Page:Mécanique céleste Vol 1.djvu/34

4 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 $$d\theta$$; 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 $$ \frac\pi 2 - d\theta $$; we shall thus have, by what precedes,

$$dx'' = z' \cdot \varphi \left( \frac\pi 2 - d\theta \right)$$;

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 $$\theta$$; we shall therefore have

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


 * (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