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DÉVELOPPEMENT DES FONCTIONS.
ANALISE TRANSCENDANTE.
Recherches d’analise, relatives au développement
des fonctions ;
Par
M. Frédéric Sarrus, professeur de mathématiques
au collége de Pezenas ;
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Soit l’équation
![{\displaystyle x=\operatorname {f} (a,b,c),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/237b46782e44c20d8ecc295331b85176a9a0d45f)
et soient
des fonctions entièrement arbitraires de
sans
on aura
![{\displaystyle \left.{\begin{aligned}&{\frac {\operatorname {d} U}{\operatorname {d} a}}={\frac {\operatorname {d} U}{\operatorname {d} x}}{\frac {\operatorname {d} x}{\operatorname {d} a}},\\\\&{\frac {\operatorname {d} U}{\operatorname {d} b}}={\frac {\operatorname {d} U}{\operatorname {d} x}}{\frac {\operatorname {d} x}{\operatorname {d} b}}\,;\end{aligned}}\right\}{\text{(1}})\qquad \left.{\begin{aligned}&{\frac {\operatorname {d} V}{\operatorname {d} a}}={\frac {\operatorname {d} V}{\operatorname {d} x}}{\frac {\operatorname {d} x}{\operatorname {d} a}},\\\\&{\frac {\operatorname {d} V}{\operatorname {d} b}}={\frac {\operatorname {d} V}{\operatorname {d} x}}{\frac {\operatorname {d} x}{\operatorname {d} b}}\,;\end{aligned}}\right\}{\text{(2}})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dcfe5930296bd8963c879d5d07739b1e2283e3ec)
les équations des deux couples donnent, par division,
![{\displaystyle {\frac {\frac {\operatorname {d} x}{\operatorname {d} a}}{\frac {\operatorname {d} x}{\operatorname {d} b}}}={\frac {\frac {\operatorname {d} U}{\operatorname {d} x}}{\frac {\operatorname {d} U}{\operatorname {d} b}}},\qquad {\frac {\frac {\operatorname {d} x}{\operatorname {d} a}}{\frac {\operatorname {d} x}{\operatorname {d} b}}}={\frac {\frac {\operatorname {d} V}{\operatorname {d} a}}{\frac {\operatorname {d} V}{\operatorname {d} b}}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/af94ac82a4cd8ca058569d63e21f32924b2533d8)
et en égalant ces deux valeurs