1191 
A, however, was an arbitrary point of the laboratory L. So we 
have proved the following : 
“If the observers in a moving laboratory L, which is without 
gravitation are to observe constancy and reversibility of the ways 
of light, it is necessary that the “world-lines” of the points of the 
laboratory are a system of oo* branches of hyperbolas, or else straight 
lines in the 2, y, éspace.” 
Without a new supposition, only in consequence of the circum- 
stance that through every pair of these world-lines — e.g. p and q 
— can always be brought a light hyperboloid //,, '), it can further 
be proved: that the co? world line hyperbolas lie in oo! surfaces, 
which pass fanlike through a straight line I of the w, y, t-space; 
they cut I in two real or conjugated imaginary points $2; and $2, 
(which may also coincide). In this way dependent on the situation 
of the points 2, and 2,, ° fields of world lines originate, which 
are of a very particular nature. *) 
§ 4. The frequency of the static fields of attraction caused by 
n centres which are stationary with respect to each other, is already 
greater than o° for n23. But the “hypothesis of equivalence’, 
cannot be satisfied in any other case than in that of the very special 
fields of attraction, which correspond to the o° fields of acceleration 
of the preceding §. 
REMARK. 
Up to now we have only used the constancy of the form of the 
rays of light. Moreover in every point of the laboratory L’ the 
velocity of the light must also be independent of the time. In order 
to introduce this condition, the measurement of time in L’ would 
have to be taken into account in the considerations, which renders 
them more intricate. 
Possibly the class of fields for which the hypothesis of equivalence 
is admissible, might then be still further limited. 
The field of hyperbnlas which in the 2, y, space represents Born’s 
“motion of hyperbolas” of a two-dimensional laboratory, is contained 
in the oo° fields of hyperbolas of § 4 as a special case. 
Moreover it satisfies (with suitable measurement of time in Ee 
the condition that the velocity of the light is independent of time. 
1) Formed by the lines of light of the signals, which may be sent from P to 
Q, and from Q to P. 
2) A proof for these theses and a classification of the above mentioned o fields 
of world lines is found in a paper by Mr. Cu. H. van Os, which will shortly appear. 
