Page:The evolution of worlds - Lowell.djvu/284

 242 with that of the Earth. Then those approaching it at any angle $$\theta$$ less than that which makes $$\theta_1 = 90^\circ$$ will be visible at sunset ; those at a greater angle, at sunrise. The angle $$\theta_1$$is given by the relation,

in which $$a$$ is the Earth's velocity, $$x$$ the meteor's, and $$\theta_1$$ is reckoned from the Earth's quit.

The portion of the celestial dome covered at sunset is, therefore,

where $$\phi$$ is the azimuth,

that at sunrise, $$\int_{0}^{180^\circ} \int_{0}^{360^\circ} \sin{\theta}\cdot d\theta \cdot d\phi $$.

If the meteors have direct motion only, $$\theta$$ can never exceed 90°, and the limits become,

for sunset, $$\int_{0}^{\theta_1} \int_{0}^{360^\circ} \sin{\theta}\cdot d\theta \cdot d\phi $$,

and for sunrise, $$\int_{0}^{90^\circ} \int_{0}^{360^\circ} \sin{\theta}\cdot d\theta \cdot d\phi $$.

The mean inclination at sunset is

in which $$\theta_1$$ must be expressed in terms of $$\theta$$, etc.

From this it appears that the relative number of bodies, travelling in all directions and at parabolic speed, which the Earth would encounter at sunrise and sunset respectively would be:—

and with the speed of the short-period comets,