Page:Mémoires de l’Académie des sciences, Tome 12.djvu/362

Si l’on fait

${\begin{alignedat}{2}{\frac {d\mathrm {V} }{dx}}=&\mathrm {M} ,\quad &{\frac {d\mathrm {V} }{dy}}=&\mathrm {N} ,\quad &{\frac {d\mathrm {V} }{dy'}}=&\mathrm {P} ,\quad &{\frac {d\mathrm {V} }{dy''}}=&\mathrm {Q} ,{\text{ etc}}.,\\&&{\frac {d\mathrm {V} }{dz}}=&n,&{\frac {d\mathrm {V} }{dz'}}=&p,&{\frac {d\mathrm {V} }{dz''}}=&q,{\text{ etc}}.,\end{alignedat}}$

de sorte que l’on ait

$\delta \mathrm {V} =\mathrm {M} \delta x+\mathrm {N} \delta y+\mathrm {P} \delta y'+\mathrm {Q} \delta y''+$etc.

$+n\delta z+p\delta z'+q\delta z''+$etc.

et que l’on pose ensuite

${\begin{aligned}\mathrm {H} &=\mathrm {N-P'+Q''-etc.} \\\mathrm {K} &=n-p'+q''-{\text{etc}}.,\\\Gamma &=\mathrm {V} _{\text{ı}}\delta x_{\text{ı}}+(\mathrm {P} _{\text{ı}}-\mathrm {Q} '_{\text{ı}}+etc.)(\delta y_{\text{ı}}-y'_{\text{ı}}\delta x_{\text{ı}})\\&+(p_{\text{ı}}-q'_{\text{ı}}+{\text{etc}}.)(\delta z_{\text{ı}}-z'_{\text{ı}}\delta x_{\text{ı}})\\&+(\mathrm {Q} _{\text{ı}}-{\text{etc}}.)(\delta y_{\text{ı}}'-y_{\text{ı}}''\delta x_{\text{ı}})-(q-{\text{etc}}.)(\delta z_{\text{ı}}'-z_{\text{ı}}''\delta x_{\text{ı}})-{\text{etc}}.,\\&-\mathrm {V} _{0}\delta x_{0}+(\mathrm {P} _{0}-\mathrm {Q} '_{0}+etc.)(\delta y_{0}-y'_{0}\delta x_{0})\\&-(p_{0}-q'_{0}+{\text{etc}}.)(\delta z_{0}-z'_{0}\delta x_{0})\\&-(\mathrm {Q} _{0}-{\text{etc}}.)(\delta y_{0}'-y_{0}''\delta x_{0})-(q-{\text{etc}}.)(\delta z_{0}'-z_{0}''\delta x_{0})-{\text{etc}}.,\end{aligned}}$

on trouvera

$\delta \mathrm {U} =\Gamma +\int _{x_{0}}^{x_{1}}\left[\mathrm {H} (\delta y-y'\delta x)+\mathrm {K} (\delta z-z'\delta x)\right]dx.\qquad$(5)

pour l’accroissement de $\mathrm {U}$ correspondant aux variations $\delta x,\,\delta y,\,\delta z,$ de $x,\,y,\,z.$ On devra, comme dans le n^o 3, ajouter à cette formule, une partie

$\delta x_{0}\int _{x_{0}}^{x_{1}}{\frac {d\mathrm {V} }{dx_{0}}}dx+\delta x_{\text{ı}}\int _{x_{0}}^{x_{1}}{\frac {d\mathrm {V} }{dx_{\text{ı}}}}dx+$etc.,