in which a is the Earth’s velocity, x the meteor’s, and θ₁ is reckoned from the Earth’s quit.

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

⌠θ₁⌠360°
⎮ ⎮ sin θ·dθ·dφ,
⌡0⌡0

where φ is the azimuth,

that at sunrise,

⌠180°⌠360°
⎮ ⎮ sin θ·dθ·dφ.
⌡θ₁⌡0

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

for sunset,

⌠θ₁⌠360°
⎮ ⎮ sin θ·dθ·dφ,
⌡0⌡0

and for sunrise,