Page:Joseph Louis de Lagrange - Œuvres, Tome 5.djvu/175

car, comme $d\mathrm {R} =0$ (38), on aura simplement

$d\mathrm {P} =\mathrm {R} ds,\quad d\mathrm {Q} =\mathrm {R} du\,;$

d’ailleurs (39)

$\mathrm {R} ={\frac {1}{\sqrt {\Delta }}}={\sqrt {r}}.$

Donc si l’on fait, pour abréger,

$(0,1)={\frac {\mathrm {T} 'rr'(r,r')_{1}}{4{\sqrt {r}}}},\quad (0,2)={\frac {\mathrm {T} ''rr''(r,r'')_{1}}{4{\sqrt {r}}}},\ldots ,$

on aura ces équations linéaires

${\begin{aligned}{\frac {ds}{dt}}+(0,1)(u-u')+(0,2)(u-u'')+\ldots =&0,\\{\frac {du}{dt}}-(0,1)(s\,-s')+(0,2)(s\,-s'')-\ldots =&0\,;\end{aligned}}$

en faisant de même

${\begin{alignedat}{2}(1,0)&={\frac {\mathrm {T} r'r(r',r)_{1}}{4{\sqrt {r'}}}},\qquad &(1,2)&={\frac {\mathrm {T} ''r'r''(r',r'')_{1}}{4{\sqrt {r'}}}},\ldots ,\\(2,0)&={\frac {\mathrm {T} r''r(r'',r)_{1}}{4{\sqrt {r''}}}},&(2,1)&={\frac {\mathrm {T} 'r''r'(r'',r')_{1}}{4{\sqrt {r''}}}},\ldots ,\\\end{alignedat}}$

$\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots ,$

on aura aussi

${\begin{aligned}{\frac {ds'}{dt}}+(1,0)(\,u'-u)+(1,2)(u'\,-u'')+\ldots &=0,\\{\frac {du'}{dt}}-(1,0)(\,s'-s\,)-(1,2)(s'\ -s'')-\ldots &=0,\\{\frac {ds''}{dt}}+(2,0)(u''-u)+(2,1)(u''-u')+\ldots &=0,\\{\frac {du''}{dt}}-(2,0)(s''-s\,)-(2,1)(s''-s'\,)-\ldots &=0,\\\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots &\ldots ,\end{aligned}}$