THEORY OF RELATIVITY 
281 
and we have: 
dx=cos tox^dxj — sin ojX^dx 2 — w(Xi sin ojx^-fx 2 cos wx^)dx 4 
dy=sin o^x^dx^+cos o)X^dx.,-^o)(x^ cos o^x^ — Xo sin o^xjdx^ 
dz=dx.. 
dt=dx. 
Substituting in the equation for ds“ we have: 
ds^= — dx^ — dx^ — dx^-f [ (1 — or) (x^4-x“)]dx^ 
+2(0X2 dxi dx^ — 2(oXi dx2 dx^ 
Comparing this with the expression for 
ds-=g^^ dxj+g^+x; .... 
we get the values of the g’s : 
g.2= —1 
£'!= [(1— <'^=) (x_f+x^^)] 
2g,4= 2a,x, 
2g.,t= — 2 wX , 
or the array is : 
— 1 0 0 0 JX 2 
0 — 1 0 — (ox^ 
0 0—1 0 
0 0 0 [(1— or) (xi+x:)] 
If we put: |o)+x^+x^)=0 then g 4 i= 1 — 2 q, repre- 
sents the potential of centrifugal force, or a special type of 
gravity-potential. 
Although the g’s represent the metrical properties of the 
space-time continuum, we do not ordinarily look upon them as the 
field produced by rotating axes. We have other means like the 
gyro-compass or Foucault’s pendulum to present such a field to 
our minds, yet the field is nevertheless defined by the g’s and as 
seen especially in this case, by the g 44 . 
It must therefor be possible to represent the gravitational 
field in the specific sense by the g’s, and Einstein has accom- 
plished this by finding the differential equations satisfied by the 
g’s representing this field. These differential equations for the 
generalized potential express the law of gravitation of Einstein 
just as the Newtonian law of gravitation is represented by: 
A^(pz=0 
