Page:SahaSpaceTime.djvu/14

 "Thermodynamics and Radiation," for "Electromagnetic phenomena", and finally for "Mechanics with the maintenance of the idea of mass."

For this last mentioned province of physics, the question may be asked: if there is a force with the components X, Y, Z (in the direction of the space-axes) at a world-point (x, y, z, t), where the velocity-vector is $$\left(\dot{x},\ \dot{y},\ \dot{z},\ \dot{t}\right)$$, then how are we to regard this force when the system of reference is changed in any possible manner? Now it is known that there are certain well-tested theorems about the ponderomotive force in electromagnetic fields, where the group Gc is undoubtedly permissible. These theorems lead us to the following simple rule; if the system of reference be changed in any way, then the supposed force is to be put as a force in the new space-coordinates in such a manner, that the corresponding vector with the components

$\dot{t}X,\ \dot{t}Y,\ \dot{t}Z,\ \dot{t}T,$

where

$T=\frac{1}{c^{2}}\ \left(\frac{\dot{x}}{\dot{t}}X+\frac{\dot{y}}{\dot{t}}Y+\frac{\dot{z}}{\dot{t}}Z\right)=\frac{1}{c^{2}}$|undefined

(the rate of which work is done at the world-point), remains unaltered. This vector is always normal to the velocity-vector at P. Such a force-vector, representing a force at P, may be called a moving force-vector at P.

Now the world-line passing through P will be described by a substantial point with the constant mechanical mass m. Let us call m-times the velocity-vector at P as the