Page:The principle of relativity (1920).djvu/157

 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 ([.x], [.y], [.z], [.t]), 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 G_{c} 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

[.t]X, [.t]Y, [.t]Z, [.t]T,

where T = 1/c^2 ([.x]/[.t] X + [.y]/[.t] Y + [.z]/[.t] Z) = 1/c^2 (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