734 
Goo = 1+ 0— § = Radnn AP, Abs (Amy AX, — Ans Am) zi zo. 
Obviously in the last term the value 0 for 7 contributes nothing 
to the sum. Because of the skew-symmetry of Rasim, in a and 6, | 
the value O can be disregarded also for 7, and the skew-symmetry 
im m and » allows us to write: 
Gop = 1+ J Ret AG Abr A”, Ae ze 27. 
Proceeding to g'o,, we get 
gon =O +0 — 4E Radmn Atn A (Am; An, — An; Am) ied — 
— 4E Rainn At, Abs (Am, AX, — An, Am,) 2F 2° — 
— sy E Rabynn A%, APs (AM: Ar — Ate Amy) 2° or. 
Taking #=0 in the first sum, this part cancels out against the 
second sum (skew-symmetry of Ray», in 4,5). The remaining part 
is taken together with the third sum, and we get 
go = 3E Robynn A% Ab, Am, An, 27 27. 
Finally for g',, we find: 
Jp = — Em + 0 — He & Radin [At, Ao, (Am An, — Anz Am) + 
+ At, Ao, (Am, Ar, — An, Am,)] 27 zt — 
— 42 Radmn (A% Abu + Aas Ab‚) (Am, An, — Att, Am) 2" 2%, 
where «,,=1 for «=r and ¢,,=0 for uv. Having regard to 
the skew-symmetries of Rasm, we reduce this expression to 
gu == — Em +4 = Rab,mn AS, Ao, Am, A”, 27 27. 
If we remember the transformation formnla for Ras mn: 
Bij,rs = = Pai Pdj Pmr Pns Ravn » 
we at once see that without lowering the degree of approximation, 
we may abridge the forms for gs into: 
g'00 == 1 a = R'2,0 Oe, 
910 == } pes: Rabe oe con 
Tig —— Eny + a ee) Rey Zo ie 
It must be noticed that these gravitation potentials depend no more 
on the time z°. The field in our geodesic falling coordinates is station- 
ary as far as our approximation goes. 
The R's are closely associated with RisMann’s measure of curvature. 
If only particles are considered moving so near the centre that the 
squares of the distances multiplied by the measure of curvature 
may be neglected altogether, then the g'ij may be considered to be 
constant and to have the homoloidal values 1, +1, —1, —1. 
Equations of motion for free particles in geodesical falling 
coordinates. 
We put forward the simplifying assumption that only particles 
