Page:The principle of relativity (1920).djvu/131

 If we allow R^0 to converge to a point, we come to the conception of an infinitely thin space-time filament. In such a case, a space-time line will be thought of as a principal line and by the term 'Proper-time' of the filament will be understood the 'Proper-time' which is laid along this principal line; under the term normal cross-section of the filament, we shall understand the cross-section upon the space which is normal to the principal line through P.

We shall now formulate the principle of conservation of mass.

To every space R at a time t, belongs a positive quantity—the mass at R at the time t. If R converges to a point (x, y, z, t), then the quotient of this mass, and the volume of R approaches a limit [mu](x, y, z, t), which is known as the mass-density at the space-time point (x, y, z, t).

The principle of conservation of mass says—that for an infinitely thin space-time filament, the product [mu]dJ, where [mu] = mass-density at the point (x, y, z, t) of the filament (i.e., the principal line of the filament), dJ = contents of the cross-section normal to the t axis, and passing through (x, y, z, t), is constant along the whole filament.

Now the contents dJ_{n} of the normal cross-section of the filament which is laid through (x, y, z, t) is

(4) dJ_{n} = (1/[sqrt](1 - u^2))dJ = -[iota][omega]_{4} dJ = (dt/dτ)dJ.

and the function [nu] = [mu]/-[iota][omega]_{4} = [mu][sqrt](1 - u^2)) = [mu]([part][tau]/[part]t. (5)

may be defined as the rest-mass density at the position