67
INDÉTERMINÉES.
![{\displaystyle Const.=\left\{{\begin{aligned}&\left[\left({\frac {\operatorname {d} V}{\operatorname {d} x'}}\right)_{1}-\left({\frac {\operatorname {d} V}{\operatorname {d} x''}}\right)_{1}'+\left({\frac {\operatorname {d} V}{\operatorname {d} x'''}}\right)_{1}''-\ldots \right]X_{1}\\\\+&\left[\left({\frac {\operatorname {d} V}{\operatorname {d} x''}}\right)_{1}-\left({\frac {\operatorname {d} V}{\operatorname {d} x'''}}\right)_{1}'+\ldots \right]X_{1}'+\left[\left({\frac {\operatorname {d} V}{\operatorname {d} x'''}}\right)_{1}-\ldots \right]X_{1}''+\ldots \\\\+&\left[\left({\frac {\operatorname {d} V}{\operatorname {d} y'}}\right)_{1}-\left({\frac {\operatorname {d} V}{\operatorname {d} y''}}\right)_{1}'+\left({\frac {\operatorname {d} V}{\operatorname {d} y'''}}\right)_{1}''-\ldots \right]Y_{1}\\\\+&\left[\left({\frac {\operatorname {d} V}{\operatorname {d} y''}}\right)_{1}-\left({\frac {\operatorname {d} V}{\operatorname {d} y'''}}\right)_{1}'+\ldots \right]Y_{1}'+\left[\left({\frac {\operatorname {d} V}{\operatorname {d} y'''}}\right)_{1}-\ldots \right]Y_{1}''+\ldots \\\\\end{aligned}}\right\}\,;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1c2e907c854472122a894786f526704dad3f6be6)
d’où, en retranchant, on conclura, pour l’équation aux limites,
![{\displaystyle 0=\left\{{\begin{aligned}&\left[\left({\frac {\operatorname {d} V}{\operatorname {d} x'}}\right)_{1}-\left({\frac {\operatorname {d} V}{\operatorname {d} x''}}\right)_{1}'+\left({\frac {\operatorname {d} V}{\operatorname {d} x'''}}\right)_{1}''-\ldots \right]X_{1}\\\\&\qquad +\left[\left({\frac {\operatorname {d} V}{\operatorname {d} x''}}\right)_{1}-\left({\frac {\operatorname {d} V}{\operatorname {d} x'''}}\right)_{1}'+\ldots \right]X_{1}'+\left[\left({\frac {\operatorname {d} V}{\operatorname {d} x'''}}\right)_{1}-\ldots \right]X_{1}''+\ldots \\\\-&\left[\left({\frac {\operatorname {d} V}{\operatorname {d} x'}}\right)_{0}-\left({\frac {\operatorname {d} V}{\operatorname {d} x''}}\right)_{0}'+\left({\frac {\operatorname {d} V}{\operatorname {d} x'''}}\right)_{0}''-\ldots \right]X_{0}\\\\&\qquad -\left[\left({\frac {\operatorname {d} V}{\operatorname {d} x''}}\right)_{0}-\left({\frac {\operatorname {d} V}{\operatorname {d} x'''}}\right)_{0}'+\ldots \right]X_{0}'-\left[\left({\frac {\operatorname {d} V}{\operatorname {d} x'''}}\right)_{0}-\ldots \right]X_{0}''-\ldots \\\\+&\left[\left({\frac {\operatorname {d} V}{\operatorname {d} y'}}\right)_{1}-\left({\frac {\operatorname {d} V}{\operatorname {d} y''}}\right)_{1}'+\left({\frac {\operatorname {d} V}{\operatorname {d} y'''}}\right)_{1}''-\ldots \right]Y_{1}\\\\&\qquad +\left[\left({\frac {\operatorname {d} V}{\operatorname {d} y''}}\right)_{1}-\left({\frac {\operatorname {d} V}{\operatorname {d} y'''}}\right)_{1}'+\ldots \right]Y_{1}'+\left[\left({\frac {\operatorname {d} V}{\operatorname {d} y'''}}\right)_{1}-\ldots \right]Y_{1}''+\ldots \\\\-&\left[\left({\frac {\operatorname {d} V}{\operatorname {d} y'}}\right)_{0}-\left({\frac {\operatorname {d} V}{\operatorname {d} y''}}\right)_{0}'+\left({\frac {\operatorname {d} V}{\operatorname {d} y'''}}\right)_{0}''-\ldots \right]Y_{0}\\\\&\qquad -\left[\left({\frac {\operatorname {d} V}{\operatorname {d} y''}}\right)_{0}-\left({\frac {\operatorname {d} V}{\operatorname {d} y'''}}\right)_{0}'+\ldots \right]Y_{0}'-\left[\left({\frac {\operatorname {d} V}{\operatorname {d} y'''}}\right)_{0}-\ldots \right]Y_{0}''-\ldots \\\\\end{aligned}}\right\}\,;(7)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b03f13594b34501b998dbfcdf6a354ed8cfbbc96)
63. Cela posé, si aucune condition n’a été prescrite relativement aux limites de l’intégrale ; c’est-à-dire, si les valeurs de ![{\displaystyle x,x',\ldots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/15f9e0f46474474503bd3659a8b34f7f714a3773)
peuvent être quelconques à ces limites, les fonctions
et par suite les dérivées
devront, à ces mêmes limites, conserver toute leur indétermination et toute leur indépendance ; l’équation (7) ne pourra donc subsister alors gu’autant que les multiplicateurs de
![{\displaystyle X_{0},X'_{0},X''_{0},\ldots Y_{0},Y'_{0},Y''_{0},\ldots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/6ac383001693818ce8f58c68ffb315dccdd17444)
![{\displaystyle X_{1},X'_{1},X''_{1},\ldots Y_{1},Y'_{1},Y''_{1},\ldots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/8c27516ab7617e69dc78230acf32ed705acc01de)