22 
MATHEMATICS: EISENHART AND VEBLEN Proc. N. A. S. 
5. Returning now to the question as to the conditions on the F's that 
they shall yield a Riemann geometry, we observe that the left hand member 
of (2.4) is the covariant derivative of g^. Thus (2.4) may be written 
gijk = 0. (5.1) 
By (4.8) 
gijkl - gijlk = gajBfkl + gia Bjki (5.2) . 
which combined with (5.1) gives 
gcj Bfkl + g^a B%i = 0. (5.3) 
If these equations be differentiated covariantly, we get 
gict BJklm + gaj Bfklm + giam B%l + gajm Bfkl = 0. (5.4) 
Hence by (5.1) 
gia Bjklm + gajBfklm = 0. (5.5) 
Proceeding in this manner we get a sequence of equations, namely 
gia Bjklmn ~1~ gaj Biklmn 0, 
(5.6) 
gia Bjklmn ■ ■ ■ ■ p -\~ gaj Bfklmn ■ ^ ■ ■ p = 0. 
If the geometry of paths is Riemannian (5.1) must be satisfied and 
hence also (5.3), (5.5) and (5.6). The equations (5.3), (5.5) and (5.6) 
are linear equations in the (n + l)n/2 functions gij with coefficients which 
are functions of the T's alone. Hence the algebraic conditions for the 
consistency of (5.3), (5.5) and (5.6) regarded as linear equations in the 
g's are necessary conditions on the T's that the geometry of paths shall 
be Riemannian. 
6. Now suppose that equations (5.3) and (5.5) are algebraically con- 
sistent in the g's, and that the rank of the matrix of the B's is such that 
the g's are determined by (5.3) to within a factor, which is at most a 
function of the x's. Let gy stand for a particular solution of (5.3) and 
(5.5). If (5.3) be differential covariantly with respect to x^, we have 
in consequence of (5.5) 
giam Bfkl + gccjm Bfkl = 0- (6.1) 
Since these equations are of the same form as (5.3), it follows from the 
above hypothesis about (5.3) that 
gijk = ^kgij, (6.2) 
where (pk is a covariant vector. Substituting these expressions in 
gijkl - gijlk = 0, (6.3) 
which follows from (5.2) and (5.3), we obtain 
_ d<pk 
