Page:Annales de mathématiques pures et appliquées, 1828-1829, Tome 19.djvu/277

$\left.{\begin{aligned}&\left({\frac {\operatorname {d} y}{\operatorname {d} t}}{\frac {\operatorname {d} ^{2}z}{\operatorname {d} t^{2}}}-{\frac {\operatorname {d} z}{\operatorname {d} t}}{\frac {\operatorname {d} ^{2}y}{\operatorname {d} t^{2}}}\right)(X-x)\\\\+&\left({\frac {\operatorname {d} z}{\operatorname {d} t}}{\frac {\operatorname {d} ^{2}x}{\operatorname {d} t^{2}}}-{\frac {\operatorname {d} x}{\operatorname {d} t}}{\frac {\operatorname {d} ^{2}z}{\operatorname {d} t^{2}}}\right)(Y-y)\\\\+&\left({\frac {\operatorname {d} x}{\operatorname {d} t}}{\frac {\operatorname {d} ^{2}y}{\operatorname {d} t^{2}}}-{\frac {\operatorname {d} y}{\operatorname {d} t}}{\frac {\operatorname {d} ^{2}x}{\operatorname {d} t^{2}}}\right)(Z-z)\end{aligned}}\right\}=0\,;\quad$(7)

et il faudra d’abord que ce plan soit perpendiculaire, en $(x,y,z),$ au plan tangent (5) au même point ; ce qui donnera, pour première équation du mouvement de la molécule

$P\left({\frac {\operatorname {d} y}{\operatorname {d} t}}{\frac {\operatorname {d} ^{2}z}{\operatorname {d} t^{2}}}-{\frac {\operatorname {d} z}{\operatorname {d} t}}{\frac {\operatorname {d} ^{2}y}{\operatorname {d} t^{2}}}\right)+Q\left({\frac {\operatorname {d} z}{\operatorname {d} t}}{\frac {\operatorname {d} ^{2}x}{\operatorname {d} t^{2}}}-{\frac {\operatorname {d} x}{\operatorname {d} t}}{\frac {\operatorname {d} ^{2}z}{\operatorname {d} t^{2}}}\right)$

$+R\left({\frac {\operatorname {d} x}{\operatorname {d} t}}{\frac {\operatorname {d} ^{2}y}{\operatorname {d} t^{2}}}-{\frac {\operatorname {d} y}{\operatorname {d} t}}{\frac {\operatorname {d} ^{2}x}{\operatorname {d} t^{2}}}\right)=0,$

ou bien

$\left(Q{\frac {\operatorname {d} ^{2}z}{\operatorname {d} t^{2}}}-R{\frac {\operatorname {d} ^{2}y}{\operatorname {d} t^{2}}}\right){\frac {\operatorname {d} x}{\operatorname {d} t}}+\left(R{\frac {\operatorname {d} ^{2}x}{\operatorname {d} t^{2}}}-P{\frac {\operatorname {d} ^{2}z}{\operatorname {d} t^{2}}}\right){\frac {\operatorname {d} y}{\operatorname {d} t}}$

$+\left(P{\frac {\operatorname {d} ^{2}y}{\operatorname {d} t^{2}}}-Q{\frac {\operatorname {d} ^{2}x}{\operatorname {d} t^{2}}}\right){\frac {\operatorname {d} z}{\operatorname {d} t}}=0.\quad$(8)

Les vîtesses de la molécule, parallèlement aux axes des $x$, des $y$ et des $z$, étant respectivement

${\frac {\operatorname {d} x}{\operatorname {d} t}},\quad {\frac {\operatorname {d} y}{\operatorname {d} t}},\quad {\frac {\operatorname {d} z}{\operatorname {d} t}}\,;$

les équations de la tangente à la trajectoire, au point $(x,y,z),$ seront

${\frac {X-x}{\frac {\operatorname {d} x}{\operatorname {d} t}}}={\frac {Y-y}{\frac {\operatorname {d} y}{\operatorname {d} t}}}={\frac {Z-z}{\frac {\operatorname {d} z}{\operatorname {d} t}}}\,;\qquad$(9)

de sorte que, si l’on représente par $\theta$ l’angle que fait cette tan-