Page:Annales de mathématiques pures et appliquées, 1828-1829, Tome 19.djvu/214

$\operatorname {Sin} .t{\sqrt {r''}}=\operatorname {Sin} .t{\sqrt {r'}}+{\frac {it}{1}}\operatorname {Cos} .t{\sqrt {r'}}-{\frac {i^{2}t^{2}}{1.2}}\operatorname {Sin} .t{\sqrt {r'}}+\ldots$

et par suite

$T''\operatorname {Cos} .t{\sqrt {r''}}+U''\operatorname {Sin} .t{\sqrt {r''}}$

$=T''\operatorname {Cos} .t{\sqrt {r'}}+U''\operatorname {Sin} .t{\sqrt {r'}}+t\left(iT''\operatorname {Cos} .t{\sqrt {r'}}-iU\right)$

$T''\operatorname {Cos} .t{\sqrt {r'}}+U''\operatorname {Sin} .t{\sqrt {a''}}=\left\{{\begin{aligned}&T''\operatorname {Cos} .t{\sqrt {r'}}+U''\operatorname {Sin} .t{\sqrt {r'}}\\\\+&{\frac {it}{1}}\left(U''\operatorname {Cos} .t{\sqrt {r'}}-T''\operatorname {Sin} .t{\sqrt {r'}}\right)\\\\+&{\frac {i^{2}t^{2}}{1.2}}\left(T''\operatorname {Cos} .t{\sqrt {r'}}-U''\operatorname {Sin} .t{\sqrt {r'}}\right)\\\\+&\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \end{aligned}}\right\}\,;$

de sorte que si l’on fait, pour abréger,

$U''={\frac {U_{1}}{i}},\quad T''={\frac {T_{1}}{i}},\quad U'+U''=U_{2},\qquad T'+T''=T_{2},$

les équations (18) et (19) deviendront

$Z=T_{2}\operatorname {Cos} .t{\sqrt {r'}}+U_{2}\operatorname {Sin} .t{\sqrt {r'}}+tU_{1}\operatorname {Cos} .t{\sqrt {r'}}+tT_{1}\operatorname {Sin} .t{\sqrt {r'}}$

$+T'''\operatorname {Cos} .t{\sqrt {r'''}}+U'''\operatorname {Sin} .t{\sqrt {r'''}}+{\frac {it^{2}}{1.2}}\Lambda ,$

$\lambda =p'\left(T_{2}+tU_{1}\right)\operatorname {Cos} .t{\sqrt {r'}}+p'\left(U_{2}+tT_{1}\right)\operatorname {Sin} .t{\sqrt {r'}}$

$+p'''T'''\operatorname {Cos} .t{\sqrt {r'''}}+p'''U'''\operatorname {Sin} .t{\sqrt {r'''}}+p'.{\frac {it^{2}}{1.2}}\Gamma ,$