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Annales de mathématiques pures et appliquées, 1820-1821, Tome 11.djvu/115
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107
DES ÉQUATIONS.
R
′
d
S
′
d
x
=
−
4
+
18
x
+
4
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2
−
6
x
3
,
S
d
R
d
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+
4
+
10
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20
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2
−
6
x
3
;
S
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d
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x
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6
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{\displaystyle {\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}}}
S
d
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x
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V
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8
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S
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V
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V
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8
x
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x
4
,
R
d
S
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S
d
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8
x
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1
+
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)
,
R
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S
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R
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8
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)
,
{\displaystyle {\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}}}
R
R
′
=
1
−
22
x
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+
9
x
4
,
S
S
′
=
1
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14
x
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+
x
4
,
{\displaystyle {\begin{aligned}RR'&=1-22x^{2}+9x^{4},\\SS'&=1-14x^{2}+\ \ x^{4},\end{aligned}}}
R
R
′
−
S
S
′
=
−
8
x
2
(
1
−
x
2
)
,
{\displaystyle RR'-SS'=-8x^{2}\left(1-x^{2}\right),}
R
+
R
′
=
2
−
6
x
2
,
{\displaystyle R+R'=2-6x^{2},}
S
(
R
+
R
′
)
=
2
+
8
x
−
4
x
2
−
24
x
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6
x
4
,
S
′
(
R
+
R
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)
=
2
−
8
x
−
4
x
2
+
24
x
3
−
6
x
4
,
{\displaystyle {\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}}}
S
(
R
R
′
−
S
S
′
)
=
−
8
x
2
(
1
−
x
2
)
(
1
+
4
x
+
x
2
)
,
S
′
(
R
R
′
−
S
S
′
)
=
−
8
x
2
(
1
−
x
2
)
(
1
−
4
x
+
x
2
)
,
{\displaystyle {\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}}}