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Joseph Louis de Lagrange - Œuvres, Tome 3.djvu/365
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Donc
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{\displaystyle \mathrm {Y} ={\Bigl [}t-f\left[(x')(y)(z)\right]{\Bigr ]}\times {\Bigl [}t-f\left[(x'')(y)(z)\right]{\Bigr ]}\times {\Bigl [}t-f\left[(x''')(y)(z)\right]{\Bigr ]},}
De là on aura
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{\displaystyle {\begin{alignedat}{3}\mathrm {Y} '\ \ =&{\Bigl [}t-f\left[(x')(x'\ \ )(z)\right]{\Bigr ]}&&\times {\Bigl [}t-f\left[(x'')(x'\ )(z)\right]{\Bigr ]}&&\times {\Bigl [}t-f\left[(x''')(x'\,\ )(z)\right]{\Bigr ]},\\\mathrm {Y} ''\ =&{\Bigl [}t-f\left[(x')(x''\ )(z)\right]{\Bigr ]}&&\times {\Bigl [}t-f\left[(x'')(x''\,)(z)\right]{\Bigr ]}&&\times {\Bigl [}t-f\left[(x''')(x''\ )(z)\right]{\Bigr ]},\\\mathrm {Y} '''=&{\Bigl [}t-f\left[(x')(x''')(z)\right]{\Bigr ]}&&\times {\Bigl [}t-f\left[(x'')(x''')(z)\right]{\Bigr ]}&&\times {\Bigl [}t-f\left[(x''')(x''')(z)\right]{\Bigr ]}.\end{alignedat}}}
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{\displaystyle {\begin{alignedat}{3}\mathrm {Z} \quad &={\Bigl [}t-f\left[(x')(x'')(z)\right]{\Bigr ]}&&\times {\Bigl [}t-f\left[(x')(x''')(z)\right]{\Bigr ]}&&\times {\Bigl [}t-f\left[(x'')(x''')(z)\right]{\Bigr ]}\\&\times {\Bigl [}t-f\left[(x'')(x')(z)\right]{\Bigr ]}&&\times {\Bigl [}t-f\left[(x''')(x')(z)\right]{\Bigr ]}&&\times {\Bigl [}t-f\left[(x''')(x'')(z)\right]{\Bigr ]}\\&\times {\Bigl [}t-f\left[(x')^{2}(z)\right]{\Bigr ]}&&\times {\Bigl [}t-f\left[(x'')^{2}(z)\right]{\Bigr ]}&&\times {\Bigl [}t-f\left[(x''')^{2}(z)\right]{\Bigr ]}.\end{alignedat}}}
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{\displaystyle {\begin{alignedat}{3}\mathrm {Z} '\ \ &={\Bigl [}t-f\left[(x')(x'')(x')\right]{\Bigr ]}&&\times {\Bigl [}t-f\left[(x')(x''')(x')\right]{\Bigr ]}&&\times {\Bigl [}t-f\left[(x'')(x''')(x')\right]{\Bigr ]}\\&\times {\Bigl [}t-f\left[(x'')(x')^{2}\right]{\Bigr ]}&&\times {\Bigl [}t-f\left[(x''')(x')^{2}\right]{\Bigr ]}&&\times {\Bigl [}t-f\left[(x''')(x'')(x')\right]{\Bigr ]}\\&\times {\Bigl [}t-f\left[(x')^{3}\right]{\Bigr ]}&&\times {\Bigl [}t-f\left[(x'')^{2}(x')\right]{\Bigr ]}&&\times {\Bigl [}t-f\left[(x''')^{2}(x')\right]{\Bigr ]},\\\\\mathrm {Z} ''\ &={\Bigl [}t-f\left[(x')(x'')^{2}\right]{\Bigr ]}&&\times {\Bigl [}t-f\left[(x')(x''')(x'')\right]{\Bigr ]}&&\times {\Bigl [}t-f\left[(x'')(x''')(x'')\right]{\Bigr ]}\\&\times {\Bigl [}t-f\left[(x'')(x')(x'')\right]{\Bigr ]}&&\times {\Bigl [}t-f\left[(x''')(x')(x'')\right]{\Bigr ]}&&\times {\Bigl [}t-f\left[(x''')(x'')^{2}\right]{\Bigr ]}\\&\times {\Bigl [}t-f\left[(x')^{2}(x'')\right]{\Bigr ]}&&\times {\Bigl [}t-f\left[(x'')^{3}\right]{\Bigr ]}&&\times {\Bigl [}t-f\left[(x''')^{2}(x'')\right]{\Bigr ]},\\\\\mathrm {Z} '''&={\Bigl [}t-f\left[(x')(x'')(x''')\right]{\Bigr ]}&&\times {\Bigl [}t-f\left[(x')(x''')^{2}\right]{\Bigr ]}&&\times {\Bigl [}t-f\left[(x'')(x''')^{2}\right]{\Bigr ]}\\&\times {\Bigl [}t-f\left[(x'')(x')(x''')\right]{\Bigr ]}&&\times {\Bigl [}t-f\left[(x''')(x')(x''')\right]{\Bigr ]}&&\times {\Bigl [}t-f\left[(x''')(x'')(x''')\right]{\Bigr ]}\\&\times {\Bigl [}t-f\left[(x')^{2}(x''')\right]{\Bigr ]}&&\times {\Bigl [}t-f\left[(x'')^{2}(x''')\right]{\Bigr ]}&&\times {\Bigl [}t-f\left[(x''')^{3}\right]{\Bigr ]}.\end{alignedat}}}
Donc enfin, si l’on multiplie ces trois quantités ensemble, et qu’on fasse,
![{\displaystyle \mathrm {Y} ={\Bigl [}t-f\left[(x')(y)(z)\right]{\Bigr ]}\times {\Bigl [}t-f\left[(x'')(y)(z)\right]{\Bigr ]}\times {\Bigl [}t-f\left[(x''')(y)(z)\right]{\Bigr ]},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3d977ae1df313afe6089b91c894c5882c6cf747f)
![{\displaystyle {\begin{alignedat}{3}\mathrm {Y} '\ \ =&{\Bigl [}t-f\left[(x')(x'\ \ )(z)\right]{\Bigr ]}&&\times {\Bigl [}t-f\left[(x'')(x'\ )(z)\right]{\Bigr ]}&&\times {\Bigl [}t-f\left[(x''')(x'\,\ )(z)\right]{\Bigr ]},\\\mathrm {Y} ''\ =&{\Bigl [}t-f\left[(x')(x''\ )(z)\right]{\Bigr ]}&&\times {\Bigl [}t-f\left[(x'')(x''\,)(z)\right]{\Bigr ]}&&\times {\Bigl [}t-f\left[(x''')(x''\ )(z)\right]{\Bigr ]},\\\mathrm {Y} '''=&{\Bigl [}t-f\left[(x')(x''')(z)\right]{\Bigr ]}&&\times {\Bigl [}t-f\left[(x'')(x''')(z)\right]{\Bigr ]}&&\times {\Bigl [}t-f\left[(x''')(x''')(z)\right]{\Bigr ]}.\end{alignedat}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1209d5fbf0e7c4c416a8178cc7b1201004e968d8)
![{\displaystyle {\begin{alignedat}{3}\mathrm {Z} \quad &={\Bigl [}t-f\left[(x')(x'')(z)\right]{\Bigr ]}&&\times {\Bigl [}t-f\left[(x')(x''')(z)\right]{\Bigr ]}&&\times {\Bigl [}t-f\left[(x'')(x''')(z)\right]{\Bigr ]}\\&\times {\Bigl [}t-f\left[(x'')(x')(z)\right]{\Bigr ]}&&\times {\Bigl [}t-f\left[(x''')(x')(z)\right]{\Bigr ]}&&\times {\Bigl [}t-f\left[(x''')(x'')(z)\right]{\Bigr ]}\\&\times {\Bigl [}t-f\left[(x')^{2}(z)\right]{\Bigr ]}&&\times {\Bigl [}t-f\left[(x'')^{2}(z)\right]{\Bigr ]}&&\times {\Bigl [}t-f\left[(x''')^{2}(z)\right]{\Bigr ]}.\end{alignedat}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c46364a955f0e63f20f908056ff6b38ac25f7ab2)
![{\displaystyle {\begin{alignedat}{3}\mathrm {Z} '\ \ &={\Bigl [}t-f\left[(x')(x'')(x')\right]{\Bigr ]}&&\times {\Bigl [}t-f\left[(x')(x''')(x')\right]{\Bigr ]}&&\times {\Bigl [}t-f\left[(x'')(x''')(x')\right]{\Bigr ]}\\&\times {\Bigl [}t-f\left[(x'')(x')^{2}\right]{\Bigr ]}&&\times {\Bigl [}t-f\left[(x''')(x')^{2}\right]{\Bigr ]}&&\times {\Bigl [}t-f\left[(x''')(x'')(x')\right]{\Bigr ]}\\&\times {\Bigl [}t-f\left[(x')^{3}\right]{\Bigr ]}&&\times {\Bigl [}t-f\left[(x'')^{2}(x')\right]{\Bigr ]}&&\times {\Bigl [}t-f\left[(x''')^{2}(x')\right]{\Bigr ]},\\\\\mathrm {Z} ''\ &={\Bigl [}t-f\left[(x')(x'')^{2}\right]{\Bigr ]}&&\times {\Bigl [}t-f\left[(x')(x''')(x'')\right]{\Bigr ]}&&\times {\Bigl [}t-f\left[(x'')(x''')(x'')\right]{\Bigr ]}\\&\times {\Bigl [}t-f\left[(x'')(x')(x'')\right]{\Bigr ]}&&\times {\Bigl [}t-f\left[(x''')(x')(x'')\right]{\Bigr ]}&&\times {\Bigl [}t-f\left[(x''')(x'')^{2}\right]{\Bigr ]}\\&\times {\Bigl [}t-f\left[(x')^{2}(x'')\right]{\Bigr ]}&&\times {\Bigl [}t-f\left[(x'')^{3}\right]{\Bigr ]}&&\times {\Bigl [}t-f\left[(x''')^{2}(x'')\right]{\Bigr ]},\\\\\mathrm {Z} '''&={\Bigl [}t-f\left[(x')(x'')(x''')\right]{\Bigr ]}&&\times {\Bigl [}t-f\left[(x')(x''')^{2}\right]{\Bigr ]}&&\times {\Bigl [}t-f\left[(x'')(x''')^{2}\right]{\Bigr ]}\\&\times {\Bigl [}t-f\left[(x'')(x')(x''')\right]{\Bigr ]}&&\times {\Bigl [}t-f\left[(x''')(x')(x''')\right]{\Bigr ]}&&\times {\Bigl [}t-f\left[(x''')(x'')(x''')\right]{\Bigr ]}\\&\times {\Bigl [}t-f\left[(x')^{2}(x''')\right]{\Bigr ]}&&\times {\Bigl [}t-f\left[(x'')^{2}(x''')\right]{\Bigr ]}&&\times {\Bigl [}t-f\left[(x''')^{3}\right]{\Bigr ]}.\end{alignedat}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/44bed8b509ddf2ea32268506c4bc5b69f797e307)
