I. Kinematical Part.Show the German source

§ 5. The Addition Theorem of Velocities.Show the German source

In the system $k$, moving along the $X$-axis of the system $K$ with the velocity $v$, let a point move according to the equations:Show the German source $$\begin{aligned} \xi &= w_\xi \tau , \\ \eta &= w_\eta \tau , \\ \zeta &= 0 , \end{aligned}$$$\xi$Position in the moving system$w_{\xi }$Velocity along x, moving system$\tau$Time in the moving system$\eta$Position in the moving system$w_{\eta }$Velocity along y, moving system$\zeta$Position in the moving systemExplain this equationShow the German source where $w_\xi$ and $w_\eta$ denote constants.Show the German source

The motion of the point relative to the system $K$ is sought.Show the German source If, with the help of the transformation equations developed in § 3, we introduce the quantities $x$, $y$, $z$, $t$ into the equations of motion of the point, we obtain:Show the German source $$\begin{aligned} x &= \frac{w_\xi + v}{1 + \frac{v w_\xi}{V^2}} t , \\ y &= \frac{\sqrt{1 - \left( \frac{v}{V} \right)^2}}{1 + \frac{v w_\xi}{V^2}} w_\eta t , \\ z &= 0 . \end{aligned}$$$x$Position in the stationary system$w_{\xi }$Velocity along x, moving system$v$Speed of the moving system$V$Speed of light$t$Time in the stationary system$y$Position in the stationary system$w_{\eta }$Velocity along y, moving system$z$Position in the stationary systemExplain this equationShow the German source The law of the parallelogram of velocities therefore holds, according to our theory, only to a first approximation.Show the German source We put:Show the German source $$\begin{aligned} U^2 &= \left( \frac{d x}{d t} \right)^2 + \left( \frac{d y}{d t} \right)^2 , \\ w^2 &= w_\xi^2 + w_\eta^2 \end{aligned}$$$U$Composed velocity (stationary system)$x$Position in the stationary system$t$Time in the stationary system$y$Position in the stationary system$w$Velocity in the moving frame$w_{\xi }$Velocity along x, moving system$w_{\eta }$Velocity along y, moving systemExplain this equationShow the German source andShow the German source $$\alpha = \operatorname{arctg} \frac{w_y}{w_x} ;$$$\alpha$Velocity direction angle: alpha$w_y$Velocity along y, moving system$w_x$Velocity along x, moving systemExplain this equationShow the German source $\alpha$ is then to be regarded as the angle between the velocities $v$ and $w$.Show the German source After a simple calculation we obtain:Show the German source $$U = \frac{\sqrt{(v^2 + w^2 + 2 v w \cos \alpha) - \left( \frac{v w \sin \alpha}{V} \right)^2}}{1 + \frac{v w \cos \alpha}{V^2}} .$$$U$Composed velocity (stationary system)$v$Speed of the moving system$w$Velocity in the moving frame$\alpha$Velocity direction angle: alpha$V$Speed of lightExplain this equationShow the German source It is noteworthy that $v$ and $w$ enter the expression for the resultant velocity symmetrically.Show the German source If $w$ also has the direction of the $X$-axis ($\Xi$-axis), we obtain:Show the German source $$U = \frac{v + w}{1 + \frac{v w}{V^2}} .$$$U$Composed velocity (stationary system)$v$Speed of the moving system$w$Velocity in the moving frame$V$Speed of lightExplain this equationShow the German source From this equation it follows that from the composition of two velocities which are smaller than $V$ there always results a velocity smaller than $V$.Show the German source For if we put $v = V - \varkappa$, $w = V - \lambda$, where $\varkappa$ and $\lambda$ are positive and smaller than $V$, then:Show the German source $$U = V \frac{2 V - \varkappa - \lambda}{2 V - \varkappa - \lambda + \frac{\varkappa \lambda}{V}} < V .$$$U$Composed velocity (stationary system)$V$Speed of light$\varkappa$Speed deficit from light speed: kappa, lambda$\lambda$Speed deficit from light speed: kappa, lambdaExplain this equationShow the German source

It follows further that the velocity of light $V$ cannot be altered by composition with a “sub-light velocity”.Show the German source For this case we obtain:Show the German source $$U = \frac{V + w}{1 + \frac{w}{V}} = V .$$$U$Composed velocity (stationary system)$V$Speed of light$w$Velocity in the moving frameExplain this equationShow the German source We could also have obtained the formula for $U$, for the case in which $v$ and $w$ have the same direction, by composing two transformations according to § 3.Show the German source If, besides the systems $K$ and $k$ figuring in § 3, we introduce a third coordinate system $k'$, in parallel motion relative to $k$, whose origin moves on the $\Xi$-axis with the velocity $w$, we obtain between the quantities $x$, $y$, $z$, $t$ and the corresponding quantities of $k'$ equations which differ from those found in § 3 only in that the quantityShow the German source $$\frac{v + w}{1 + \frac{v w}{V^2}}$$$v$Speed of the moving system$w$Velocity in the moving frame$V$Speed of lightExplain this equationShow the German source takes the place of “$v$”; one sees from this that such parallel transformations — as must be the case — form a group.Show the German source

We have now derived the theorems, necessary for our purposes, of the kinematics corresponding to our two principles, and we proceed to show their application in electrodynamics.Show the German source

II. Electrodynamical Part.Show the German source

In conclusion I remark that in working on the problem treated here my friend and colleague M. Besso stood loyally by my side, and that I owe him several valuable suggestions.Show the German source

Bern, June 1905.Show the German source

(Received 30 June 1905.)Show the German source