Page:The principle of relativity (1920).djvu/133

 which is the condition of continuity

([part]μu_{x}/[part]x) + ([part]μu_{y}/[part]y) + ([part]μu_{z}/[part]z) + ([part]μ/[part]t) = 0.

Further let us form the integral

N = [integral] [integral][integral][integral] νdxdydzdt  (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 dτ 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 νdJ_{n} = dm and write τ^0, τ^l for the 'Proper-time' of the upper and lower boundary of the sichel.

Then the integral (7) can be denoted by

[integral][integral] νdJ_{n} dτ = [integral] (τ^l-τ^0) 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 + δx, y + δy, z + δz, t + δt, when the parameter has