529 
from zero, this is not true for large values of 7. It is found, how- 
ever, that also the rigorous equation’) is satisfied by 
OEE NE Pe EP as Sah ine a (3) 
k being an arbitrary constant. It will be seen that both (1) and (2) 
are special cases of the general formula (3). The flaw in the argu- 
ment used above was that (1) was considered to be the solution, 
instead of (3). KinsreiN’s theory in fact requires that g,, shall be of 
the form (3), but it does not prescribe the constant of integration. 
Newron’s theory did, and even therein lay its absolute character, 
Einstein's theory however is relative: in it the differential equation 
is the fundamental one, and the choice of the constants of inte- 
eration remains free, 
The constants of integration must, in a given system of reference, 
be so determined that the observed relative motions are correctly 
represented. In a true theory of relativity not only does the trans- 
formed general solution satisfy the invariant differential equation, 
but the particular solution which agrees with observed phenomena 
in one system must by the transformation give the particular 
solution which does so in the new system. Consequently the con- 
stants of integration must also be transformed and will generally 
be different in different systems. 
Suppose that we have originally taken a system of coordinates 
relatively to which the earth has a rotation w,. This w, is, of course, 
entirely arbitrary and, as our coordinate axes cannot be observed, 
it must in a true theory of relativity disappear from the final for- 
mulae. Now we have g,, =r’, and to determine #, we transform 
to axes relatively to which the earth has no rotation by #=9—w,t. 
Then in the new system 
Dis = (£ aan w,) r°, 
Observation shows that the correct value of 9',, in this system 
is —wr’, therefore 
B Ned ap ty hg ee he (4) 
If we use this value of /, the final formulas for the motion of 
bodies relatively to the earth contain only the observable quantity w. 
The relativity of the theory is thus seen to be based on the free 
choice of the constant of integration /. No value of & is a priori 
1) The second term, which is small also if 7 is large, has a more complicated 
form in the rigorous solution. It does not interest us here, as it is of the order 
of the mass of the earth. By the “rigorous equation” is meant the complete 
equation for 934, not reduced to its linear terms, but in which the other gij are 
supposed to be known functions of the coordinates, 
34 
Proceedings Royal Acad. Amsterdam. Vol. XIX. 
