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Lagrange - Œuvres (1867) vol. 1.djvu/340
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{\displaystyle {\begin{aligned}&+{\frac {1}{8}}\left[(x')+{\frac {d(x)}{dt}}\right]^{\left(\mathrm {X} +t{\sqrt {c}},\mathrm {Y} +t{\sqrt {\frac {c}{6}}},\mathrm {Z} -t{\sqrt {\frac {c}{6}}}\right)}\\&+{\frac {1}{8}}\left[(x')+{\frac {d(x)}{dt}}\right]^{\left(\mathrm {X} +t{\sqrt {c}},\mathrm {Y} -t{\sqrt {\frac {c}{6}}},\mathrm {Z} +t{\sqrt {\frac {c}{6}}}\right)}\\&+{\frac {1}{8}}\left[(x')+{\frac {d(x)}{dt}}\right]^{\left(\mathrm {X} +t{\sqrt {c}},\mathrm {Y} +t{\sqrt {\frac {c}{6}}},\mathrm {Z} +t{\sqrt {\frac {c}{6}}}\right)}\\&+{\frac {1}{8}}\left[(y')+{\frac {d(y)}{dt}}\right]^{\left(\mathrm {X} -t{\sqrt {\frac {c}{2}}},\mathrm {Y} -t{\sqrt {\frac {c}{2}}},\mathrm {Z} -t{\sqrt {\frac {c}{6}}}\right)}\\&+{\frac {1}{8}}\left[(y')+{\frac {d(y)}{dt}}\right]^{\left(\mathrm {X} -t{\sqrt {\frac {c}{2}}},\mathrm {Y} -t{\sqrt {\frac {c}{2}}},\mathrm {Z} +t{\sqrt {\frac {c}{6}}}\right)}\\&+{\frac {1}{8}}\left[(y')+{\frac {d(y)}{dt}}\right]^{\left(\mathrm {X} +t{\sqrt {\frac {c}{2}}},\mathrm {Y} +t{\sqrt {\frac {c}{2}}},\mathrm {Z} -t{\sqrt {\frac {c}{6}}}\right)}\\&+{\frac {1}{8}}\left[(y')+{\frac {d(y)}{dt}}\right]^{\left(\mathrm {X} +t{\sqrt {\frac {c}{2}}},\mathrm {Y} +t{\sqrt {\frac {c}{2}}},\mathrm {Z} +t{\sqrt {\frac {c}{6}}}\right)}\\&+{\frac {1}{8}}\left[(y')+{\frac {d(y)}{dt}}\right]^{\left(\mathrm {X} -t{\sqrt {\frac {c}{2}}},\mathrm {Y} +t{\sqrt {\frac {c}{2}}},\mathrm {Z} -t{\sqrt {\frac {c}{6}}}\right)}\\&+{\frac {1}{8}}\left[(y')+{\frac {d(y)}{dt}}\right]^{\left(\mathrm {X} -t{\sqrt {\frac {c}{2}}},\mathrm {Y} +t{\sqrt {\frac {c}{2}}},\mathrm {Z} +t{\sqrt {\frac {c}{6}}}\right)}\\&+{\frac {1}{8}}\left[(y')+{\frac {d(y)}{dt}}\right]^{\left(\mathrm {X} +t{\sqrt {\frac {c}{2}}},\mathrm {Y} -t{\sqrt {\frac {c}{2}}},\mathrm {Z} -t{\sqrt {\frac {c}{6}}}\right)}\\&+{\frac {1}{8}}\left[(y')+{\frac {d(y)}{dt}}\right]^{\left(\mathrm {X} +t{\sqrt {\frac {c}{2}}},\mathrm {Y} -t{\sqrt {\frac {c}{2}}},\mathrm {Z} +t{\sqrt {\frac {c}{6}}}\right)}\\&+{\frac {1}{8}}\left[(z')+{\frac {d(z)}{dt}}\right]^{\left(\mathrm {X} -t{\sqrt {\frac {c}{2}}},\mathrm {Y} -t{\sqrt {\frac {c}{6}}},\mathrm {Z} -t{\sqrt {\frac {c}{2}}}\right)}\\&+{\frac {1}{8}}\left[(z')+{\frac {d(z)}{dt}}\right]^{\left(\mathrm {X} -t{\sqrt {\frac {c}{2}}},\mathrm {Y} +t{\sqrt {\frac {c}{6}}},\mathrm {Z} -t{\sqrt {\frac {c}{2}}}\right)}\\\end{aligned}}}