Page:Annales de mathématiques pures et appliquées, 1822-1823, Tome 13.djvu/35

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31

INDÉTERMINÉES.

Const.=\left\{{\begin{aligned}&\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}\\\\+&\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)_{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}\\\\+&\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\},

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\}\,;

d’où, en retranchant,

0=\left\{{\begin{aligned}&\left[\left({\tfrac {\operatorname {d} V}{\operatorname {d} x'}}\right)_{1}-\left({\tfrac {\operatorname {d} V}{\operatorname {d} x''}}\right)_{1}'+\left({\tfrac {\operatorname {d} V}{\operatorname {d} x'''}}\right)_{1}''-\ldots \right]Y_{1}\\\\&+\left[\left({\tfrac {\operatorname {d} V}{\operatorname {d} x''}}\right)_{1}-\left({\tfrac {\operatorname {d} V}{\operatorname {d} x'''}}\right)_{1}'+\ldots \right]X_{1}'+\left[\left({\tfrac {\operatorname {d} V}{\operatorname {d} x'''}}\right)_{1}-\ldots \right]X_{1}''+\ldots \\\\&-\left[\left({\tfrac {\operatorname {d} V}{\operatorname {d} x'}}\right)_{0}-\left({\tfrac {\operatorname {d} V}{\operatorname {d} x''}}\right)_{0}'+\left({\tfrac {\operatorname {d} V}{\operatorname {d} x'''}}\right)_{0}''-\ldots \right]X_{0}\\\\&-\left[\left({\tfrac {\operatorname {d} V}{\operatorname {d} x''}}\right)_{0}-\left({\tfrac {\operatorname {d} V}{\operatorname {d} x'''}}\right)_{0}'+\ldots \right]X_{0}'-\left[\left({\tfrac {\operatorname {d} V}{\operatorname {d} x'''}}\right)_{0}-\ldots \right]X_{0}''-\ldots \\\\&\left[\left({\tfrac {\operatorname {d} V}{\operatorname {d} y'}}\right)_{1}-\left({\tfrac {\operatorname {d} V}{\operatorname {d} y''}}\right)_{1}'+\left({\tfrac {\operatorname {d} V}{\operatorname {d} y'''}}\right)_{1}''-\ldots \right]Y_{1}\\\\&+\left[\left({\tfrac {\operatorname {d} V}{\operatorname {d} y''}}\right)_{1}-\left({\tfrac {\operatorname {d} V}{\operatorname {d} y'''}}\right)_{1}'+\ldots \right]Y_{1}'+\left[\left({\tfrac {\operatorname {d} V}{\operatorname {d} y'''}}\right)_{1}-\ldots \right]Y_{1}''+\ldots \\\\&-\left[\left({\tfrac {\operatorname {d} V}{\operatorname {d} y'}}\right)_{0}-\left({\tfrac {\operatorname {d} V}{\operatorname {d} y''}}\right)_{0}'+\left({\tfrac {\operatorname {d} V}{\operatorname {d} y'''}}\right)_{0}''-\ldots \right]Y_{0}\\\\&-\left[\left({\tfrac {\operatorname {d} V}{\operatorname {d} y''}}\right)_{0}-\left({\tfrac {\operatorname {d} V}{\operatorname {d} y'''}}\right)_{0}'+\ldots \right]Y_{0}'-\left[\left({\tfrac {\operatorname {d} V}{\operatorname {d} y'''}}\right)_{0}-\ldots \right]Y_{0}''-\ldots \end{aligned}}\right\}\,;(\mathrm {XIX} )