Page:Annales de mathématiques pures et appliquées, 1820-1821, Tome 11.djvu/112

104

INTÉGRATION

$4V^{2}S'{\frac {\operatorname {d} ^{2}q'}{\operatorname {d} x^{2}}}+2V\left\{2{\frac {S'\operatorname {d} V-V\operatorname {d} S'}{\operatorname {d} x}}-S'(R+R')\right\}{\frac {\operatorname {d} q'}{\operatorname {d} x}}$

$+\left\{2V{\frac {R'\operatorname {d} S'-S'\operatorname {d} R'}{\operatorname {d} x}}+S'(RR'-SS')\right\}q'=0\,;$

de sorte qu’en posant, pour abréger,

${\begin{aligned}2(S\operatorname {d} V-V\operatorname {d} S)&-S(R+R')\operatorname {d} x=T\operatorname {d} x,\\2(S'\operatorname {d} V-V\operatorname {d} S')&-S'(R+R')\operatorname {d} x=T'\operatorname {d} x\,;\\2V(R\operatorname {d} S-S\operatorname {d} R)&+S(RR'-SS')\operatorname {d} x=U\operatorname {d} x,\\2V(S'\operatorname {d} S'-R'\operatorname {d} R')&+S'(RR'-SS')\operatorname {d} x=U'\operatorname {d} x\,;\end{aligned}}$

les quatre équations à résoudre seront

${\begin{alignedat}{3}&4V^{2}S{\frac {\operatorname {d} ^{2}p}{\operatorname {d} x^{2}}}&+&2VT{\frac {\operatorname {d} p}{\operatorname {d} x}}\ \ +Up&=&0,\\&4V^{2}S'{\frac {\operatorname {d} ^{2}q}{\operatorname {d} x^{2}}}&+&2VT'{\frac {\operatorname {d} q}{\operatorname {d} x}}\ +U'q&=&0,\\&4V^{2}S{\frac {\operatorname {d} ^{2}p'}{\operatorname {d} x^{2}}}&+&2VT{\frac {\operatorname {d} p'}{\operatorname {d} x}}\ +Up'&=&0,\\&4V^{2}S'{\frac {\operatorname {d} ^{2}q'}{\operatorname {d} x^{2}}}&+&2VT'{\frac {\operatorname {d} q'}{\operatorname {d} x}}+U'q'&=&0.\end{alignedat}}$

13. Au moyen de ces quatre équations, on déterminera $p,q,$ $p',q',$ d’où on conclura (10) $P,Q,P',Q'$ qui (5), substitués dans la formule