Page:Annales de mathématiques pures et appliquées, 1829-1830, Tome 20.djvu/175

 Les corrections sont expliquées en page de discussion

$z(z+A)\left(x{\frac {\operatorname {d} ^{2}x}{\operatorname {d} s^{2}}}+y{\frac {\operatorname {d} ^{2}y}{\operatorname {d} s^{2}}}\right)\operatorname {Sin} .^{2}\omega \operatorname {Cos} .^{2}\omega$

$+\left(x^{2}+y^{2}\right)\left\{(z+A){\frac {\operatorname {d} ^{2}z}{\operatorname {d} s^{2}}}-1\right\}\operatorname {Cos} .^{4}\omega$

$+\left\{\left(x^{2}+y^{2}\right)\operatorname {Cos} .^{4}\omega +z^{2}\operatorname {Sin} .^{4}\omega \right\}\left({\frac {\operatorname {d} z}{\operatorname {d} s}}\right)^{2}=0.$

En mettant d’abord, dans ces deux équations, pour $\left(x^{2}+y^{2}\right)\operatorname {Cos} .^{2}\omega$ sa valeur $z^{2}\operatorname {Sin} .^{2}\omega ,$ elles deviendront

$Nz\operatorname {Sin} .\omega =$

$(z+A)\left\{\left(x{\frac {\operatorname {d} ^{2}x}{\operatorname {d} s^{2}}}+y{\frac {\operatorname {d} ^{2}y}{\operatorname {d} s^{2}}}\right)\operatorname {Cos} .^{2}\omega -z{\frac {\operatorname {d} ^{2}z}{\operatorname {d} s^{2}}}\operatorname {Sin} .^{2}\omega \right\}+\operatorname {Sin} .^{2}\omega ,$ (29)

$(z+A)\left(x{\frac {\operatorname {d} ^{2}x}{\operatorname {d} s^{2}}}+y{\frac {\operatorname {d} ^{2}y}{\operatorname {d} s^{2}}}+z{\frac {\operatorname {d} ^{2}z}{\operatorname {d} s^{2}}}\right)\operatorname {Cos} .^{2}\omega +z\left({\frac {\operatorname {d} z}{\operatorname {d} s}}\right)^{2}$

$=z\operatorname {Cos} .^{2}\omega \,;$ (30)

mais, par deux différentiations, on tire de l’équation (28)

$\left(x{\frac {\operatorname {d} x}{\operatorname {d} s}}+y{\frac {\operatorname {d} y}{\operatorname {d} s}}\right)\operatorname {Cos} .^{2}\omega =z{\frac {\operatorname {d} z}{\operatorname {d} s}}\operatorname {Sin} .^{2}\omega$

$\left(x{\frac {\operatorname {d} ^{2}x}{\operatorname {d} s^{2}}}+y{\frac {\operatorname {d} ^{2}y}{\operatorname {d} s^{2}}}\right)\operatorname {Cos} .^{2}\omega =$

$z{\frac {\operatorname {d} ^{2}z}{\operatorname {d} s^{2}}}\operatorname {Sin} .^{2}\omega -\left\{\left({\frac {\operatorname {d} x}{\operatorname {d} s}}\right)^{2}+\left({\frac {\operatorname {d} y}{\operatorname {d} s}}\right)^{2}\right\}\operatorname {Cos} .^{2}\omega +\left({\frac {\operatorname {d} z}{\operatorname {d} s}}\right)^{2}\operatorname {Sin} .^{2}\omega ,$

ou bien encore

$\left(x{\frac {\operatorname {d} ^{2}x}{\operatorname {d} s^{2}}}+y{\frac {\operatorname {d} ^{2}y}{\operatorname {d} s^{2}}}\right)\operatorname {Cos} .^{2}\omega =$

$z{\frac {\operatorname {d} ^{2}z}{\operatorname {d} s^{2}}}\operatorname {Sin} .^{2}\omega -\left\{1-\left({\frac {\operatorname {d} z}{\operatorname {d} s}}\right)^{2}\right\}\operatorname {Cos} .^{2}\omega +\left({\frac {\operatorname {d} z}{\operatorname {d} s}}\right)^{2}\operatorname {Sin} .^{2}\omega ,$

ou bien enfin

$\left(x{\frac {\operatorname {d} ^{2}x}{\operatorname {d} s^{2}}}+y{\frac {\operatorname {d} ^{2}y}{\operatorname {d} s^{2}}}\right)\operatorname {Cos} .^{2}\omega =z{\frac {\operatorname {d} ^{2}z}{\operatorname {d} s^{2}}}\operatorname {Sin} .^{2}\omega +\left({\frac {\operatorname {d} z}{\operatorname {d} s}}\right)^{2}-\operatorname {Cos} .^{2}\omega \,;$

ce qui donnera, en substituant dans les équations (29) et (30)

$Nz\operatorname {Sin} .\omega =(z+A)\left\{\left({\frac {\operatorname {d} z}{\operatorname {d} s}}\right)^{2}-\operatorname {Cos} .^{2}\omega \right\}+z\operatorname {Sin} .^{2}\omega ,\qquad$(31)