280 
PAUL BIEFELD 
The g’s are functions of the infinitesimal coordinates; mathe- 
matically they are quadratic functions of the infinitesimals. 
The /XV may be represented by the symmetrical array: 
we need only: 
11 
12 
13 
14 
21 
22 
23 
24 
31 
32 
33 
34 
41 
42 
43 
44 
11 
12 
13 
14 
22 
23 
24 
33 
34 
but as the /xv's = 
/XV s = v/x « 
44 which are the double subscripts 
occurring in the above expansion. 
In case of the absence of a field of force, the equation becomes 
that of the restricted theory : 
ds^=-dx2-dx“-dx^-|-dx2 
1 2 ;! ' 4 
in which x^, x^. x. are imaginary space coordinates and dx^ the 
real time coordinate that can be measured directly by means of 
a clock. The g’s in this case are represented then by the fol- 
lowing array: 
—10 0 0 
0—1 0 0 
0 0—1 0 
0 0 0+1 
As a concrete example we will consider the transformation to 
rotating axes, the rotating system mentioned on page 278. 
The infinitesimal space-time interval of the frame of refer- 
ence is represented by : 
ds“=-dx^-dy--dz 2 +dt- 
let Xi, X 2 , X 3 , X 4 be any functions of x, y, z and t then : 
dfi ^ 5fi 6'fi dt^ 
dx=: — dXi -1 dx 2 4 dx 3 -j dx^ . . . 
o'Xi ('^Xo 5 x 3 ■' ax. 
and this will lead us to the expressions holding in the general 
theory namely : 
ds^— g as given above. 
/X V 
1-4 
The relations for transforming to rotating axes are : 
x=Xi cos 0 JX 4 — X 2 sin o)X 4 
y=Xi sin ( 0 X 4 +X 2 cos wX^ 
Z = X3 
t^x^; 
