Page:TolmanEquations.djvu/3

 equations that, if a point has a uniform acceleration $$\left(\ddot{x},\ddot{y},\ddot{z}\right)$$ with respect to an observer in system S, it will not in general have a uniform acceleration $$\left(\ddot{x}',\ddot{y}',\ddot{z}'\right)$$ in another system S', since the acceleration in system S' depends not only on the constant acceleration but also on the velocity in system S which is necessarily varying.

We may next obtain transformation equations for a useful function of the velocity, namely, $$\frac{1}{\sqrt{1-q^{2}/c^{2}}}$$, where we have placed $$q^{2}=\dot{x}^{2}+\dot{y}^{2}+\dot{z}^{2}$$. By substitution of equations (6), (7), and (8) and simplification we obtain

It has been shown in an earlier article that the principles of non-Newtonian mechanics lead to the equation $$m=\frac{m_{0}}{\sqrt{1-q^{2}/c^{2}}}$$ for the mass of a moving body, where $$m_{0}$$ is the mass of the body at rest and $$q$$ is its velocity. By substitution of equation (12) we may obtain the following equation for transforming measurements of mass from one system of coordinates to the other:

where $$m$$ is the mass of the body and $$x$$ the X component of its velocity as measured in system S and $$m'$$ its mass as measured in system S'.

By differentiation of equation (13) and simplification we may obtain the following transformation equation for the rate at which the mass of a body is changing owing to change in velocity :

$$\dot{x}$$ and $$\ddot{x}$$ are the X components of the velocity and acceleration of the body in question as measured in system S.

We are now in a position to obtain transformation equations for the force acting on a particle. The force