Page:Annales de mathématiques pures et appliquées, 1820-1821, Tome 11.djvu/110

102

INTÉGRATION

des deux premières équations, puis ensuite des deux dernières, il viendra

${\begin{aligned}2V\operatorname {d} (P+Q)&=(P+Q)\operatorname {d} V-X'(P-Q)\operatorname {d} x+2(XQ-X''P)\operatorname {d} x,\\2V\operatorname {d} (P-Q)&=(P-Q)\operatorname {d} V-X'(P+Q)\operatorname {d} x+2(XQ+X''P)\operatorname {d} x,\\2V\operatorname {d} (P'+Q')&=(P'+Q')\operatorname {d} V-X'(P'-Q')\operatorname {d} x+2(XQ'-X''P')\operatorname {d} x,\\2V\operatorname {d} (P'-Q')&=(P'-Q')\operatorname {d} V-X'(P'+Q')\operatorname {d} x+2(XQ'+X''P')\operatorname {d} x,\\\end{aligned}}$

Posant donc

$\left.{\begin{aligned}P+Q&=2p,\\P-Q&=2q,\\P'+Q'&=2p',\\P'-Q'&=2q'\,;\\\end{aligned}}\right\}\quad {\rm {{d'o{\grave {u}}}\quad \left\{{\begin{aligned}P&=p+q,\\Q&=p-q,\\P'&=p'+q',\\Q'&=p'-q'\,;\\\end{aligned}}\right.}}$

ces équations deviendront

${\begin{aligned}2V\operatorname {d} p&=p\left[\operatorname {d} V+(X-X'')\operatorname {d} x\right]-q(X+X'+X'')\operatorname {d} x,\\2V\operatorname {d} q&=q\left[\operatorname {d} V-(X-X'')\operatorname {d} x\right]+p(X-X'+X'')\operatorname {d} x,\\2V\operatorname {d} p'&=p'\left[\operatorname {d} V+(X-X'')\operatorname {d} x\right]-q'(X+X'+X'')\operatorname {d} x,\\2V\operatorname {d} q'&=q'\left[\operatorname {d} V-(X-X'')\operatorname {d} x\right]+p'(X-X'+X'')\operatorname {d} x.\\\end{aligned}}$

11. En posant donc, pour abréger,

$\operatorname {d} V+(X-X'')\operatorname {d} xR\operatorname {d} x,\qquad X+X'+X''=+S,$