Vol.. 8, 1922 
MATHEMATICS: L. P. EISENHART 
211 
6. Suppose now that the geometry is Riemannian, the fundamental 
form being 
ds^ = gij dx' dx' {gij = gji) (6.1) 
Since 
2^ [bx^ dx' bxV' 
equations (5.1) are in this case equivalent to 
T7+vf-^« = 2,.„i^,log^ (6.2) 
dx^ ()x^ dx bx^ 
for a, j = 1, . . n. When we take o: = / = 1, we find gn = \l/^^{x^, . .x^). 
When we take / = 1, o: =|= 1, we find 
^ = $ M + ^ ^. (6.3) 
dx'\xP J dx" 
2 ()x' 
Consequently gi^ are given by quadratures, and likewise g^^j (a 4= 1, 
j ifi l)from (6.2). 
In particular if \l/ = const., by interchanging a and / in (6.2), we find 
that bgaj/bx^ = 0 for all values of a and /. Moreover (6.2) becomes 
^11^=^^, (6.4) 
c>x' dx'' 
As a first consequence of this equation we have that gn is a constant which 
may be taken equal to unity. Again (6.4) are the necessary and sufficient 
conditions that 
gi^dx'' = dx' + dip (x\ . ., x^). 
If then x'^ is replaced by x' — (p {x^, . . x^), the form (6.1) becomes 
ds^ = dx\ + gij dx^ dx^ (i, / = 2, . . n), (6.5) 
where gy are independent of x''-. A space with linear element (6.5) is the 
most general which admits a translation into itself. (Bianchi, Teoria dei 
gruppi continui, Pisa, 1918, p. 500.) The space-time manifold of four 
dimensions used by Einstein in his cosmological considerations is of the 
type (6.5), x'^ being the coordinate of time. 
In the case of m{<n) fields of parallel vectors for which T^j = 0, (^ = 1, 
, .m; i, j = 1, . . , n), these equations are equivalent to 
^• = 0, ^ = (6.6) 
bx^ bx' bx' 
From the first of these it follows that all of the functions g^ are indepen- 
dent of x', . . x^. From the second of (6.6) we find that gp^ (pq — 1, 
