Page:Lagrange - Œuvres (1867) vol. 1.djvu/664

$+\left[{\mathfrak {A}}{\frac {d\mathrm {U} '}{dt}}+\beta {\frac {d\mathrm {V} }{dt}}+(m-{\frac {\mathrm {N} '(\mathrm {M+N} h')}{8h'\left(m-k'\right)}}y\right.$

${\begin{aligned}&+\left({\frac {(\mathrm {M+N} h')m}{4h'(m-k')}}+h\alpha -2\mathrm {H} {\mathfrak {A}}\right)\mathrm {u} '\\&+\left.\left.\left({\frac {\mathrm {N} (\mathrm {M+N} h')}{8h'(m-k')}}+(h-4\mathrm {H} )\beta \right)\mathrm {v} \right]\right\rangle e^{-(h+im)t{\sqrt {-1}}}\end{aligned}}$

${\begin{aligned}=\alpha g'&+\beta g+\left[\left({\frac {(\mathrm {M+N} h')m}{4h'(m-k')}}+h\alpha -\mathrm {H} {\mathfrak {A}}\right)f'+(m-(h-2h')\beta \right]{\sqrt {-1}}\\&+\int \left[\beta \mathrm {T} \left(\cos 2\mathrm {H} t+\sin 2\mathrm {H} t{\sqrt {-1}}\right)\right.\end{aligned}}$

$\left.+\mathrm {T} '\left(\alpha \cos \mathrm {H} t+{\mathfrak {A}}\sin \mathrm {H} t{\sqrt {-1}}\right)\right]e^{-(h+im)t{\sqrt {-1}}}dt.$

On traitera cette équation comme on a fait ci-devant l’équation $(f),$ et supposant, pour abréger,

${\begin{aligned}\varphi &=\sin \left[(h+im)t\right]\int \mathrm {T} \cos \left[(2h'-h+im)t\right]dt\\&-\cos \left[(h+im)t\right]\int \mathrm {T} \sin \left[(2h'-h+im)t\right]dt,\\\\\chi &=\sin \left[(h+im)t\right]\int \mathrm {T} '\cos \left[(h'+im)t\right]dt\\&-\cos \left[(h+im)t\right]\int \mathrm {T} '\sin \left[(h'+im)t\right]dt,\\\\\psi &=\sin \left[(h+im)t\right]\int \mathrm {T} '\cos \left[(2h-h'+im)t\right]dt\\&-\cos \left[(h+im)t\right]\int \mathrm {T} '\sin \left[(2h-h'+im)t\right]dt,\end{aligned}}$

et de plus

${\begin{aligned}\gamma &=m-{\frac {\mathrm {N} '(\mathrm {M+N} h')}{8h'(m-k')}},\\\delta &={\frac {(\mathrm {M+N} h')m}{4h'(m-k')}}+h\alpha -2\mathrm {H} {\mathfrak {A}},\\\varepsilon &={\frac {\mathrm {N} '(\mathrm {M+N} h')}{8h'(m-k')}}+(h-4\mathrm {H} )\beta ,\\\zeta &=\alpha g'+\beta g,\\\eta &=\left[{\frac {(\mathrm {M+N} h')m}{4h'(m-k')}}+h\alpha -\mathrm {H} {\mathfrak {A}}\right]f'+\left[m-(h-2h')\beta \right]f,\end{aligned}}$