λ = 0
then on the basis of (10) and (11), we have the expression (12):—
= -∫∫∫∫ ∑ ω_{_h_}((∂ξ_{_h_}/∂_x₁_)ω₁ + (∂ξ_{_h_}/∂_x₂_)ω₂
+(∂ξ_{_h_}/∂_x₃_)ω₃ + (∂ξ_{_h_}/∂_x₄_)ω₄)
_dx dy dz dt_
for the system (_x₁_ _x₂_ _x₃_ _x₄_) on the boundary of the _sichel_,
(δ_x₁_ δ_x₂_ δ_x₃_ δ_x₄_) shall vanish for every value of λ and
therefore ξ₁, ξ₂, ξ₃, ξ₄ are nil. Then by partial integration, the
integral is transformed into the form
∫∫∫∫ ∑ ξ_{_h_}(∂νω_{_h_}ω₁/∂_x₁_ + ∂νω_{_h_}ω₂/∂_x₂_ +
∂νω_{_h_}ω₃/∂_x₃_ + ∂νω_{_h_}ω₄/∂_x₄_)
_dx dy dz dt_
the expression within the bracket may be written as
= ω_{_h_} ∑ ∂νω_{_k_}/∂_x__{_k_} + ν∑ω_{_k_}∂ω_{_h_}/∂_x__{_k_}.
The first sum vanishes in consequence of the continuity equation (_b_).
The second may be written as
(∂ω_{_h_}/∂_x₁_)(_dx₁_/_d_τ) + (∂ω_{_h_}/∂_x₂_)(_dx₂_/_d_τ) +
(∂ω_{_h_}/∂_x₃_)(_dx₃_/_d_τ) + (∂ω_{_h_}/∂_x₄_)(_dx₄_/_d_τ)
= _d_ω_{_h_}/_d_τ = (_d_/_d_τ)(_dx__{_h_}/_d_τ)
whereby (_d_/_d_τ) is meant the differential quotient in the direction
of the space-time line at any position. For the differential quotient
(12), we obtain the final expression
(14) ∫∫∫∫ ν((∂ω₁/∂τ)ξ₁ + (∂ω₂/∂τ)ξ₂ + (∂ω₃/∂τ)ξ₃ + (∂ω₄/∂τ)ξ₄)
_dx dy dz dt_.
For a virtual displacement in the _sichel_ we have postulated the
condition that the points supposed to be substantial shall advance
normally to the curves giving their actual motion, which is λ = 0; this
condition denotes that the ξ_{_h_} is to satisfy the condition
_w₁_ξ₁ + _w₂_ξ₂ + _w₃_ξ₃ + _w₄_ξ₄ = 0. (15)
Let us now turn our attention to the Maxwellian tensions in the
electrodynamics of stationary bodies, and let us consider the results in
§ 12 and 13; then we find that Hamilton’s Principle can be reconciled to
the relativity postulate for continuously extended elastic media.
At every space-time point (as in § 13), let a space time matrix of the
2nd kind be known
(16) S =
| S₁₁ S₁₂ S₁₃ S₁₄ | = | X_{_x_} Y_{_x_} Z_{_x_} -_i_T_{_x_} |
| S₂₁ S₂₂ S₂₃ S₂₄ | = | X_{_y_} Y_{_y_} Z_{_y_} -_i_T_{_y_} |
| S₃₁ S₃₂ S₃₃ S₃₄ | = | X_{_z_} Y_{_z_} Z_{_z_} -_i_T_{_z_} |
| S₄₁ S₄₂ S₄₃ S₄₄ | = | -_i_X_{_t_} -_i_Y_{_t_} -_i_Z_{_t_} T_{_t_}
|
where X_{_n_} Y_{_x_} .....X_{_z_}, T_{_t_} are real magnitudes.
For a virtual displacement in a space-time sichel (with the previously
applied designation) the value of the integral
(17) W + δW = ∫∫∫∫ (∑S_{_h k_} (∂(_x__{_k_} +
δ_x__{_k_}))/∂_x__{_h_} _dx dy dz dt_
extended over the whole range of the _sichel_, may be called the
tensional work of the virtual displacement.