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Lagrange - Œuvres (1867) vol. 1.djvu/333
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{\displaystyle {\frac {1}{4}}\mathrm {L} ^{(\mathrm {X,Y} +qt,\mathrm {Z} -rt)}+{\frac {1}{4}}\mathrm {L} ^{(\mathrm {X,Y} -qt,\mathrm {Z} +rt)}+{\frac {1}{4}}\mathrm {L} ^{(\mathrm {X,Y} +qt,\mathrm {Z} -rt)}+{\frac {1}{4}}\mathrm {L} ^{(\mathrm {X,Y} -qt,\mathrm {Z} -rt)}}
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{\displaystyle =\mathrm {L} +{\frac {1}{2}}t^{2}\left(q^{2}{\frac {d^{2}\mathrm {L} }{d\mathrm {Y} ^{2}}}+r^{2}{\frac {d^{2}\mathrm {L} }{d\mathrm {Z} ^{2}}}\right)}
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{\displaystyle +{\frac {1}{2.3.4}}t^{4}\left(q^{4}{\frac {d^{4}\mathrm {L} }{d\mathrm {Y} ^{4}}}+6q^{2}r^{2}{\frac {d^{4}\mathrm {L} }{d\mathrm {Y} ^{2}d\mathrm {Z} ^{2}}}+r^{4}{\frac {d^{4}\mathrm {L} }{d\mathrm {Z} ^{4}}}\right),}
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{\displaystyle {\begin{aligned}&\quad \ {\frac {1}{8}}\mathrm {L} ^{(\mathrm {X} +pt,\mathrm {Y} +qt,\mathrm {Z} +rt)}+{\frac {1}{8}}\mathrm {L} ^{(\mathrm {X} +pt,\mathrm {Y} -qt,\mathrm {Z} +rt)}\\&+{\frac {1}{8}}\mathrm {L} ^{(\mathrm {X} -pt,\mathrm {Y} -qt,\mathrm {Z} +rt)}+{\frac {1}{8}}\mathrm {L} ^{(\mathrm {X} -pt,\mathrm {Y} -qt,\mathrm {Z} -rt)}\\&-{\frac {1}{8}}\mathrm {L} ^{(\mathrm {X} +pt,\mathrm {Y} -qt,\mathrm {Z} +rt)}-{\frac {1}{8}}\mathrm {L} ^{(\mathrm {X} +pt,\mathrm {Y} -qt,\mathrm {Z} -rt)}\\&-{\frac {1}{8}}\mathrm {L} ^{(\mathrm {X} -pt,\mathrm {Y} +qt,\mathrm {Z} +rt)}-{\frac {1}{8}}\mathrm {L} ^{(\mathrm {X} -pt,\mathrm {Y} +qt,\mathrm {Z} -rt)}\\=&{\frac {1}{2}}2pqt^{2}{\frac {d^{2}\mathrm {L} }{d\mathrm {X} d\mathrm {Y} }}+{\frac {1}{2.3.4}}t^{4}\left(4p^{3}q{\frac {d^{4}\mathrm {L} }{d\mathrm {X} ^{3}d\mathrm {Y} }}+4pq^{3}{\frac {d^{4}\mathrm {L} }{d\mathrm {X} d\mathrm {Y} ^{3}}}\right.\\\end{aligned}}}
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{\displaystyle +\left.12pqr^{2}{\frac {d^{4}\mathrm {L} }{d\mathrm {X} d\mathrm {Y} d\mathrm {Z} ^{2}}}\right),}
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{\displaystyle {\begin{aligned}&\quad \ {\frac {1}{8}}\mathrm {L} ^{(\mathrm {X} +pt,\mathrm {Y} +qt,\mathrm {Z} +rt)}+{\frac {1}{8}}\mathrm {L} ^{(\mathrm {X} +pt,\mathrm {Y} -qt,\mathrm {Z} +rt)}\\&+{\frac {1}{8}}\mathrm {L} ^{(\mathrm {X} -pt,\mathrm {Y} +qt,\mathrm {Z} -rt)}+{\frac {1}{8}}\mathrm {L} ^{(\mathrm {X} -pt,\mathrm {Y} -qt,\mathrm {Z} -rt)}\\&-{\frac {1}{8}}\mathrm {L} ^{(\mathrm {X} +pt,\mathrm {Y} +qt,\mathrm {Z} -rt)}-{\frac {1}{8}}\mathrm {L} ^{(\mathrm {X} +pt,\mathrm {Y} -qt,\mathrm {Z} -rt)}\\&-{\frac {1}{8}}\mathrm {L} ^{(\mathrm {X} -pt,\mathrm {Y} +qt,\mathrm {Z} +rt)}-{\frac {1}{8}}\mathrm {L} ^{(\mathrm {X} -pt,\mathrm {Y} -qt,\mathrm {Z} +rt)}\\=&{\frac {1}{2}}2prt^{2}{\frac {d^{2}\mathrm {L} }{d\mathrm {X} d\mathrm {Z} }}+{\frac {1}{2.3.4}}t^{4}\left(4p^{3}r{\frac {d^{4}\mathrm {L} }{d\mathrm {X} ^{3}d\mathrm {Z} }}+4pr^{3}{\frac {d^{4}\mathrm {L} }{d\mathrm {X} d\mathrm {Z} ^{3}}}\right.\\\end{aligned}}}
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{\displaystyle +\left.12pq^{2}r{\frac {d^{4}\mathrm {L} }{d\mathrm {X} d\mathrm {Y} ^{2}d\mathrm {Z} }}\right).}
On formera de pareilles formules à l’égard des expressions
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{\displaystyle \mathrm {M} ^{(\mathrm {X} +pt,\mathrm {Y} +qt,\mathrm {Z} +rt)}\quad {\text{et}}\quad \mathrm {N} ^{(\mathrm {X} +pt,\mathrm {Y} +qt,\mathrm {Z} +rt)}.}
