Page:Lagrange - Œuvres (1867) vol. 1.djvu/334

et en donnant des valeurs convenables aux constantes indéterminées $p,q,r$ on changera, par les substitutions, l’équation (F) en celle-ci :

(L)$\quad \mathrm {L} +{\frac {1}{2}}t^{2}ck\mathrm {L} +{\frac {1}{2.3.4}}t^{4}c^{2}k^{2}\mathrm {L}$

${\begin{aligned}=&\mathrm {L} ^{(\mathrm {X,Y,Z} )}\\&-{\frac {1}{4}}\mathrm {L} ^{\left(\mathrm {X} ,\mathrm {Y} +t{\sqrt {\frac {c}{6}}},\mathrm {Z} +t{\sqrt {\frac {c}{6}}}\right)}-{\frac {1}{4}}\mathrm {L} ^{\left(\mathrm {X} ,\mathrm {Y} -t{\sqrt {\frac {c}{6}}},\mathrm {Z} +t{\sqrt {\frac {c}{6}}}\right)}\\&-{\frac {1}{4}}\mathrm {L} ^{\left(\mathrm {X} ,\mathrm {Y} +t{\sqrt {\frac {c}{6}}},\mathrm {Z} -t{\sqrt {\frac {c}{6}}}\right)}-{\frac {1}{4}}\mathrm {L} ^{\left(\mathrm {X} ,\mathrm {Y} -t{\sqrt {\frac {c}{6}}},\mathrm {Z} -t{\sqrt {\frac {c}{6}}}\right)}\\&+{\frac {1}{8}}\mathrm {L} ^{\left(\mathrm {X} +t{\sqrt {c}},\mathrm {Y} +t{\sqrt {\frac {c}{6}}},\mathrm {Z} +t{\sqrt {\frac {c}{6}}}\right)}+{\frac {1}{8}}\mathrm {L} ^{\left(\mathrm {X} +t{\sqrt {c}},\mathrm {Y} -t{\sqrt {\frac {c}{6}}},\mathrm {Z} +t{\sqrt {\frac {c}{6}}}\right)}\\&+{\frac {1}{8}}\mathrm {L} ^{\left(\mathrm {X} +t{\sqrt {c}},\mathrm {Y} +t{\sqrt {\frac {c}{6}}},\mathrm {Z} -t{\sqrt {\frac {c}{6}}}\right)}+{\frac {1}{8}}\mathrm {L} ^{\left(\mathrm {X} +t{\sqrt {c}},\mathrm {Y} -t{\sqrt {\frac {c}{6}}},\mathrm {Z} -t{\sqrt {\frac {c}{6}}}\right)}\\&+{\frac {1}{8}}\mathrm {L} ^{\left(\mathrm {X} -t{\sqrt {c}},\mathrm {Y} +t{\sqrt {\frac {c}{6}}},\mathrm {Z} +t{\sqrt {\frac {c}{6}}}\right)}+{\frac {1}{8}}\mathrm {L} ^{\left(\mathrm {X} -t{\sqrt {c}},\mathrm {Y} -t{\sqrt {\frac {c}{6}}},\mathrm {Z} +t{\sqrt {\frac {c}{6}}}\right)}\\&+{\frac {1}{8}}\mathrm {L} ^{\left(\mathrm {X} -t{\sqrt {c}},\mathrm {Y} +t{\sqrt {\frac {c}{6}}},\mathrm {Z} -t{\sqrt {\frac {c}{6}}}\right)}+{\frac {1}{8}}\mathrm {L} ^{\left(\mathrm {X} -t{\sqrt {c}},\mathrm {Y} -t{\sqrt {\frac {c}{6}}},\mathrm {Z} -t{\sqrt {\frac {c}{6}}}\right)}\\&+{\frac {1}{8}}\mathrm {M} ^{\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}}\mathrm {M} ^{\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}}\mathrm {M} ^{\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}}\mathrm {M} ^{\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}}\mathrm {M} ^{\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}}\mathrm {M} ^{\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}}\mathrm {M} ^{\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}}\mathrm {M} ^{\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}}\mathrm {N} ^{\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}}\mathrm {N} ^{\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}}$