Earthquakes and other earth movements

by John Milne

• t₁ = t - m
• t₂ = t - p
• t₃ = t - q
• t₄ = t - r.

Subtracting equation No. 1 from each of the equations 2, 3, 4, and 5, we obtain,

• a₁² + b₁² - 2a₁ x - 2b₁ y = v² (t₁² - t²) = v² (m² - 2t m)
• a₂² + b₂² - 2a₂ x - 2b₂ y = v² (t₂² - t²) = v² (p² - 2t p)
• a₃² + b₃² - 2a₃ x - 2b₃ y = v² (t₃² - t²) = v² (q² - 2t q)
• a₄² + b₄² - 2a₄ x - 2b₄ y = v² (t₄² - t²) = v² (r² - 2t r)

Now let v² = u, and 2v² t = w.

Then

1. 2a₁ x + 2b₁ y + u m² - n m = a₁² + b₁²
2. 2a₂ x + 2b₂ y + u p² - n p = a₂² + b₂²
3. 2a₃ x + 2b₃ y + u q² - n q = a₃² + b₃²
4. 2a₄ x + 2b₄ y + u r² - n r = a₄² + b₄²

We have here four simple equations containing the four unknown quantities x, y, u, and w.

x and y determine the origin of the shock. The absolute velocity v equals √ u. From v and w we obtain t. Substituting x, y, v, and t in the first equation we obtain z.

We have here assumed that the points of observation have all about the same elevation above sea level.

If it is thought necessary to take these elevations into account, a sixth equation may be introduced.