Page:Joseph Louis de Lagrange - Œuvres, Tome 6.djvu/597

du trinôme $\psi '',$ on aura, comme l’on sait,

${\begin{aligned}\varpi =&-{\frac {q+q'}{2}}+{\frac {\sqrt {(q-q')^{2}-4p'}}{2}},\\\rho =&-{\frac {q+q'}{2}}-{\frac {\sqrt {(q-q')^{2}-4p'}}{2}},\end{aligned}}$

et la fraction ${\frac {p\psi '}{\psi ''}}$ se décomposera en ces deux-ci

${\frac {\mathrm {F} }{1-\pi y}}+{\frac {\mathrm {G} }{1-\rho y}},$

en faisant

${\begin{aligned}\mathrm {F} =&-{\frac {p(q'+\varpi )}{\rho -\varpi }}={\frac {p}{2}}\left[1-{\frac {q-q'}{\sqrt {(q-q')^{2}-4p'}}}\right],\\\mathrm {G} =&-{\frac {p(q'+\rho \ )}{\varpi -\rho }}={\frac {p}{2}}\left[1+{\frac {q-q'}{\sqrt {(q-q')^{2}-4p'}}}\right].\end{aligned}}$

Introduisons maintenant dans ces formules les valeurs trouvées ci-dessus, et prenons d’abord celles qui résultent de la première série.

On aura donc

${\begin{alignedat}{3}q+q'\quad =&0{,}83950,\\q-q'\quad =&0{,}18906,\qquad &\log =&9{,}2765997,\qquad &2\log =&8{,}5531994,\\(q-q')^{2}=&0,03574,\\-4p'=&{\underline {4{,}92492}},\\&4{,}96066&\log =&0{,}6955394,&{\frac {1}{2}}\log =&0{,}3477697.\end{alignedat}}$

${\begin{aligned}{\sqrt {(q-q')^{2}-4p'}}=&2{,}22726,\\q+q'=&{\underline {0{,}83950}}.\\\mathrm {Somme} \ldots \ldots \ldots \ \ &3{,}06675.\\\mathrm {Diff{\acute {e}}rence} \ldots \ldots \ \ &1{,}38775.\\\end{aligned}}$

Donc

$\varpi =0{,}69387,\quad \rho =-1{,}53337\,;$