(4) |
and the function
(5) |
may be defined as the rest-mass density at the position x, y, z, t. Then the principle of conservation of mass can be formulated in this manner: —
For an infinitely thin space-time filament, the product of the rest-mass density and the contents of the normal cross-section is constant along the whole filament.
In any space-time filament, let us consider two cross-sections and , 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 . The finite range enclosed between and shall be called a space-time sickle[WS 1] is the lower boundary, and is the upper boundary of the sickle.
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 sickle, there corresponds an exit point of the same by the upper boundary, whereby for both, the product taken in the sense of (4) and (5), has got the same value. Therefore the difference of the two integrals (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
the integration being extended over the whole range of the sickle, and (comp. (67), § 12)
If the sickle reduces to a point, then the differential equation
(6) | , |
which is the condition of continuity
- ↑ Saha used the German word "Sichel" in this edition.