In any space-time filament, let us consider two cross-sections Q° and Q′, which have only the points on the boundary common to each other; let the space-time lines inside the filament have a larger value of _t_ on Q′ than on Q°. The finite range enclosed between Q° and Q′ shall be called a space-time _sichel_,[29] Q′ is the lower boundary, and Q′ is the upper boundary of the _sichel_. If we decompose a filament into elementary space-time filaments, then to an entrance-point of an elementary filament through the lower boundary of the _sichel_, there corresponds an exit point of the same by the upper boundary, whereby for both, the product νdJ_{n} taken in the sense of (4) and (5), has got the same value. Therefore the difference of the two integrals ∫ν_dJ__{n} (the first being extended over the upper, the second upon the lower boundary) vanishes. According to a well-known theorem of Integral Calculus the difference is equivalent to ∫∫∫∫ lor ν[=ω] _dx dy dz dt_, the integration being extended over the whole range of the _sichel_, and (comp. (67), § 12) lor ν[=ω] = (∂νω₁/∂_x₁_) + (∂νω₂/∂_x₂_) + (∂νω₃/∂_x₃_) + (∂νω₄/∂_x₄_). If the _sichel_ reduces to a point, then the differential equation lor ν[=ω] = 0, (6) which is the condition of continuity (∂μ_u__{_x_}/∂_x_) + (∂μ_u__{_y_}/∂_y_) + (∂μ_u__{_z_}/∂_z_) + (∂μ/∂_t_) = 0. Further let us form the integral N = ∫ ∫∫∫ ν _dx dy dz dt_ (7) extending over the whole range of the space-time _sichel_. We shall decompose the _sichel_ into elementary space-time filaments, and every one of these filaments in small elements _d_τ of its proper-time, which are however large compared to the linear dimensions of the normal cross-section; let us assume that the mass of such a filament ν_d_J_{_n_} = _dm_ and write τ⁰, τ^l for the ‘Proper-time’ of the upper and lower boundary of the _sichel_. Then the integral (7) can be denoted by ∫∫ ν_d_J_{_n_} _d_τ = ∫ (τ′-τ⁰) _dm_. taken over all the elements of the sichel. Now let us conceive of the space-time lines inside a space-time _sichel_ as material curves composed of material points, and let us suppose that they are subjected to a continual change of length inside the sichel in the following manner. The entire curves are to be varied in any possible manner inside the _sichel_, while the end points on the lower and upper boundaries remain fixed, and the individual substantial points upon it are displaced in such a manner that they always move forward normal to the curves. The whole process may be analytically represented by means of a parameter λ, and to the value λ = 0, shall correspond the actual curves inside the _sichel_. Such a process may be called a virtual displacement in the sichel.