Page:Mémoires de l’Académie des sciences, Tome 10.djvu/572

${\begin{aligned}\mathrm {U} _{1}=&{\frac {a'\alpha '}{2{\sqrt {a^{2}\left(\delta ^{2}-\alpha '^{2}\right)+a'^{2}\alpha '^{2}}}}}e^{{\frac {a'}{a}}(x-h){\sqrt {\delta ^{2}-\alpha '^{2}}}}\\\mathrm {V} _{1}=&{\frac {1}{2{\sqrt {a^{2}\left(\delta ^{2}-\alpha '^{2}\right)+a'^{2}\alpha '^{2}}}}}\\&\qquad \qquad \times \left[a'\alpha '\cos .\alpha '(x-h)+a{\sqrt {\delta ^{2}-\alpha '^{2}}}\sin .\alpha '(x-h)\right].\end{aligned}}$

Le radical qui se trouve au dénominateur de ces expressions est la même chose que

${\frac {1}{a'}}{\sqrt {a^{2}-a'^{2}}}{\sqrt {a^{2}\left(\beta ^{2}+\gamma ^{2}\right)-a'^{2}\alpha '^{2}}}\,;$

à cause de $l=\infty ,$ la quantité $\Delta _{1}$ se réduit à $\Delta _{1}={\frac {1}{8}}l\,;$ cela étant, les formules (17) se changeront en celles-ci :

$\left.{\begin{aligned}u=&{\frac {2}{\pi \left(a^{2}-a'^{2}\right)}}\int _{0}^{\delta }\left(p_{1}\cos .\lambda t+\mathrm {P} _{1}{\frac {\sin .\lambda t}{\lambda }}\right){\frac {d\alpha '}{a^{2}\left(\beta ^{2}+\gamma ^{2}\right)-a'^{2}\alpha '^{2}}},\\v=&{\frac {2}{\pi \left(a^{2}-a'^{2}\right)}}\int _{0}^{\delta }\left(q_{1}\cos .\lambda t+\mathrm {Q} _{1}{\frac {\sin .\lambda t}{\lambda }}\right){\frac {d\alpha '}{a^{2}\left(\beta ^{2}+\gamma ^{2}\right)-a'^{2}\alpha '^{2}}},\end{aligned}}\right\}$

(19)

où l’on a fait, pour abréger,

${\begin{aligned}p_{1}=&a'^{4}\alpha '^{2}e^{{\frac {a'}{a}}(x-h){\sqrt {\delta ^{2}-\alpha '^{2}}}}\int _{-\infty }^{h}f(x',y',z')e^{{\frac {a'}{a}}(x'-h){\sqrt {\delta ^{2}-\alpha '^{2}}}}dx'\\&+a^{2}a'\alpha 'e^{{\frac {a'}{a}}(x-h){\sqrt {\delta ^{2}-\alpha '^{2}}}}\int _{h}^{\infty }f'(x',y',z')\left[a'\alpha '\cos .\alpha '(x'-h)\right.\\&\qquad \qquad \qquad \qquad \qquad \qquad +\left.a{\sqrt {\delta ^{2}-\alpha '^{2}}}\sin .\alpha '(x'-h)\right]dx',\\q_{1}=&{\frac {a'^{5}\alpha '}{a^{2}}}\left[a'\alpha '\cos .\alpha '(x'-h)\right.\\&+\left.{\sqrt {\delta ^{2}-\alpha '^{2}}}\sin .\alpha '(x'-h)\right]\int _{-\infty }^{h}f'(x',y',z')e^{{\frac {a'}{a}}(x'-h){\sqrt {\delta ^{2}-\alpha '^{2}}}}dx'\\&+a'^{2}\left[a'\alpha '\cos .\alpha '(x-h)\right.\\&+\left.{\sqrt {\delta ^{2}-\alpha '^{2}}}\sin .\alpha '(x-h)\right]\int _{h}^{\infty }f'(x',y',z')\left[a'\alpha '\cos .\alpha '(x'-h)\right.\\&\qquad \qquad \qquad \qquad \qquad \qquad \qquad +\left.a{\sqrt {\delta ^{2}-\alpha '^{2}}}\sin .\alpha '(x'-h)\right]dx',\\\end{aligned}}$