X_{_x_} + Y_{_y_} + Z_{_z_} + T_{_t_} + X_{_x_} (∂δ_x_)/∂_x_ +
X_{_y_} (∂δ_x_)/∂_y_ + ... Z_{_z_} (∂δ_z_)/∂_z_
- X_{_t_} (∂δ_x_/∂_t_ - ... + T_{_x_} (∂δ_t_)/∂_x_ + ... T_{_t_}
(∂δ_t_)/∂_t_
we can now postulate the following _minimum principle in mechanics_.
_If any space-time Sichel be bounded, then for each virtual displacement
in the Sichel, the sum of the mass-works, and tension works shall always
be an extremum for that process of the space-time line in the Sichel
which actually occurs._
The meaning is, that for each virtual displacement,
([_d_(·δN + δW)]/_d_λ)_{λ = 0} = 0 (18)
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 ∂_w__{_h_}/∂τ = K_{_h_} + χ_w__{_h_} (_h_ = 1, 2, 3, 4)
whence K_{_h_} = ∂S_{1 _h_}/∂_x₁_ + ∂S_{2 _h_}/∂_x₂_ + ∂S_{3
_h_}/∂_x₃_ + ∂S_{4 _h_}/∂_x₄_, (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__ẇ_ = - 1. By
multiplying (19) by _w__{_h_}, and summing the four, we obtain X = K_ẇ_,
and therefore clearly K + (K_ẇ_)_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