valeur de
nous avons
![{\displaystyle q={\frac {1}{2}}\left(rr''\right)^{\frac {1}{2}}\!\left(\cos {\frac {1}{2}}(v''\!\!-\!v)+{\frac {1-\operatorname {tang} ^{2}(45^{\circ }\!\!-\!z)\operatorname {cotang} ^{2}\!{\dfrac {1}{4}}(v''\!\!-\!v)}{1+\operatorname {tang} ^{2}(45^{\circ }\!\!-\!z)\operatorname {cotang} ^{2}\!{\dfrac {1}{4}}(v''\!\!-\!v)}}\right)\!,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bc895dc9d2ed1cc540602f6b6b7f7a4eaab28188)
en réduisant au même dénominateur dans la parenthèse et en substituant
à
il vient
![{\displaystyle q={\frac {1}{2}}\left(rr''\right)^{\frac {1}{2}}\!\left({\frac {2\cos ^{2}\!{\dfrac {1}{4}}(v''\!\!-\!v)-2\operatorname {tang} ^{2}(45^{\circ }\!\!-\!z)\operatorname {cotang} ^{2}\!{\dfrac {1}{4}}(v''\!-v)\sin ^{2}(v''\!-v)}{1+\operatorname {tang} ^{2}(45^{\circ }\!\!-\!z)\operatorname {cotang} ^{2}\!{\dfrac {1}{4}}(v''\!\!-\!v)}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/641dfd3c5c098fdb05acd6e56098ba86f64ed4ea)
ou
![{\displaystyle {\begin{aligned}q=\left(rr''\right)^{\frac {1}{2}}&\sin ^{2}\!{\frac {1}{4}}(v''\!\!-\!v)\cos ^{2}\!{\frac {1}{4}}(v''\!\!-\!v)\\.&\left[{\frac {1-\operatorname {tang} ^{2}(45^{\circ }\!\!-\!z)}{\sin ^{2}\!{\dfrac {1}{4}}(v''\!\!-\!v)+\operatorname {tang} ^{2}(45^{\circ }\!\!-\!z)\cos ^{2}\!{\dfrac {1}{4}}(v''\!\!-\!v)}}\right].\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/857f77622dea8d13f3511985ff1e0dae709a1863)
Mais on a
![{\displaystyle \operatorname {tang} ^{2}(45^{\circ }\!-z)={\frac {\left(r''^{\frac {1}{2}}-r^{\frac {1}{2}}\right)^{2}}{\left(r''^{\frac {1}{2}}+r^{\frac {1}{2}}\right)^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/00d2a6a1618687f93266fbe997c7a263f2663515)
Il vient donc en substituant cette valeur
![{\displaystyle {\begin{aligned}q=&\left(rr''\right)^{\frac {1}{2}}\sin ^{2}\!{\frac {1}{4}}(v''\!\!-\!v)\cos ^{2}{\frac {1}{4}}(v''\!\!-\!v)\\.&\left[{\frac {\left(r''^{\frac {1}{2}}+r^{\frac {1}{2}}\right)^{2}-\left(r''^{\frac {1}{2}}-r^{\frac {1}{2}}\right)^{2}}{\left(r''^{\frac {1}{2}}+r^{\frac {1}{2}}\right)^{2}\sin ^{2}\!{\dfrac {1}{4}}(v''\!\!-\!v)+\left(r''^{\frac {1}{2}}-r^{\frac {1}{2}}\right)^{2}\cos ^{2}\!{\dfrac {1}{4}}(v''\!\!-\!v)}}\right]\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7ab63c996435d23f22953ff2c706ae153b58cc45)
ou, en développant les carrés dans la parenthèse et réduisant,
![{\displaystyle {\begin{aligned}q&={\frac {rr''\sin ^{2}\!{\dfrac {1}{2}}(v''\!-v)}{r''+r-2\left(rr''\right)^{\frac {1}{2}}\left(\cos ^{2}\!{\dfrac {1}{4}}(v''\!-v)-\sin ^{2}\!{\dfrac {1}{4}}(v''\!-v)\right)}}\\&={\frac {rr''\sin ^{2}\!{\dfrac {1}{2}}(v''\!-v)}{r''+r-2\left(rr''\right)^{\frac {1}{2}}\cos {\dfrac {1}{2}}(v''\!-v)}}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/742763743ad2d69c19228bcf81caf9275ad3b1d7)
d’où l’on obtient