Page:Joseph Louis de Lagrange - Œuvres, Tome 6.djvu/211

ensuite

${\begin{array}{|r|r|r|r|r|r|}\hline (1,2)&(1,3)&(1,4)&(2,3)&(2,4)&(3,4)\\\hline -1{,}578&-0{,}213&-0{,}004&-1{,}155&-0{,}162&-0{,}765\\\hline (2,1)&(3,1)&(4,1)&(3,2)&(4,2)&(4,3)\\\hline -2{,}365&-0{,}363&-10{,}040&-1{,}414&-0{,}289&-1{,}331\\\hline \end{array}}$

Donc l’équation (Z) de l’Article XCIX deviendra

${\begin{aligned}(\varepsilon )\quad &\mathrm {K_{1}K_{2}K_{3}K_{4}-2{,}532K_{1}K_{2}-0{,}468K_{1}K_{3}-1{,}633K_{1}K_{4}} \\&\quad -\mathrm {0{,}040K_{2}K_{3}-0{,}077K_{2}K_{4}-3{,}732K_{3}K_{4}} \\&\quad +\mathrm {1{,}013K_{1}+1{,}641K_{2}+2{,}593K_{3}+1{,}373K_{4}-5{,}599} =0,\end{aligned}}$

où il n’y aura plus qu’à substituer au lieu de $\mathrm {K_{1},K_{2},K_{3},K_{4}}$ leurs valeurs

$2{\frac {{\cfrac {\rho }{\mu _{1}}}-{\text{ϐ}}_{1}}{\chi _{1}}},\quad 2{\frac {{\cfrac {\rho }{\mu _{2}}}-{\text{ϐ}}_{2}}{\chi _{2}}},\quad 2{\frac {{\cfrac {\rho }{\mu _{3}}}-{\text{ϐ}}_{3}}{\chi _{3}}},\quad 2{\frac {{\cfrac {\rho }{\mu _{4}}}-{\text{ϐ}}_{4}}{\chi _{4}}}.$

Second Cas.

On aura ici (Article CX)

$(1,2)={\overset {\circ }{\Gamma }}_{1}(a_{1},a_{2}),\quad (1,3)={\overset {\circ }{\Gamma }}_{1}(a_{1},a_{3}),\ldots ,\quad (2,1)={\overset {\circ }{\Gamma }}_{1}(a_{2},a_{1}),\ldots .$

Donc, faisant successivement $q={\frac {a_{1}}{a_{2}}},\ q={\frac {a_{1}}{a_{3}}},\ldots ,$ on aura (Articles XXXII et XLIII)

$(1,2)=q^{2}\mathrm {B} ,\quad (1,3)=q^{2}\mathrm {B} ,\ldots \,;$

ensuite

$(2,1)=q\mathrm {B} ,\quad (3,1)=q\mathrm {B} ,\ldots .$