Vol.. 8, 1922 MATHEMATICS: EISENHART AND VEBLEN 
21 
paths is Riemannian is one of the inverse problems of the Calculus of 
Variations. But it does not seem hitherto to have been solved. Before 
considering it further we must mention a few general theorems, which 
have already been noted more or less explicitly by Weyl and Kddington. 
4. If we put %' = (p{x\ i"), thus introducing a new set of coordinates, 
the equations (1.1) become 
where 
d^x^ , d^^d^''^ bx^ 
bx^d^''^ ''dx'dx' %x'- ^^-^^ 
Expressing the conditions of integrabiUty of these equations regarded 
as differential equations for determining the x's as functions of the x's, 
when the F's and r's are known, we obtain 
where 
~B',rs= ^Bljk (4.3) 
T^p qs qr i -p^ — (A 
^qrs ^ y ^ ^ \ ^ ar ^ qs ^ as ^ qr- V^'^y 
bx bx 
Equation (4.3) may also be written 
bx bx^ bx bx^ 
This states that the functions B are the components in the coordinate 
system i of a tensor, B, which is contravariant of the first order and co- 
variant of the third order, with respect to the group of all transformations 
of the paths into themselves. This tensor is called the curvature tensor 
of the manifold. 
From (4.4) it follows that 
Btrs + B',sr = 0, (4.6) 
and 
Btrs + Btsq-hB%r=0. (4.7) 
The theory of covariant differentiation (cf. Ricci and Levi-Civita in 
Math. Ann., 54 (1901) can be generalized at once to the geometry of 
paths by replacing the Christoffel symbols {^i } by the functions Tjk in 
all formulas. In particular it is easily proved by means of (4.1) and 
(4.2) that the operation of covariant differentiation converts any tensor 
into a tensor of higher order. The formulas for covariant differentiation 
of sums and products of tensors also generalize without change, and also 
the theorem that if a^- is any tensor and aijki is its second covariant de- 
rivative, 
O'ijki ~ <^ijik = ci„jBiki + ai^Bfki- ' (4.8) 
