753 
BREED ay Pe BGs … Kl) 
§ 3. In Ernsrem’s theory the gravitation field is determined by 
certain characteristic quantities gap, functions of #,, #,, #, «,, among 
which there are 10 different ones, as 
OL eee Ca ye Pa Cuaron rsi Ne (5) 
A point of fundamental importance is the connection between 
these quantities and the corresponding coefficients 9',,, with which 
we are concerned, when by an arbitrary substitution 2,,..., are 
changed for other coordinates a',,...2',. This connection is defined 
by the condition that 
de = gd +... +9,,d2,’ + 29,,dz7,dr,+... 
or shorter 
ds? = (ab) gap dea day 
be an invariant. 
Putting 
Clg SAAD) Pap degen NOL Teas GB) 
we find 
Jar = = (cd) prea ab Gad nS) eg oe Wellin hs (7) 
Instead of (6) we shall also write 
det’ = 2 (6) rig dey, 
so that the set of quantities 25, may be called the inverse of the 
set Pas. Similarly, we introduce a set of quantities ys, the inverse 
of the set gap. *) ‘ 
We remark here that in virtue of (5) and (7) g'5a=g'as and that 
likewise Ys Yab: 
Our formulae will also contain the determinant of the quantities 
Jab, Which we shall denote by g, and the determinant p of the 
coefficients pay (absolute value: |p|). The determinant g is always 
negative. 
We may now, as has been shown by Einstein, deduce the motion 
of a material point in a gravitation field from the principle expressed 
by (3) if we take for the Lagrangian function 
ds EEN Eat 20S. 
eN OT NE 
dt 
1) Suppose 
Bq == 2b) 408s 
to follow from the equations 
Sa — = (drapes ; 
then the set Mab is the inverse of the set nab. 
48 
Proceedings Royal Acad. Amsterdam. Vol. XIX. 
