Vol.. 8, 1922 MA THEM A TICS: L. P. EISENHART 
209 
Consequently whenever there exists a field of vectors satisfying (3.1), all 
the vectors are parallel to one another for any curve, and thus there is a 
significant direction at each point of the space. 
For the second covariant derivatives of any contravariant vector we 
have the identity 
Aj, - Ai,- = - A- Bijk, (3.3) 
where 
Bajk = — — T + ^ck r^; ~ r«i ^'^ky (3.4) 
that is the B's are the components of the curvature tensor, as defined in the 
former paper. From (3.3) it follows that the conditions of integrability of 
(3.1) are 
; A- Bijk = 0. (3.5) 
From this equation it -follows that a necessary and sufficient condition 
that (3.1) be completely integrable, that is that there exists a field of vec- 
tors parallel to any given vector is Bl,jk = 0. From the results of 
the former paper it follows that in this case the space is euclidean. 
If the space is not euclidean, a necessary condition that the A's given by 
(3.5) shall satisfy (3.2) is 
= 0, (3.6) 
where B]^jki is the covariant derivative of Bl^jk. 
Suppose now that the rank of the matrix of equations (3.5) is such that 
these equations admit a set of solutions A'^ determined to within a scalar 
factor, and that these functions satisfy (3.6). Differentiating (3.5) covari- 
antly with respect to x^ and taking account of (3.6), we have that the 
functions must satisfy (3.5). In consequence of the above assumptions, 
it follows that 
At = A-n (3.7) 
where (pi is a covariant vector. Substituting in (3.3) and making use of 
(3.5), we find that <pi is a gradient, and consequently (3.7) is of the form 
(3.2) . 
The case when equations (3.5) admit m {<n) sets of solutions, in terms 
of which any set of solutions is linearly expressible can be handled by a 
method similar to that used in §7 of the former paper. In this case any 
vector at a point P in the m-fold bundle of vectors determined by the m 
vectors at P is parallel to a vector in the corresponding bundle at any 
other point of the space. 
4. In the preceding section we have given the conditions for one or 
more fields of vectors in invariantive form. Now we shall show how all 
such fields may be obtained by making a suitable choice of co5rdinates. 
