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

with that of the Earth. Then those approaching it at any angle ${\displaystyle \theta }$ less than that which makes ${\displaystyle \theta _{1}=90^{\circ }}$ will be visible at sunset ; those at a greater angle, at sunrise. The angle ${\displaystyle \theta _{1}}$is given by the relation,

in which ${\displaystyle a}$ is the Earth's velocity, ${\displaystyle x}$ the meteor's, and ${\displaystyle \theta _{1}}$ is reckoned from the Earth's quit.

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

where ${\displaystyle \phi }$ is the azimuth,

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

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

for sunset, ${\displaystyle \int _{0}^{\theta _{1}}\int _{0}^{360^{\circ }}\sin {\theta }\cdot d\theta \cdot d\phi }$,

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

The mean inclination at sunset is

in which ${\displaystyle \theta _{1}}$ must be expressed in terms of ${\displaystyle \theta }$, 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,