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EXPOSANTS CARACTERISTIQUES.
bien tous les déterminants contenus dans la matrice
![{\displaystyle \left|\left|{\frac {d\mathrm {F} _{i}}{dx_{k}}}\right|\right|\qquad (i=1,\,2,\,\ldots ,\,p\,;\;\;k=1,\,\ldots ,\,n)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6d61a95e9a7fa1c469165877b5b49c582ed9b9a9)
seront nuls pour tous les points de la solution périodique génératrice.
Supposons, en effet, pour fixer les idées,
![{\displaystyle n=4,\quad p=2.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3b03205116752096ebf06d2dbb0f65ad1479a3d2)
Nous aurons alors les équations suivantes
![{\displaystyle {\begin{array}{c}\left.{\begin{aligned}{\dfrac {d\mathrm {F} _{1}}{dx_{1}}}{\dfrac {d\psi _{1}}{d\beta _{i}}}+{\dfrac {d\mathrm {F} _{1}}{dx_{2}}}{\dfrac {d\psi _{2}}{d\beta _{i}}}+{\dfrac {d\mathrm {F} _{1}}{dx_{3}}}{\dfrac {d\psi _{3}}{d\beta _{i}}}+{\dfrac {d\mathrm {F} _{1}}{dx_{4}}}{\dfrac {d\psi _{4}}{d\beta _{i}}}&=0,\\{\dfrac {d\mathrm {F} _{2}}{dx_{1}}}{\dfrac {d\psi _{1}}{d\beta _{i}}}+{\dfrac {d\mathrm {F} _{2}}{dx_{2}}}{\dfrac {d\psi _{2}}{d\beta _{i}}}+{\dfrac {d\mathrm {F} _{2}}{dx_{3}}}{\dfrac {d\psi _{3}}{d\beta _{i}}}+{\dfrac {d\mathrm {F} _{2}}{dx_{4}}}{\dfrac {d\psi _{4}}{d\beta _{i}}}&=0\end{aligned}}\right\}\;(i=1,\,2,\,3,\,4)\\[1.5ex]\quad \qquad {\begin{aligned}{\dfrac {d\mathrm {F} _{1}}{dx_{1}}}{\dfrac {d\psi _{1}}{d\tau }}+{\dfrac {d\mathrm {F} _{1}}{dx_{2}}}{\dfrac {d\psi _{2}}{d\tau }}+{\dfrac {d\mathrm {F} _{1}}{dx_{3}}}{\dfrac {d\psi _{3}}{d\tau }}+{\dfrac {d\mathrm {F} _{1}}{dx_{4}}}{\dfrac {d\psi _{4}}{d\tau }}&=0,\\{\dfrac {d\mathrm {F} _{2}}{dx_{1}}}{\dfrac {d\psi _{1}}{d\tau }}+{\dfrac {d\mathrm {F} _{2}}{dx_{2}}}{\dfrac {d\psi _{2}}{d\tau }}+{\dfrac {d\mathrm {F} _{2}}{dx_{3}}}{\dfrac {d\psi _{3}}{d\tau }}+{\dfrac {d\mathrm {F} _{2}}{dx_{4}}}{\dfrac {d\psi _{4}}{d\tau }}&=0.\end{aligned}}\end{array}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/de3bba506bb5e0a1c117bad9c63ba70f8bfa9697)
De ces équations il est permis de conclure :
Ou bien que tous les déterminants contenus dans la matrice
![{\displaystyle \left|\left|{\begin{array}{ccccc}{\dfrac {d\mathrm {F} _{1}}{dx_{1}}}&{\dfrac {d\mathrm {F} _{1}}{dx_{2}}}&{\dfrac {d\mathrm {F} _{1}}{dx_{3}}}&{\dfrac {d\mathrm {F} _{1}}{dx_{4}}}\\{\dfrac {d\mathrm {F} _{2}}{dx_{1}}}&{\dfrac {d\mathrm {F} _{2}}{dx_{2}}}&{\dfrac {d\mathrm {F} _{2}}{dx_{3}}}&{\dfrac {d\mathrm {F} _{2}}{dx_{4}}}\\\end{array}}\right|\right|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9e273a68decca2af2cd616fe46f6fd15e796cc46)
sont nuls à la fois ; ou bien que tous les déterminants contenus dans la matrice
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sont nuls à la fois, ainsi que leurs mineurs du premier ordre.