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

${\frac {1}{\sqrt {1-q^{2}/c^{2}}}}={\frac {\left(1-{\frac {v{\dot {x}}}{c^{2}}}\right)\kappa }{\sqrt {1-q^{2}/c^{2}}}}$

(12)

It has been shown in an earlier article^[1] 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:

$m'=\left(1-{\frac {v{\dot {x}}}{c^{2}}}\right)\kappa m,$

(13)

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 :

${\dot {m}}'={\dot {m}}-{\frac {mv}{c^{2}}}{\ddot {x}}\left(1-{\frac {v{\dot {x}}}{c^{2}}}\right)^{-1}$

(14)

${\dot {x}}$ 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

1. Phil. Mag. xxiii. p. 375 (1912).