1150 
the points VA Bee G of the laboratory at other moments as 
many times as they like. Geometrical representation in the 2, y, f-space : 
The world-lines a,b,...f,g are intersected by all the oo! lines of 
light s,,s,...; they all lie on the ruled surface formed by the 
oc’ light lines. 
In agreement with condition |B] of $ 2 the observers of the 
laboratory Z must then moreover as often as they like be able to 
send optical signals S’,, S’,... in opposite direction G, F,... BA. 
In the z,y,t-space again oo' light lines s’,,s',,... correspond with 
‘his, which all intersect the world lines a, b,... f,g. Hence the world 
lines a,b,...g,h all lie on a surface covered by two systems each 
of o! light lines. If we then bear in mind that the light lines all 
make an angle of 45° C. with the ¢-axis, it is easy to see that such 
a surface must necessarily be an equilateral hyperboloid of revolution 
with the axis of revolution // to the t-axis; i.e. the equation of this 
surface has the form: 
Ate? Pty Bete Cy DEES Oe ee 
In particular the case may also present itself that A = 0, ie. 
that the hyperboloid degenerates into a plane. 
Such hyperboloids will be briefly called “ight-hyperboloids”’. Ac- 
cordingly the world lines a,b,...f,g of the points A, B,...F,G 
of the laboratory Z lie on a common “light hyperboloid” Hes. 
Now the observers might just as well have sent a light signal 
instead of from A to B, from A to any other point B’ of the 
laboratory. In exactly the same way we see then that also the two 
world lines a and 5 must lie on a common light hyperboloid Hy. 
Let the equation of this be: 
Ala? st) Bat Cy4 DiL E20... 
So the world line a lies at the same time on two different light- 
hyperboloids He, and Hy; it is the section of both, and this is 
necessarily a plane section. (Multiply equation (1) by A’ and equa- 
tion (2) by A, and subtract). If we now bear in mind that the point 
A of the laboratory must never have a greater velocity than that 
of light, of all the plane sections of a light-hyperboloid only two 
types deserve consideration: hyperbolas the two branches of which 
run from {= —o to t—= + w, and as limiting case the light lines 
of the hyperboloid. (In other words the sections with planes which 
1 cut the gorge circle of the hyperboloid, and 2 make an angle of 
< 45° with the t-axis. As besides, the case may occur that the light 
hyperboloids which pass through the world line a, degenerate to 
planes, the world line mn may also be a straight line, making an 
angle with the taxis, which is smaller than 45°. 
