Page:Joseph Louis de Lagrange - Œuvres, Tome 5.djvu/194

$(1,0),\ldots$ pour désigner les mêmes quantités que dans le n^o 40, faisons de plus

${\begin{aligned}&[0,1]=\mathrm {T} '{\frac {\left(r^{2}+r'^{2}\right)(r,r')_{1}-3rr'(r,r')}{2{\sqrt {r}}}},\\&[0,2]=\mathrm {T} ''{\frac {\left(r^{2}+r''^{2}\right)(r,r'')_{1}-3rr''(r,r'')}{2{\sqrt {r}}}},\\&\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \,;\end{aligned}}$

nous aurons ces équations

${\begin{aligned}{\frac {dx}{dt}}&-\left[(0,1)+(0,2)+\ldots \right]y+[0,1]y'+[0,2]y''+\ldots =0,\\{\frac {dy}{dt}}&+\left[(0,1)+(0,2)+\ldots \right]x-[0,1]x'-[0,2]x''-\ldots =0\,;\end{aligned}}$

et, faisant pareillement

${\begin{aligned}&[1,0]=T'\ \ {\frac {\left(r'^{2}+r^{2}\right)(r',r)_{1}-3r'r(r',r)}{2{\sqrt {r'}}}},\\&[1,2]=T''{\frac {\left(r'^{2}+r''^{2}\right)(r',r'')_{1}-3r'r''(r',r'')}{2{\sqrt {r'}}}},\\&\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots ,\\&[2,0]=T\ {\frac {\left(r''^{2}+r^{2}\right)(r'',r)_{1}-3r''r(r'',r)}{2{\sqrt {r''}}}},\\&[2,1]=T'{\frac {\left(r''^{2}+r'^{2}\right)(r'',r')_{1}-3r''r'(r'',r')}{2{\sqrt {r''}}}},\\&\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots ,\end{aligned}}$

on aura aussi

${\begin{aligned}{\frac {dx'}{dt}}&-\left[(1,0)+(1,2)+\ldots \right]y'+[1,0]y+[1,2]y''+\ldots =0,\\{\frac {dy'}{dt}}&+\left[(1,0)+(1,2)+\ldots \right]x'-[1,0]x-[1,2]x''-\ldots =0\,;\\{\frac {dx''}{dt}}&-\left[(2,0)+(2,1)+\ldots \right]y''+[2,0]y+[2,1]y'\ +\ldots =0,\\{\frac {dy''}{dt}}&+\left[(2,0)+(2,1)+\ldots \right]x''-[2,0]x-[2,1]x'\ -\ldots =0\,;\\\ldots &\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots .\end{aligned}}$