Page:The principle of relativity (1920).djvu/139

 Let x, y, z, t, denote a point on a principal line chosen in any manner within the filament. We shall form the equations (21) for the points of the normal cross section of the filament through x, y, z, t, and integrate them, multiplying by the elementary contents of the cross section over the whole space of the normal section. If the integrals of the right side be R_{x} R_{y} R_{z} R_{t} and if m be the constant mass of the filament, we obtain

(22) m d/dτ dx/dτ = R_{x}, m d/dτ dy/dτ = R_{y}, m d/dτ dz/dτ = R_{z}, m d/dτ dt/dτ = R_{t}

R is now a space-time vector of the 1st kind with the components (R_{x} R_{y} R_{z} R_{t}) which is normal to the space-time vector of the 1st kind w,—the velocity of the material point with the components

dx/dτ, dy/dτ, dz/dτ, i dt/dτ.

We may call this vector R ''the moving force of the material point''.

If instead of integrating over the normal section, we integrate the equations over that cross section of the filament which is normal to the t axis, and passes through (x, y, z, t), then [See (4)] the equations (22) are obtained, but

are now multiplied by dτ/dt; in particular, the last equation comes out in the form,

m d/dt (dt/dτ) = w_{x} R_{x} dτ/dt + w_{y} R_{y} dτ/dt + w_{z} R_{z} dτ/dt.

The right side is to be looked upon ''as the amount of work done per unit of time'' at the material point. In this