Page:Mécanique céleste Vol 1.djvu/34

Let us now determine the angle $\theta$. If we increase the force $x$ by the differential $dx$, without varying the force $y$, that angle will be diminished by the infinitely small quantity $d\theta$;^[1] now we may conceive the force $dx$ to be resolved into two other forces, the one $dx'$ in the direction $z$, and the other $dx''$ perpendicular to $z$; the point $M$ will then be acted upon by the two forces $z+dx'$ and $dx''$, perpendicular to each other, and the resultant of these two forces, which we shall call $z'$, will make with $dx''$ the angle ${\frac {\pi }{2}}-d\theta$;^[2] we shall thus have, by what precedes,

[4] $dx''=z'\cdot \varphi \left({\frac {\pi }{2}}-d\theta \right)$;

consequently the function $\varphi \left({\frac {\pi }{2}}-d\theta \right)$ is infinitely small, and of the form $-\ kd\theta$, $k$ being a constant quantity, independent of the angle $\theta$;^[3]</ref> we shall therefore have

[5] ${\frac {dx''}{z'}}=-kd\theta$,

• (5) The resultant of the forces $x$, $y$, is, by hypothesis, in the direction $AZ$, and, by increasing the force $x$ by $dx$, the forces become equal to $z$ in the direction $AZ$, and $dx$ in the direction $AX$, and the resulting force $z'$, must evidently fall between $AZ$, $AX$, on a line as $AG$, forming with $AZ$ an infinitely small angle $ZAG$, represented by $d\theta$. Then the force $dx$, in the direction $AX$, may be resolved into two forces, the one $dx'$ in the direction $AZ$, the other $dx''$ in the direction $AE$, and as this last force is inclined to $AX$ by the angle $XAE={\frac {\pi }{2}}-\theta$, we shall have as above $dx''=dx.\varphi \left({\frac {\pi }{2}}-\theta \right)$; or by substituting the preceding value of $\varphi \left({\frac {\pi }{2}}-\theta \right)={\frac {y}{z}},dx''={\frac {ydx}{z}}$.

f (6) This angle is equal to GAE.== '^—dd ; and if the force z' in the direction A G is resolved into two forces in tiie directions A Z, AE, the last will (by tiie nature of the function cp) be represented hy z' .cp( -^ — dd.

X (7) Because (p r- — d& contains only die quantities ^, <?^, but does not explicitiy con-

tain ^. Moreover, the function (p ( 1^ — di being developed m the usual manner, according

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