ou, en vertu des formules (2),
![{\displaystyle s={\frac {v}{\operatorname {d} v}}{\sqrt {\operatorname {d} t^{2}+\operatorname {d} u^{2}+\operatorname {d} v^{2}}},\qquad }](https://wikimedia.org/api/rest_v1/media/math/render/svg/95701d1ee1d3b91da57974c245df130d0b370865)
(4)
d’où
![{\displaystyle \operatorname {d} s={\sqrt {\operatorname {d} t^{2}+\operatorname {d} u^{2}+\operatorname {d} v^{2}}}-{\frac {v}{\operatorname {d} v}}\operatorname {d} v.{\sqrt {\operatorname {d} t^{2}+\operatorname {d} u^{2}+\operatorname {d} v^{2}}},\qquad }](https://wikimedia.org/api/rest_v1/media/math/render/svg/8813b94b4a06d86ed81cf4523746abf1f93d1fee)
(5)
mais on doit avoir d’ailleurs
![{\displaystyle \operatorname {d} s={\sqrt {\operatorname {d} t^{2}+\operatorname {d} u^{2}+\operatorname {d} v^{2}}}\,;\qquad }](https://wikimedia.org/api/rest_v1/media/math/render/svg/7af70daf0201cfe7e58d7b90dfac480ee5838516)
(6)
donc
![{\displaystyle d.{\sqrt {\operatorname {d} t^{2}+\operatorname {d} u^{2}+\operatorname {d} v^{2}}}=0,\quad }](https://wikimedia.org/api/rest_v1/media/math/render/svg/961f90092f924979cbdbc97937090348bfe181d3)
d’où
![{\displaystyle \quad {\sqrt {\operatorname {d} t^{2}+\operatorname {d} u^{2}+\operatorname {d} v^{2}}}=A\operatorname {d} v,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b46465aa25665ee9d5687a3993dd12cc20b1979f)
étant la variable indépendante. Cela donne en quarrant et transformant la constante
![{\displaystyle \operatorname {d} t^{2}+\operatorname {d} u^{2}=B\operatorname {d} v^{2}.\qquad }](https://wikimedia.org/api/rest_v1/media/math/render/svg/e7595aea91d7439368dc2cb47634925586188d27)
(7)
mais, en vertu de l’équation (1)
![{\displaystyle v={\frac {\sqrt {t^{2}+u^{2}}}{\operatorname {Tang} .\alpha }},\qquad }](https://wikimedia.org/api/rest_v1/media/math/render/svg/2bdceb9cac8bdeec8cefb1612ec30d219292a1cf)
(8)
d’où
![{\displaystyle \operatorname {d} v={\frac {t\operatorname {d} t+u\operatorname {d} u}{\operatorname {Tang} .\alpha .{\sqrt {t^{2}+u^{2}}}}},\quad }](https://wikimedia.org/api/rest_v1/media/math/render/svg/a8b0016b4c0862dbb46fd581e1809baac1a682c1)
et
![{\displaystyle \quad \operatorname {d} v^{2}={\frac {(t\operatorname {d} t+u\operatorname {d} u)^{2}}{\left(t^{2}+u^{2}\right)\operatorname {Tang} .^{2}\alpha .}}\,;\qquad }](https://wikimedia.org/api/rest_v1/media/math/render/svg/39b938d90e1832cd03116cfd353c25512a29f8e8)
(9)
donc, en substituant dans (7) et transformant ancore la constante