Let the point (_x_, _y_, _z_, _t_) in the sichel λ = 0 have the values _x_ + δ_x_, _y_ + δ_y_, _z_ + δ_z_, _t_ + δ_t_, when the parameter has the value λ; these magnitudes are then functions of (_x_, _y_, _z_, _t_, λ). Let us now conceive of an infinitely thin space-time filament at the point (_x_ _y_ _z_ _t_) with the normal section of contents _d_J_{_n_} and if _d_J_{_n_} + δ_d_J_{_n_} be the contents of the normal section at the corresponding position of the varied filament, then according to the principle of conservation of mass—(ν + _d_ν being the rest-mass-density at the varied position), (8) (ν + δν) (_d_J_{_n_} + δ_d_J_{_n_}) = ν_d_J_{_n_} = _dm_. In consequence of this condition, the integral (7) taken over the whole range of the _sichel_, varies on account of the displacement as a definite function N + δN of λ, and we may call this function N + δN as the _mass action_ of the virtual displacement. If we now introduce the method of writing with indices, we shall have (9) _d_(_x__{_h_} + δ_x__{_h_}) = _dx__{_h_} + ∑_{_k_} ∂δ_x__{_h_}/∂_x__{_k_} + ∂δ_x__{_h_}/∂λ _d_λ _k_ = 1, 2, 3, 4 _h_ = 1, 2, 3, 4 Now on the basis of the remarks already made, it is clear that the value of N + δN, when the value of the parameter is λ, will be:— (10) N + δN = ∫∫∫∫ ((ν_d_(τ + δτ))/_d_τ)_dx_ _dy_ _dz_ _dt_, the integration extending over the whole sichel _d_(τ + δτ) where _d_(τ + δτ) denotes the magnitude, which is deduced from √(-(_dx₁_ + _d_δ_x₁_)² - (_dx₂_ + _d_δ_x₂_)² - (_dx₃_ + _d_δ_x₃_)² - (_dx₄_ + _d_δ_x₄_)²) by means of (9) and _dx₁_ = ω₁ _d_τ, _dx₂_ = ω₂ _d_τ, _dx₃_ = ω₃ _d_τ, _dx₄_ = ω₄ _d_τ, _d_λ = 0 therefore:— (11) (_d_(τ + δτ))/_d_τ = √( -∑(ω_{_h_} + ∑(∂δ_x__{_h_}/∂_x__{_k_})ω_{_k_})²) _k_ = 1, 2, 3, 4. _h_ = 1, 2, 3, 4. We shall now subject the value of the differential quotient (12) ((_d_(N + δN))/_d_λ) (λ = 0) to a transformation. Since each δ_x__{_h_} as a function of (_x_, _y_, _z_, _t_) vanishes for the zero-value of the parameter λ, so in general _d_δ_x__{_k_}/(∂_x__{_h_} = 0, for λ = 0. Let us now put (∂δ_x__{_h_}/∂λ) = ξ_{_h_} (_h_ = 1, 2, 3, 4) (13) 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_}.