Page:Joseph Louis de Lagrange - Œuvres, Tome 3.djvu/135

${\begin{aligned}\mathrm {M} ''\ ={\frac {n^{2}}{\mathrm {Z} }}{\frac {d\mathrm {Z} }{dz}}&\left[{\frac {1}{(1+m)^{2}}}+{\frac {\mathrm {Z} }{(1+m){\sqrt {-1}}}}+{\frac {\mathrm {Z} ^{2}}{\left({\sqrt {-1}}\right)^{2}}}\right.\\&+{\frac {n^{2}\mathrm {Z} ^{3}}{(1+m)\left({\sqrt {-1}}\right)^{3}}}+{\frac {n^{4}\mathrm {Z} ^{4}}{(1+m)^{2}({\sqrt {-1}})^{4}}}+{\frac {n^{6}\mathrm {Z} ^{5}}{(1+m)^{3}({\sqrt {-1}})^{5}}}+\ldots \\&-\left.{\frac {n^{2}\mathrm {Z} }{(1+m)^{3}{\sqrt {-1}}}}+{\frac {n^{4}\mathrm {Z} ^{2}}{(1+m)^{4}({\sqrt {-1}})^{2}}}-{\frac {n^{6}\mathrm {Z} ^{3}}{(1+m)^{5}({\sqrt {-1}})^{3}}}+\ldots \right],\\\\\mathrm {M} '''={\frac {n^{3}}{\mathrm {Z} }}{\frac {d\mathrm {Z} }{dz}}&\left[-{\frac {1}{(1+m)^{3}}}-{\frac {\mathrm {Z} }{(1+m)^{2}{\sqrt {-1}}}}-{\frac {\mathrm {Z} ^{2}}{(1+m)\left({\sqrt {-1}}\right)^{2}}}-{\frac {\mathrm {Z} ^{3}}{\left({\sqrt {-1}}\right)^{3}}}\right.\\&-{\frac {n^{2}\mathrm {Z} ^{4}}{(1+m)\left({\sqrt {-1}}\right)^{4}}}-{\frac {n^{4}\mathrm {Z} ^{5}}{(1+m)^{2}({\sqrt {-1}})^{5}}}-{\frac {n^{6}\mathrm {Z} ^{6}}{(1+m)^{3}({\sqrt {-1}})^{6}}}-\ldots \\&+\left.{\frac {n^{2}\mathrm {Z} }{(1+m)^{4}{\sqrt {-1}}}}-{\frac {n^{4}\mathrm {Z} ^{2}}{(1+m)^{5}({\sqrt {-1}})^{2}}}+{\frac {n^{6}\mathrm {Z} ^{3}}{(1+m)^{6}({\sqrt {-1}})^{3}}}-\ldots \right],\\&\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \\\\\mathrm {N} '\ \ ={\frac {n}{\mathrm {Z} }}{\frac {d\mathrm {Z} }{dz}}&\left[-{\frac {1}{1+m}}+{\frac {\mathrm {Z} }{\sqrt {-1}}}-{\frac {n^{2}\mathrm {Z} ^{2}}{(1+m)({\sqrt {-1}})^{2}}}+{\frac {n^{4}\mathrm {Z} ^{3}}{(1+m)^{2}({\sqrt {-1}})^{2}}}-\ldots \right.\\&\left.-{\frac {n^{2}\mathrm {Z} }{(1+m)^{2}{\sqrt {-1}}}}-{\frac {n^{4}\mathrm {Z} ^{2}}{(1+m)^{3}({\sqrt {-1}})^{2}}}-{\frac {n^{6}\mathrm {Z} ^{3}}{(1+m)^{4}({\sqrt {-1}})^{3}}}-\ldots \right],\\\\\mathrm {N} ''\ ={\frac {n^{2}}{\mathrm {Z} }}{\frac {d\mathrm {Z} }{dz}}&\left[{\frac {1}{(1+m)^{2}}}-{\frac {\mathrm {Z} }{(1+m){\sqrt {-1}}}}+{\frac {\mathrm {Z} ^{2}}{\left({\sqrt {-1}}\right)^{2}}}\right.\\&-{\frac {n^{2}\mathrm {Z} ^{3}}{(1+m)\left({\sqrt {-1}}\right)^{3}}}+{\frac {n^{4}\mathrm {Z} ^{4}}{(1+m)^{2}({\sqrt {-1}})^{4}}}-{\frac {n^{6}\mathrm {Z} ^{5}}{(1+m)^{3}({\sqrt {-1}})^{5}}}+\ldots \\&+\left.{\frac {n^{2}\mathrm {Z} }{(1+m)^{3}{\sqrt {-1}}}}+{\frac {n^{4}\mathrm {Z} ^{2}}{(1+m)^{4}({\sqrt {-1}})^{2}}}+{\frac {n^{6}\mathrm {Z} ^{3}}{(1+m)^{5}({\sqrt {-1}})^{3}}}+\ldots \right],\\\\\mathrm {N} '''={\frac {n^{3}}{\mathrm {Z} }}{\frac {d\mathrm {Z} }{dz}}&\left[-{\frac {1}{(1+m)^{3}}}+{\frac {\mathrm {Z} }{(1+m)^{2}{\sqrt {-1}}}}-{\frac {\mathrm {Z} ^{2}}{(1+m)\left({\sqrt {-1}}\right)^{2}}}+{\frac {\mathrm {Z} ^{3}}{\left({\sqrt {-1}}\right)^{3}}}\right.\\&-{\frac {n^{2}\mathrm {Z} ^{4}}{(1+m)\left({\sqrt {-1}}\right)^{4}}}+{\frac {n^{4}\mathrm {Z} ^{5}}{(1+m)^{2}({\sqrt {-1}})^{5}}}-{\frac {n^{6}\mathrm {Z} ^{6}}{(1+m)^{3}({\sqrt {-1}})^{6}}}+\ldots \\&-\left.{\frac {n^{2}\mathrm {Z} }{(1+m)^{4}{\sqrt {-1}}}}-{\frac {n^{4}\mathrm {Z} ^{2}}{(1+m)^{5}({\sqrt {-1}})^{2}}}-{\frac {n^{6}\mathrm {Z} ^{3}}{(1+m)^{6}({\sqrt {-1}})^{3}}}-\ldots \right],\\&\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \end{aligned}}$