Page:Joseph Louis de Lagrange - Œuvres, Tome 12.djvu/157

149

SECONDE PARTIE. — SECTION VII.

mais les valeurs $\cos \mathrm {I} ''_{_{'}},\,\cos \mathrm {I} '''_{_{'}},\,\cos \mathrm {I} '''_{_{''}},\ldots$ deviendront, par les mêmes substitutions,

${\begin{aligned}\cos \mathrm {I} ''_{_{'}}\ =&{\sqrt {1-p'^{2}\ -q'^{2}\ }}{\sqrt {1-p''^{2}\ -q''^{2}\ }}+p'p''\ \ +q'q'',\\\cos \mathrm {I} '''_{_{'}}=&{\sqrt {1-p'^{2}\ -q'^{2}\ }}{\sqrt {1-p'''^{2}-q'''^{2}}}+p'p'''\ +q'q''',\\\cos \mathrm {I} '''_{_{''}}=&{\sqrt {1-p''^{2}-q''^{2}}}{\sqrt {1-p'''^{2}-q'''^{2}}}+p''p'''+q''q''',\\\ldots \ldots \ldots &\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots .\end{aligned}}$

Faisant ces substitutions et exécutant les différentiations relatives à $p',\,q',\,p'',\,q'',\ldots ,$

${\begin{aligned}{\frac {dp'}{dt}}=&-{\frac {\mathrm {m} ''\ a'a''\ [a',a'']_{1}}{4{\sqrt {\mathrm {g} 'a'}}}}\left(q'\ {\sqrt {1-p''^{2}\ -q''^{2}\ }}-q''{\sqrt {1-p'^{2}-q'^{2}}}\right)\\&-{\frac {\mathrm {m} '''a'a'''[a',a''']_{1}}{4{\sqrt {\mathrm {g} 'a'}}}}\left(q'\ {\sqrt {1-p'''^{2}-q'''^{2}}}-q'''{\sqrt {1-p'^{2}-q'^{2}}}\right)\\&-\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots ,\\{\frac {dq'}{dt}}=&+{\frac {\mathrm {m} ''\ a'a''\ [a',a'']_{1}}{4{\sqrt {\mathrm {g} 'a'}}}}\left(p'\ {\sqrt {1-p''^{2}\ -q''^{2}\ }}-p''{\sqrt {1-p'^{2}-q'^{2}}}\right)\\&+{\frac {\mathrm {m} '''a'a'''[a',a''']_{1}}{4{\sqrt {\mathrm {g} 'a'}}}}\left(p'\ {\sqrt {1-p'''^{2}-q'''^{2}}}-p'''{\sqrt {1-p'^{2}-q'^{2}}}\right)\\&+\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots ,\\{\frac {dp''}{dt}}=&-{\frac {\mathrm {m} '\ \,a'a''\ [a',a''\ ]_{1}}{4{\sqrt {\mathrm {g} ''a''}}}}\left(q''{\sqrt {1-p'^{2}\ \ -q'^{2}\ \ }}-q'{\sqrt {1-p''^{2}-q''^{2}}}\right)\\&-{\frac {\mathrm {m} '''a'a'''[a',a''']_{1}}{4{\sqrt {\mathrm {g} ''a''}}}}\ \left(q''{\sqrt {1-p'''^{2}-q'''^{2}}}-q'''{\sqrt {1-p''^{2}-q''^{2}}}\right)\\&-\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots ,\\{\frac {dq''}{dt}}=&+{\frac {\mathrm {m} '\ a'a''\ [a'\ ,a''\ ]_{1}}{4{\sqrt {\mathrm {g} ''a''}}}}\left(p''\ {\sqrt {1-p'^{2}\ \ -q'^{2}\ }}-p'\ {\sqrt {1-p''^{2}-q''^{2}}}\right)\\&+{\frac {\mathrm {m} '''a''a'''[a'',a''']_{1}}{4{\sqrt {\mathrm {g} ''a''}}}}\left(p''{\sqrt {1-p'''^{2}-q'''^{2}}}-p'''{\sqrt {1-p''^{2}-q''^{2}}}\right)\\&+\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots ,\end{aligned}}$

· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·

108. Ces équations ont lieu, quelles que soient les valeurs des variables $p',\,q',\,p'',\,q'',\ldots$ parce que nos formules ne supposent point