Page:TolmanNon2.djvu/4

Remembering that these were bodies which had the same mass when at rest, we see that the mass of a body is inversely proportional to ${\sqrt {1-{\frac {u^{2}}{c^{2}}}}}$, where $u$ is its velocity, and have thus derived the desired relation,-

$m={\frac {m_{0}}{\sqrt {1-{\frac {u^{2}}{c^{2}}}}}}$

Collision of Any Type.

A treatment of the general case of any type of collision between any two bodies elastic or otherwise is also possible and leads to the same conclusion as to the desirability of using the expression $m_{0}/{\sqrt {1-{\tfrac {u^{2}}{c^{2}}}}}$ for the mass of a moving body.

For the mass on of a body moving with the velocity $u$ let us write the equation $m=m_{0}f\left(u^{2}\right)$ where $f()$is the function whose form we wish to determine. The mass is written as a function of the square of the velocity, since from the homogeneity of space the mass will be independent of the direction of the velocity, and the mass is made proportional to the mass at rest since a moving body may evidently be divided into parts without change in mass.

Let us now consider two bodies having the masses $m_{0}$ and $n_{0}$ when at rest, moving with the velocities u and v before collision and with the velocities U and V after a collision has taken place.

From the principle of the conservation of mass we have,-

${\begin{array}{l}m_{0}f\left(u_{x}^{2}+u_{y}^{2}+u_{z}^{2}\right)+n_{0}f\left(v_{x}^{2}+v_{y}^{2}+v_{z}^{2}\right)\\\\\qquad =m_{0}f\left(U_{x}^{2}+U_{y}^{2}+U_{z}^{2}\right)+n_{0}f\left(V_{x}^{2}+V_{y}^{2}+V_{z}^{2}\right)\end{array}}$

(1)

and from the principle of the conservation of momentum,

${\begin{array}{l}m_{0}f\left(u_{x}^{2}+u_{y}^{2}+u_{z}^{2}\right)u_{x}+n_{0}f\left(v_{x}^{2}+v_{y}^{2}+v_{z}^{2}\right)v_{x}\\\\\qquad =m_{0}f\left(U_{x}^{2}+U_{y}^{2}+U_{z}^{2}\right)U_{x}+n_{0}f\left(V_{x}^{2}+V_{y}^{2}+V_{z}^{2}\right)V_{x},\end{array}}$

(2)

${\begin{array}{l}m_{0}f\left(u_{x}^{2}+u_{y}^{2}+u_{z}^{2}\right)u_{y}+n_{0}f\left(v_{x}^{2}+v_{y}^{2}+v_{z}^{2}\right)v_{y}\\\\\qquad =m_{0}f\left(U_{x}^{2}+U_{y}^{2}+U_{z}^{2}\right)U_{y}+n_{0}f\left(V_{x}^{2}+V_{y}^{2}+V_{z}^{2}\right)V_{y},\end{array}}$

(3)

${\begin{array}{l}m_{0}f\left(u_{x}^{2}+u_{y}^{2}+u_{z}^{2}\right)u_{z}+n_{0}f\left(v_{x}^{2}+v_{y}^{2}+v_{z}^{2}\right)v_{z}\\\\\qquad =m_{0}f\left(U_{x}^{2}+U_{y}^{2}+U_{z}^{2}\right)U_{z}+n_{0}f\left(V_{x}^{2}+V_{y}^{2}+V_{z}^{2}\right)V_{z}.\end{array}}$

(4)

These velocities $u_{x},u_{y},u_{z},v_{x},v_{y},v_{z},U_{x}$, &c., are measured with respect to some definite system of "space time" coordinates. An observer moving past this system of co-ordinates with the velocity $\phi$ in the X direction would find