Page:Annales de mathématiques pures et appliquées, 1820-1821, Tome 11.djvu/115

107

DES ÉQUATIONS.

${\begin{aligned}R'{\frac {\operatorname {d} S'}{\operatorname {d} x}}&=-4+18x+4x^{2}-6x^{3},\\\\S{\frac {\operatorname {d} R}{\operatorname {d} x}}\ \ &=+4+10x-20x^{2}-6x^{3}\,;\\\\S'{\frac {\operatorname {d} R'}{\operatorname {d} x}}&=-4+10x+20x^{2}-6x^{3},\end{aligned}}$

${\begin{aligned}S{\frac {\operatorname {d} V}{\operatorname {d} x}}-V{\frac {\operatorname {d} S}{\operatorname {d} x}}&=1-4x^{2}-8x^{3}-x^{4},\\\\S'{\frac {\operatorname {d} V}{\operatorname {d} x}}-V{\frac {\operatorname {d} S'}{\operatorname {d} x}}&=1-4x^{2}+8x^{3}-x^{4},\\\\R{\frac {\operatorname {d} S}{\operatorname {d} x}}-S{\frac {\operatorname {d} R}{\operatorname {d} x}}&=8x(1+2x),\\\\R'{\frac {\operatorname {d} S'}{\operatorname {d} x}}-S'{\frac {\operatorname {d} R'}{\operatorname {d} x}}&=8x(1-2x),\end{aligned}}$

${\begin{aligned}RR'&=1-22x^{2}+9x^{4},\\SS'&=1-14x^{2}+\ \ x^{4},\end{aligned}}$

$RR'-SS'=-8x^{2}\left(1-x^{2}\right),$

$R+R'=2-6x^{2},$

${\begin{aligned}S(R+R')&=2+8x-4x^{2}-24x^{3}-6x^{4},\\S'(R+R')&=2-8x-4x^{2}+24x^{3}-6x^{4},\end{aligned}}$

${\begin{aligned}S(RR'-SS')&=-8x^{2}\left(1-x^{2}\right)\left(1+4x+x^{2}\right),\\S'(RR'-SS')&=-8x^{2}\left(1-x^{2}\right)\left(1-4x+x^{2}\right),\end{aligned}}$