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Annales de mathématiques pures et appliquées, 1822-1823, Tome 13.djvu/68
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64
INTÉGRALES
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{\displaystyle {\begin{array}{l}\left({\frac {\operatorname {d} V}{\operatorname {d} x}}\right)X=\left({\frac {\operatorname {d} V}{\operatorname {d} x}}\right)X,\\\\\left({\frac {\operatorname {d} V}{\operatorname {d} x'}}\right)X'=\left[\left({\frac {\operatorname {d} V}{\operatorname {d} x'}}\right)X\right]'-\left({\frac {\operatorname {d} V}{\operatorname {d} x'}}\right)'X,\\\\\left({\frac {\operatorname {d} V}{\operatorname {d} x''}}\right)X''=\left[\left({\frac {\operatorname {d} V}{\operatorname {d} x''}}\right)X'\right]'-\left[\left({\frac {\operatorname {d} V}{\operatorname {d} x''}}\right)'X\right]'+\left({\frac {\operatorname {d} V}{\operatorname {d} x''}}\right)''X,\\\\\left({\frac {\operatorname {d} V}{\operatorname {d} x'''}}\right)X'''=\left[\left({\frac {\operatorname {d} V}{\operatorname {d} x'''}}\right)X''\right]'-\left[\left({\frac {\operatorname {d} V}{\operatorname {d} x'''}}\right)'X'\right]'+\left[\left({\frac {\operatorname {d} V}{\operatorname {d} x'''}}\right)''X\right]'-\left({\frac {\operatorname {d} V}{\operatorname {d} x'''}}\right)'''X,\\\\\end{array}}}
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{\displaystyle {\begin{array}{l}\left({\frac {\operatorname {d} V}{\operatorname {d} y}}\right)Y=\left({\frac {\operatorname {d} V}{\operatorname {d} y}}\right)Y,\\\\\left({\frac {\operatorname {d} V}{\operatorname {d} y'}}\right)Y'=\left[\left({\frac {\operatorname {d} V}{\operatorname {d} y'}}\right)Y\right]'-\left({\frac {\operatorname {d} V}{\operatorname {d} y'}}\right)'Y,\\\\\left({\frac {\operatorname {d} V}{\operatorname {d} y''}}\right)Y''=\left[\left({\frac {\operatorname {d} V}{\operatorname {d} y''}}\right)Y'\right]'-\left[\left({\frac {\operatorname {d} V}{\operatorname {d} y''}}\right)'Y\right]'+\left({\frac {\operatorname {d} V}{\operatorname {d} y''}}\right)''Y,\\\\\left({\frac {\operatorname {d} V}{\operatorname {d} y'''}}\right)Y'''=\left[\left({\frac {\operatorname {d} V}{\operatorname {d} y'''}}\right)Y''\right]'-\left[\left({\frac {\operatorname {d} V}{\operatorname {d} y'''}}\right)'Y'\right]'+\left[\left({\frac {\operatorname {d} V}{\operatorname {d} y'''}}\right)''Y\right]'-\left({\frac {\operatorname {d} V}{\operatorname {d} y'''}}\right)'''Y,\\\\\end{array}}}
Ce qui donnera, en substituant,
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{\displaystyle (3)\ 0=\left\{{\begin{aligned}&\left[\left({\frac {\operatorname {d} V}{\operatorname {d} x}}\right)-\left({\frac {\operatorname {d} V}{\operatorname {d} x'}}\right)'+\left({\frac {\operatorname {d} V}{\operatorname {d} x''}}\right)''-\left({\frac {\operatorname {d} V}{\operatorname {d} x'''}}\right)'''+\ldots \right]X\\\\+&\left[\left({\frac {\operatorname {d} V}{\operatorname {d} y}}\right)-\left({\frac {\operatorname {d} V}{\operatorname {d} y'}}\right)'+\left({\frac {\operatorname {d} V}{\operatorname {d} y''}}\right)''-\left({\frac {\operatorname {d} V}{\operatorname {d} y'''}}\right)'''+\ldots \right]Y\end{aligned}}\right\}}
![{\displaystyle {\begin{array}{l}\left({\frac {\operatorname {d} V}{\operatorname {d} x}}\right)X=\left({\frac {\operatorname {d} V}{\operatorname {d} x}}\right)X,\\\\\left({\frac {\operatorname {d} V}{\operatorname {d} x'}}\right)X'=\left[\left({\frac {\operatorname {d} V}{\operatorname {d} x'}}\right)X\right]'-\left({\frac {\operatorname {d} V}{\operatorname {d} x'}}\right)'X,\\\\\left({\frac {\operatorname {d} V}{\operatorname {d} x''}}\right)X''=\left[\left({\frac {\operatorname {d} V}{\operatorname {d} x''}}\right)X'\right]'-\left[\left({\frac {\operatorname {d} V}{\operatorname {d} x''}}\right)'X\right]'+\left({\frac {\operatorname {d} V}{\operatorname {d} x''}}\right)''X,\\\\\left({\frac {\operatorname {d} V}{\operatorname {d} x'''}}\right)X'''=\left[\left({\frac {\operatorname {d} V}{\operatorname {d} x'''}}\right)X''\right]'-\left[\left({\frac {\operatorname {d} V}{\operatorname {d} x'''}}\right)'X'\right]'+\left[\left({\frac {\operatorname {d} V}{\operatorname {d} x'''}}\right)''X\right]'-\left({\frac {\operatorname {d} V}{\operatorname {d} x'''}}\right)'''X,\\\\\end{array}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/790e73ccd927d67913676560551117276fb55066)
![{\displaystyle {\begin{array}{l}\left({\frac {\operatorname {d} V}{\operatorname {d} y}}\right)Y=\left({\frac {\operatorname {d} V}{\operatorname {d} y}}\right)Y,\\\\\left({\frac {\operatorname {d} V}{\operatorname {d} y'}}\right)Y'=\left[\left({\frac {\operatorname {d} V}{\operatorname {d} y'}}\right)Y\right]'-\left({\frac {\operatorname {d} V}{\operatorname {d} y'}}\right)'Y,\\\\\left({\frac {\operatorname {d} V}{\operatorname {d} y''}}\right)Y''=\left[\left({\frac {\operatorname {d} V}{\operatorname {d} y''}}\right)Y'\right]'-\left[\left({\frac {\operatorname {d} V}{\operatorname {d} y''}}\right)'Y\right]'+\left({\frac {\operatorname {d} V}{\operatorname {d} y''}}\right)''Y,\\\\\left({\frac {\operatorname {d} V}{\operatorname {d} y'''}}\right)Y'''=\left[\left({\frac {\operatorname {d} V}{\operatorname {d} y'''}}\right)Y''\right]'-\left[\left({\frac {\operatorname {d} V}{\operatorname {d} y'''}}\right)'Y'\right]'+\left[\left({\frac {\operatorname {d} V}{\operatorname {d} y'''}}\right)''Y\right]'-\left({\frac {\operatorname {d} V}{\operatorname {d} y'''}}\right)'''Y,\\\\\end{array}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/29a7c08c11041dcd6afb5c63e211a9e60e173972)
![{\displaystyle (3)\ 0=\left\{{\begin{aligned}&\left[\left({\frac {\operatorname {d} V}{\operatorname {d} x}}\right)-\left({\frac {\operatorname {d} V}{\operatorname {d} x'}}\right)'+\left({\frac {\operatorname {d} V}{\operatorname {d} x''}}\right)''-\left({\frac {\operatorname {d} V}{\operatorname {d} x'''}}\right)'''+\ldots \right]X\\\\+&\left[\left({\frac {\operatorname {d} V}{\operatorname {d} y}}\right)-\left({\frac {\operatorname {d} V}{\operatorname {d} y'}}\right)'+\left({\frac {\operatorname {d} V}{\operatorname {d} y''}}\right)''-\left({\frac {\operatorname {d} V}{\operatorname {d} y'''}}\right)'''+\ldots \right]Y\end{aligned}}\right\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c2b06e9b84fea7ed596599817ee288b07b212050)
