Page:The principle of relativity (1920).djvu/155

 at the point R of the surface, then the vectors shall be called normal to each other. Accordingly

c^2tt_{1} - xx_{1} - yy_{1} - zz_{1} = 0,

which is the condition that the vectors with the components (x, y, z, t) and (x_{1} y_{1} z_{1} t_{1}) are normal to each other.

For the measurement of vectors in different directions, the unit measuring rod is to be fixed in the following manner;—a space-like vector from 0 to -F = I is always to have the measure unity, and a time-like vector from O to +F = 1, t > 0 is always to have the measure 1/c.

Let us now fix our attention upon the world-line of a substantive point running through the world-point (x, y, z, t); then as we follow the progress of the line, the quantity

dτ = (1/c) [sqrt](c^2dt^2 - dx^2 - dy^2 - dz^2),

corresponds to the time-like vector-element (dx, dy, dz, dt).

The integral τ = [integral]dτ, taken over the world-line from

any fixed initial point P_{0} to any variable final point P, may be called the "Proper-time" of the substantial point at P_{0} upon the world-line. We may regard (x, y, z, t), i.e., the components of the vector OP, as functions of the

"proper-time" τ; let ([.x], [.y], [.z], [.t]) denote the first differential-quotients,

and ([..x]?], [..y], [..z], [..t]) the second differential quotients

of (x, y, z, t) with regard to τ, then these may respectively