732 
bm 
a 
et — 2% — J Aa; zi—_ 4 & | Abi Am; 24 23 — 
—42{ 55)" — LL 
Own a a 3 
— $F Quy Ab, (Am, An, — AX, Am) 2H 2” 2% — 
— 1S Qty my Ao, (Am, Ar, — AX, Am) 2 2° a 
By Q%m, we have denoted the same form within brackets which 
is found in the foregoing line. Note the symmetry possessed by 
Qo in the suffixes 6 and m. If we put 
Ra mn zE Q™,mn Tia Q% nm ’ 
then Rn, is the same as a four-index symbol of Riemann: 
bn 
8 
| 4% Am; Any, zi 23 oie aes 
ms 
a 
RE mn = ‚ba, mn}, 
and for its covariant components we have the identities which will 
be used in the following: 
Rabd,mn == Roajmn == TT Rab‚nm = Rynn,at ’ 
and 
Rad,mn oF Rom,an mk Rina, bn =); 
We proceed to show that the above transformation actually affords 
the geodesic falling coordinates alluded to in the introduction. 
The axis of 2° coincides with a particle's track. Put every 2*=0, 
_and we get 
bm 
wt — et —= At 2°—} > Ab, Am, 2°2° — 4E Q%) mn ALAT, A" zz es oe 
As a second approximation, this is the equation for the geodesic 
line starting from the point-instant «*, with initial direction para- 
meters A‘, and where z° is the interval along the arc. Thus our 
time-axis is along a particle’s track. Denote the second member of 
this equation by &. 
The axes of space are everywhere geodesics, as far as the approxi- 
mation goes, and perpendicular among themselves and to the axes 
of time. For put z° =r and let the other coordinates vanish with 
the exception of one 2”; on rearranging terms we get 
ga — wt, — Et = Aa, z4 — 
—=|" 
a 
b 
—4 =| ij Ab, Am, z# 2# — § 3 Woman AO, A™, A", rz Ze 
a 
Ab, Am, zee — bk DB Qay mn Abn Am, A”, rra 
Dt + = Q% mn Ab, An, An, ze 2h 2h, 
