Page:VaricakRel1910b.djvu/3

${\frac {U}{c}}={\frac {1}{3\cdot 10^{3}}}+{\frac {1}{3\cdot 3^{3}\cdot 10^{9}}}+{\frac {1}{5\cdot 3^{5}\cdot 10^{15}}}+\dots$

and the representative length U is by the way about 3mm greater then 100km. The velocity of 100000 km/sec corresponds to a length of 103700 km.

We additionally consider two velocities of β-rays calculated in the famous experiments of Kaufmann, and to which the velocity relations 0,7202 and 0,9326 are connected. They amount ca. 216060 and 279780 km/sec and they were represented by the lengths of 272400 and 503400 km.

For v = c we have U = ∞.

In graphical illustration these relations can be easily summarized.

If we take u as abscissa and ${\tfrac {v}{c}}$ as ordinate, then (1) will be represented by curve T. The straight line G or the first term in the infinite row (2), corresponds to the ordinary definition $u={\tfrac {v}{c}}$. The straight line is the inflexion tangent of T in O, so it fits well to the curve in the very far surrounding of the coordinate origin.

Fig. 1 can provide us good services for the composition of velocities. In addition to the resultant length U we can immediately take the corresponding velocity v from the figure.

If the velocities $v_{1}$ and $v_{2}$ enclose the angle α = 0, i.e. if they lie in the same direction, then $u=u_{1}+u_{2}$. The resultant velocity follows from the formula

$\operatorname {th} \,u={\frac {\operatorname {th} \,u_{1}+\operatorname {th} \,u_{2}}{1+\operatorname {th} \,u_{1}\operatorname {th} \,u_{2}}}$

or

$v={\frac {v_{1}+v_{2}}{1+{\frac {v_{1}v_{2}}{c^{2}}}}}$

Although the resultant is smaller as the sum of the components, it will be represented (as in ordinary mechanics) by a length equal to the sum of the lengths representing the components. It is namely in this case $U=U_{1}+U_{2}$.

The figure in the 6th Göttingen lecture by Poincaré is not in agreement with this definition.^[1]

If we compose two equal velocities to which corresponds the length $U_{1}$, then the resultant will be represented by the length $2U_{1}$.

2. Lorentz-Einstein transformation as translation

We take this transformation in the form as it was given by me.^[2]

$\left.{\begin{array}{c}l'=-x\ \operatorname {sh} \,u+l\ \operatorname {ch} \,\ u\\x'=x\ \operatorname {ch} \,u-l\ \operatorname {sh} \,\ u\\y'=y,\ z'=z.\end{array}}\right\}$

(4)

The variables x, y, z, l I interpret as homogeneous Weierstrass coordinates in Lobachevskian three-dimensional space. If X, Y, Z are Lobachevskian right angled coordinates, ξ, η ζ the perpendiculars that fall from point M upon the planes of the right angles coordinate system, and r is the distance of that point from the coordinate origin, then we get the following equations from the easily constructed figure

$x=\operatorname {sh} \,\xi =\operatorname {sh} \,X\ \operatorname {ch} \,Y\ \operatorname {ch} \,Z,$

(5)

$y=\operatorname {sh} \,\eta =\operatorname {sh} \,Y\ \operatorname {ch} \,Z,$

(6)

$z=\operatorname {sh} \,\zeta =\operatorname {sh} \,Z,$

(7)

$l=\operatorname {ch} \,r=\operatorname {ch} \,X\ \operatorname {ch} \,Y\ \operatorname {ch} \,Z.$

(8)

These Weierstrass coordinates satisfy, as it is easily seen, the quadratic equation^[3]

$l^{2}-x^{2}-y^{2}-z^{2}=1$

(9)

This is therefore the invariant of first kind of the transformation defined by equations (4), or by

$Uf=-l{\frac {\partial f}{\partial x}}-x{\frac {\partial f}{\partial l}}$

(10)

If we take

$y=y'=c_{1},\ z=z'=c_{2}.$

(11)

then we get to distance areas^[4]

1. H. Poincaré, Sechs Vorträge aus der reinen Mathematik und mathematischen Physik, 1910, p. 52
2. This journal. 11, 93, 1910
3. For these coordinates see Liebmann, Nichteuklidische Geometrie, p. 166, and Killing, Die nichteuklidischen Raumformen.
4. The general equation of the distance area was given by me in a treatise in "Rad jugoslavenske akademije" 175, 215-240, 1908. There, the middle plane was defined by the ends ξ, η and the angle α. For the determination of the plane see my work: Zur nichteuklidischen analytischen Geometrie in Attil del IV congresso dei matematici, Roma 1909, 2, 213-226.