∫∫ 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.
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.