Page:Annales de mathématiques pures et appliquées, 1822-1823, Tome 13.djvu/82

78

INTÉGRALES

73. Quant à l’équation aux limites, elle sera évidemment ici

$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 \\\\&\left[\left({\frac {\operatorname {d} V}{\operatorname {d} z'}}\right)_{1}-\left({\frac {\operatorname {d} V}{\operatorname {d} z''}}\right)_{1}'+\left({\frac {\operatorname {d} V}{\operatorname {d} z'''}}\right)_{1}''-\ldots \right]Z_{1}\\\\&\qquad +\left[\left({\frac {\operatorname {d} V}{\operatorname {d} z''}}\right)_{1}-\left({\frac {\operatorname {d} V}{\operatorname {d} z'''}}\right)_{1}'+\ldots \right]Z_{1}'+\left[\left({\frac {\operatorname {d} V}{\operatorname {d} z'''}}\right)_{1}-\ldots \right]Z_{1}''+\ldots \\\\-&\left[\left({\frac {\operatorname {d} V}{\operatorname {d} z'}}\right)_{0}-\left({\frac {\operatorname {d} V}{\operatorname {d} z''}}\right)_{0}'+\left({\frac {\operatorname {d} V}{\operatorname {d} z'''}}\right)_{0}''-\ldots \right]Z_{0}\\\\&\qquad -\left[\left({\frac {\operatorname {d} V}{\operatorname {d} z''}}\right)_{0}-\left({\frac {\operatorname {d} V}{\operatorname {d} z'''}}\right)_{0}'+\ldots \right]Z_{0}'-\left[\left({\frac {\operatorname {d} V}{\operatorname {d} z'''}}\right)_{0}-\ldots \right]Z_{0}''-\ldots \\\\\end{aligned}}\right\}\,;$

les indices $0$ et $1$ ayant ici la même signification que ci-dessus ; et voici l’usage que l’on fera de cette équation.

74. Si aucune condition n’a été prescrite relativement aux limites de l’intégrale ; c’est-à-dire, si à ces limites, les valeurs de $x,x',x'',\ldots ,y,y',y'',$$\ldots ,z,z',z'',\ldots$ peuvent être quelconques, les fonctions $X,Y,Z,$ et par suite leurs dérivées $X',Y',Z',X'',Y'',Z'',\ldots$ devront, à ces mêmes limites, conserver toute leur indétermination et toute leur indépendante ; l’équation (19) ne pourra donc subsister alors qu’autant que les multiplicateurs de

$X_{0},X'_{0},X''_{0},\ldots Y_{0},Y'_{0},Y''_{0},\ldots Z_{0},Z'_{0},Z''_{0},\ldots$

$X_{1},X'_{1},X''_{1},\ldots Y_{1},Y'_{1},Y''_{1},\ldots Z_{1},Z'_{1},Z''_{1},\ldots$

seront séparément nuls ; on devra donc avoir séparément