composée de ces prismes
![{\displaystyle ymnz\mu \nu +ylnz\lambda \nu +lmn\lambda \mu \nu -ylmz\lambda \mu ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e09d47c0e393ad61a027f5eaad035ee39c21b118)
et, prenant la solidité de chaque part, on trouvera cette solidité
![{\displaystyle \left.{\begin{alignedat}{4}&+{\frac {1}{3}}(yz&&+l\lambda &&+n\nu &&)\Delta yln\\&+{\frac {1}{3}}(yz&&+m\mu &&+n\nu &&)\Delta ymn\\&+{\frac {1}{3}}(l\lambda &&+m\mu &&+n\nu &&)\Delta lmn\\&-{\frac {1}{3}}(yz&&+l\lambda &&+m\mu &&)\Delta ylm\end{alignedat}}\right\}=\left\{{\begin{alignedat}{4}&+{\frac {1}{3}}(3z&&+\mathrm {S} \alpha &&+\quad \mathrm {V} \gamma &&)\Delta yln\\&+{\frac {1}{3}}(3z&&+\mathrm {T} \beta &&+\quad \mathrm {V} \gamma &&)\Delta ymn\\&+{\frac {1}{3}}(3z&&+\mathrm {T} \alpha &&+\mathrm {T} \beta +\mathrm {V} \gamma &&)\Delta lmn\\&-{\frac {1}{3}}(3z&&+\mathrm {S} \alpha &&+\quad \mathrm {T} \beta &&)\Delta lm\end{alignedat}}\right\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/41439a47e33eebfe8e052dd498dc55233f0170bd)
![{\displaystyle =\left\{{\begin{aligned}&-{\frac {1}{3}}\mathrm {S} \alpha \Delta ymn,\\&-{\frac {1}{3}}\mathrm {T} \beta \Delta yln,\\&+{\frac {1}{3}}\mathrm {V} \gamma \Delta ylm.\end{aligned}}\right.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/26fb96d549f2ecf57677a96af45fdb54447f2d54)
Ensuite, on trouve les aires de ces triangles, à cause de
comme il suit :
![{\displaystyle {\begin{aligned}\Delta ymn=&{\frac {1}{2}}x\mathrm {M} (2y+\mathrm {Q} \beta )+{\frac {1}{2}}\mathrm {MN} (2y+\mathrm {Q} \beta +\mathrm {R} \gamma )-{\frac {1}{2}}x\mathrm {N} (2y+\mathrm {R} \gamma )\\=&{\frac {1}{2}}\mathrm {Q} \beta \times x\mathrm {N} -{\frac {1}{2}}\mathrm {R} \gamma \times x\mathrm {M} ={\frac {1}{2}}\beta \gamma \mathrm {\left(NQ-MR\right)} ,\\\Delta yln\ \ =&{\frac {1}{2}}x\mathrm {N} \,(2y+\mathrm {R} \gamma )+{\frac {1}{2}}\mathrm {LN} \ (2y+\mathrm {P} \alpha +\mathrm {R} \gamma )-{\frac {1}{2}}x\mathrm {L} (2y+\mathrm {P} \alpha )\\=&{\frac {1}{2}}\mathrm {R} \gamma \,\times x\mathrm {L} -{\frac {1}{2}}\mathrm {P} \alpha \times x\mathrm {N} \,={\frac {1}{2}}\alpha \gamma \mathrm {\left(LR-NP\right)} ,\\\Delta ylm\,=&{\frac {1}{2}}x\mathrm {M} (2y+\mathrm {Q} \beta )+{\frac {1}{2}}\mathrm {LM} \,(2y+\mathrm {P} \alpha +\mathrm {Q} \beta )-{\frac {1}{2}}x\mathrm {L} (2y+\mathrm {P} \alpha )\\=&{\frac {1}{2}}\mathrm {Q} \beta \times x\mathrm {L} \ -{\frac {1}{2}}\mathrm {P} \alpha \times x\mathrm {M} ={\frac {1}{2}}\alpha \beta \mathrm {\left(LQ-MP\right)} .\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d106bb60231de6296f817f234d6495b23a91b3f3)
De là, nous tirons la solidité de notre pyramide
dans l’état d’agitation
![{\displaystyle -{\frac {1}{6}}\alpha \beta \gamma \mathrm {S(NQ-MR)-{\frac {1}{6}}\alpha \beta \gamma T(LR-NP)+{\frac {1}{6}}\alpha \beta \gamma V(LQ-MP)} ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2d4effeac4ccf618777bee66ac1a7a7891650ff0)
et, partant, la densité du milieu agité en
sera
![{\displaystyle 1:\mathrm {\left(LQV-MPV+MRS-NQS+NPT-LRT\right)} ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2147659f9dc8e1c4f8c53b3f955c23a0bf3c7df2)
et, posant
pour la hauteur de la colonne qui y balance l'élasticité, nous aurons
![{\displaystyle \Pi =h:\mathrm {\left(LQV-MPV+MRS-NQS+NPT-LRT\right)} ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0bf5b9bd44d23740e51065e018fb389322e4d82d)
laquelle étant une fonction des trois variables
posons
![{\displaystyle d\Pi =\mathrm {E} d\mathrm {X} +\mathrm {F} d\mathrm {Y} +\mathrm {G} d\mathrm {Z} ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bb8f244be259cbf424296dd216833a97672d86bd)
de sorte que
![{\displaystyle \mathrm {E={\frac {\partial \Pi }{\partial X}},\qquad F={\frac {\partial \Pi }{\partial Y}},\qquad G={\frac {\partial \Pi }{\partial Z}}} .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3df7aadbd103fd32ae829912fc703e9e20e9fd9e)
Soit, pour abréger,
![{\displaystyle \mathrm {LQV-MPV+MRS-NQS+NPT-LRT=K} ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6b478a37c245ef52d84cc30c1e5898d5d9e84150)