Page:Lagrange - Œuvres (1867) vol. 1.djvu/336

cette manière, toutes les quantités exprimées généralement par

$\mathrm {L} ^{\left(\mathrm {X} +pt,\mathrm {Y} +qt,\mathrm {Z} +rt\right)},\quad \mathrm {M} ^{\left(\mathrm {X} +pt,\mathrm {Y} +qt,\mathrm {Z} +rt\right)},\quad \mathrm {N} ^{\left(\mathrm {X} +pt,\mathrm {Y} +qt,\mathrm {Z} +rt\right)},$

redeviendront

$\mathrm {L^{(X,Y,Z)},\ M^{(X,Y,Z)},\ N^{(X,Y,Z)}} \quad {\text{ou simplement}}\quad \mathrm {L,\ M,\ N} \,;$

mais à leur tour les quantités $(x),(y),(z),$ qui les multiplient, se changeront en

$(x)^{\left(\mathrm {X} -pt,\mathrm {Y} -qt,\mathrm {Z} -rt\right)},\quad (y)^{\left(\mathrm {X} -pt,\mathrm {Y} -qt,\mathrm {Z} -rt\right)},\quad (z)^{\left(\mathrm {X} -pt,\mathrm {Y} -qt,\mathrm {Z} -rt\right)}.$

Après ces transformations, on joindra ensemble tous les termes de l’équation qui se trouvent multipliés par $\mathrm {L,M,N} ,$ et on décomposera ensuite cette équation en trois portions, dont chacune devra se vérifier séparément et indépendamment des valeurs de $\mathrm {L,M}$ et $\mathrm {N} .$ On aura donc par là

${\begin{aligned}(\mathrm {P} )\quad x=&(x)^{(\mathrm {X,Y,Z} )}\\&-{\frac {1}{4}}(x)^{\left(\mathrm {X} ,\mathrm {Y} -t{\sqrt {\frac {c}{6}}},\mathrm {Z} -t{\sqrt {\frac {c}{6}}}\right)}-{\frac {1}{4}}(x)^{\left(\mathrm {X} ,\mathrm {Y} +t{\sqrt {\frac {c}{6}}},\mathrm {Z} -t{\sqrt {\frac {c}{6}}}\right)}\\&-{\frac {1}{4}}(x)^{\left(\mathrm {X} ,\mathrm {Y} -t{\sqrt {\frac {c}{6}}},\mathrm {Z} +t{\sqrt {\frac {c}{6}}}\right)}-{\frac {1}{4}}(x)^{\left(\mathrm {X} ,\mathrm {Y} +t{\sqrt {\frac {c}{6}}},\mathrm {Z} +t{\sqrt {\frac {c}{6}}}\right)}\\&+{\frac {1}{8}}(x)^{\left(\mathrm {X} -t{\sqrt {c}},\mathrm {Y} -t{\sqrt {\frac {c}{6}}},\mathrm {Z} -t{\sqrt {\frac {c}{6}}}\right)}+{\frac {1}{8}}(x)^{\left(\mathrm {X} -t{\sqrt {c}},\mathrm {Y} +t{\sqrt {\frac {c}{6}}},\mathrm {Z} -t{\sqrt {\frac {c}{6}}}\right)}\\&+{\frac {1}{8}}(x)^{\left(\mathrm {X} -t{\sqrt {c}},\mathrm {Y} -t{\sqrt {\frac {c}{6}}},\mathrm {Z} +t{\sqrt {\frac {c}{6}}}\right)}+{\frac {1}{8}}(x)^{\left(\mathrm {X} -t{\sqrt {c}},\mathrm {Y} +t{\sqrt {\frac {c}{6}}},\mathrm {Z} +t{\sqrt {\frac {c}{6}}}\right)}\\&+{\frac {1}{8}}(x)^{\left(\mathrm {X} +t{\sqrt {c}},\mathrm {Y} -t{\sqrt {\frac {c}{6}}},\mathrm {Z} -t{\sqrt {\frac {c}{6}}}\right)}+{\frac {1}{8}}(x)^{\left(\mathrm {X} +t{\sqrt {c}},\mathrm {Y} +t{\sqrt {\frac {c}{6}}},\mathrm {Z} -t{\sqrt {\frac {c}{6}}}\right)}\\&+{\frac {1}{8}}(x)^{\left(\mathrm {X} +t{\sqrt {c}},\mathrm {Y} -t{\sqrt {\frac {c}{6}}},\mathrm {Z} +t{\sqrt {\frac {c}{6}}}\right)}+{\frac {1}{8}}(x)^{\left(\mathrm {X} +t{\sqrt {c}},\mathrm {Y} +t{\sqrt {\frac {c}{6}}},\mathrm {Z} +t{\sqrt {\frac {c}{6}}}\right)}\\&+{\frac {1}{8}}(y)^{\left(\mathrm {X} -t{\sqrt {\frac {c}{2}}},\mathrm {Y} -t{\sqrt {\frac {c}{2}}},\mathrm {Z} -t{\sqrt {\frac {c}{6}}}\right)}+{\frac {1}{8}}(y)^{\left(\mathrm {X} -t{\sqrt {\frac {c}{2}}},\mathrm {Y} -t{\sqrt {\frac {c}{2}}},\mathrm {Z} +t{\sqrt {\frac {c}{6}}}\right)}\end{aligned}}$