Page:The principle of relativity (1920).djvu/142

 Now by (OA´/B^*D^*) is to be understood the ratio of the two vectors in question. It is clear that this proposition at once shows the covariant character with respect to a Lorentz-group.

Let us now ask how the space-time filament of F behaves when the material point F^* has a uniform translatory motion, i.e., the principal line of the filament of F^* is a line. Let us take the space time null-point in this, and by means of a Lorentz-transformation, we can take this axis as the t-axis. Let x, y, z, t, denote the point B, let τ^* denote the proper time of B^*, reckoned from O. Our proposition leads to the equations

(25) d^2x/dτ^2 = - m^*x/(t - tau ?]^*)^3, d^2y/dτ^2 = - m^*y/(t - τ^*)^3

d^2z/dτ^2 = -m^*z/(t - tau ?]*)^3, (26) d^2t/dτ^2 = -m^ (t - τ^*)^2 d(t - τ^*)/dt

where (27) x^2 + y^2 + z^2 = (t - τ^*)^2

and (28) (dx/dτ)^2 + (dy/dτ)^2 + (dz/dτ)^2 = (dt/dτ)^2 - 1

In consideration of (27), the three equations (25) are of the same form as the equations for the motion of a material point subjected to attraction from a fixed centre according to the Newtonian Law, only that instead of the time t, the proper time τ of the material point occurs. The fourth equation (26) gives then the connection between proper time and the time for the material point.

Now for different values of τ´, the orbit of the space-point (x y z) is an ellipse with the semi-major axis a and the eccentricity e. Let E denote the excentric anomaly, Τ