![{\displaystyle \operatorname {S} \left({\frac {\operatorname {d} ^{2}x'\delta x'+\operatorname {d} ^{2}y'\delta y'+\operatorname {d} ^{2}z'\delta z'}{\operatorname {d} t^{2}}}\right)\mathrm {D} m=-g\iiint \delta z'\Delta \operatorname {d} x'\operatorname {d} y'\operatorname {d} z'.\quad }](https://wikimedia.org/api/rest_v1/media/math/render/svg/712596fec653b57d6639fbae2d398f71b865d4a3)
(8)
Présentement, les équations (2) nous donneront, comme cas particulier,
![{\displaystyle {\begin{aligned}&\delta X=\alpha +\nu Y-\mu Z,\\&\delta Y=\beta +\lambda Z-\nu X,\\&\delta Z=\gamma +\mu X-\lambda Y,\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/21ffca2df3bf2722eea61229b10b5791fe922f1f)
lesquelles retranchées des équations (2) donnent
![{\displaystyle {\begin{aligned}&\nu (y-Y)-\mu (z-Z)=\delta x-\delta X,\\&\lambda (z-Z)-\nu (x-X)=\delta y-\delta Y,\\&\mu (x-X)-\lambda (y-Y)=\delta z-\delta Z\,;\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f0be6f3f2f33a4036ab6d955c655d5748fa29f4d)
ou, ce qui revient au même,
![{\displaystyle {\begin{aligned}&\delta x'=\nu y'-\mu z',\\&\delta y'=\lambda z'-\nu x',\\&\delta z'=\mu x'-\lambda y',\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0429c8cd314c99683283502fdaede5ceee6d33fc)
d’après les valeurs
posées ci-dessus.
Au moyen de ces valeurs, l’équation (8) deviendra, en mettant
hors du signe d’intégration,
![{\displaystyle \left.{\begin{aligned}&\lambda \operatorname {S} \left({\frac {z'\operatorname {d} ^{2}y'-y'\operatorname {d} ^{2}z'}{\operatorname {d} t^{2}}}\right)\mathrm {D} m\\\\+&\mu \operatorname {S} \left({\frac {x'\operatorname {d} ^{2}z'-z'\operatorname {d} ^{2}x'}{\operatorname {d} t^{2}}}\right)\mathrm {D} m\\\\+&\nu \operatorname {S} \left({\frac {y'\operatorname {d} ^{2}x'-x'\operatorname {d} ^{2}y'}{\operatorname {d} t^{2}}}\right)\mathrm {D} m\end{aligned}}\right\}={\begin{aligned}&-g\mu \iiint \Delta x'\operatorname {d} x'\operatorname {d} y'\operatorname {d} z'\\&+g\lambda \iiint \Delta y'\operatorname {d} x'\operatorname {d} y'\operatorname {d} z'\,;\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fa7ba013377fccc4ea3eced0b5d8317d878062c5)