By applying the methods of the Calculus of Variations, the following four differential equations at once follow from this minimal principle by means of the transformation (14), and the condition (15).
(19) ν [part]w_{h}/[part]τ = K_{h} + χw_{h} (h = 1, 2, 3 4)
whence K_{h} = [part]S_{1 h}/[part]x_{1} + [part]S_{2 h}/[part]x_{2} + [part]S_{3 h}/[part]x_{3} + [part]S_{4 h}/[part]x_{4}, (20)
are components of the space-time vector 1st kind K = lor S, and X is a factor, which is to be determined from the relation w[=w] = - 1. By multiplying (19) by w_{h}, and summing the four, we obtain X = Kw̄, and therefore clearly K + (K[=w])w will be a space-time vector of the 1st kind which is normal to w. Let us write out the components of this vector as
X, Y, Z, ·iT
Then we arrive at the following equation for the motion of matter,
(21) ν d/dτ (dx/dτ) = X, ν ·d/dτ (dy/dτ) = Y, ν d/dτ (dz/dτ) = Z,
ν d/dτ (dx/dτ) = T?], and we have also
(dx/dτ)^2 + (dy/dτ)^2 + (dz/dτ)^2 > (dt/dτ)^2 = -1,
and X dx/dτ + Y dx/dτ + Z dz/dτ = T dt/dτ.
On the basis of this condition, the fourth of equations (21) is to be regarded as a direct consequence of the first three.
From (21), we can deduce the law for the motion of a material point, i.e., the law for the career of an infinitely thin space-time filament.
