térale nulle ; et on a, entre ces quatorze parties, les dix relations suivantes :
1.
o ![{\displaystyle \left\{{\begin{array}{ll}(pp')=3\left({\sqrt[{3}]{T''}}+{\sqrt[{3}]{T'''}}\right){\sqrt[{3}]{T''T'''}},&(p''p''')=3\left({\sqrt[{3}]{T}}+{\sqrt[{3}]{T'}}\right){\sqrt[{3}]{TT'}},\\\\(pp'')=3\left({\sqrt[{3}]{T'}}+{\sqrt[{3}]{T'''}}\right){\sqrt[{3}]{T'T'''}},&(p'p''')=3\left({\sqrt[{3}]{T}}+{\sqrt[{3}]{T''}}\right){\sqrt[{3}]{TT''}},\\\\(p'p'')=3\left({\sqrt[{3}]{T}}+{\sqrt[{3}]{T'''}}\right){\sqrt[{3}]{TT'''}},&(pp''')=3\left({\sqrt[{3}]{T'}}+{\sqrt[{3}]{T''}}\right){\sqrt[{3}]{T'T''}},\end{array}}\right.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6b467ce738452901f4a9f34e8d402be9a6d57609)
2.
o ![{\displaystyle \left\{{\begin{aligned}P^{3}\ \ &=216T'T''T''',\\\\P'^{3}\,\ &=216T''T'''T,\\\\P''^{3}\ &=216T'''TT',\\\\P'''^{3}&=216TT'T''.\end{aligned}}\right.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4c801929489e28cf754eac82be11e93cedb3a60d)
Démonstration. Ces relations résultent tout naturellement de ce que les quatre systèmes de sept corps
![{\displaystyle {\begin{aligned}&P,T',T'',T''',(p'p''),(p'p'''),(p''p'''),\\\\&P',T'',T''',T,(pp''),(pp'''),(p''p'''),\\\\&P'',T''',T,T',(pp'),(pp'''),(p'p'''),\\\\&P''',T,T',T'',(pp'),(pp''),(p'p''),\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f151d45d5002425a484164e9babd4b46ad9f3ee6)
se trouvent, les uns par rapport aux autres, dans le cas du théorème IV.