Page:The Solar System - Six Lectures - Lowell.djvu/138



by the sphere within it. It is therefore pulled by the force $$\scriptstyle \frac{m}{r^{2}}=\frac{\delta r^{3}}{r^{2}}=\delta r$$, where $$\scriptstyle \delta$$ is the density.

Since the force is thus linear it may be resolved into two harmonic motions and becomes motion in an ellipse with the acceleration directed to the centre, or elliptic harmonic motion whose equation is expressed in vector coördinates:—

whence: $$ \scriptstyle \begin{cases} \rho  & = a \cos{(nt+e)} + b \sin{(nt + e)} \\ \rho' & = n [ a sin{(nt+e)} - b \cos{(nt+e)} ] \\ \rho'' & = n^{2} [ a \cos{(nt + e)} + b \sin{(nt + e)} ] =-n^{2}p \\ \end{cases} $$

The form of the ellipse depends upon the amount and direction of the initial velocity of the particle.

This equation shows, first, that the period of rotation is the same for all the particles; and second, that the angular speed in such different nebulae is as the square root of their densities.

When the mass has practically collected in the centre, the force is $$\scriptstyle \frac{m}{r^{2}}$$, or the ordinary law of gravitation, giving elliptic motion with acceleration directed to the focus, or elliptic motion par excellence.

At any intermediate stage of the process he supposes the force to be represented by $$\scriptstyle f=\alpha r+\frac{\beta}{r^{2}}$$ $$\scriptstyle\alpha$$ gradually dying out and $$\scriptstyle\beta$$ increasing as centralization goes on.