Page:The principle of relativity (1920).djvu/230

 can take any values; This signifies that any velocity

v = [sqrt]((dx_{1}/dx_{4})^2 + (dx_{2}/dx_{4})^2 + (dx_{3}/dx_{4})^2)

can appear which is less than the velocity of light in vacuum (v < 1). If we finally limit ourselves to the consideration of the case when v is small compared to the velocity of light, it signifies that the components

dx_{1}/ds, dx_{2}/ds, dx_{3}/ds,

can be treated as small quantities, whereas dx_{4}/ds is equal to 1, up to the second-order magnitudes (the second point of view for approximation).

Now we see that, according to the first view of approximation, the magnitudes Γ_{μν}^τ's are all small quantities of at least the first order. A glance at (46) will also show, that in this equation according to the second view of approximation, we are only to take into account those terms for which μ = ν = 4.

By limiting ourselves only to terms of the lowest order we get instead of (46), first, the equations:—

d^2x_{τ}/dt^2 = Γ_{4 4}^τ, where ds = dx_{4} = dt,

or by limiting ourselves only to those terms which according to the first stand-point are approximations of the first order,

d^2x_{τ}/dt^2 = [44 τ] (τ = 1, 2, 3)

d^2x_{4}/dt^2 = -[4^4 4].