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208 MA THEM A TICS: L. P. EISENHART Proc. N. A. 
The components dx^ds of the vector tangent to a path being contra- 
variant we put dx^ Ids = A!' . In the former paper we observed that the 
theory of covariant differentiation can be generahzed to the geometry of 
paths by replacing the Chris toff el symbols \^\\ by Tj^. Thus the quanti- 
ties 
Aj=^+C.A« (2.2) 
QX^ 
are the covariant derivatives oi ; they are the components of a mixed 
tensor of the second order. Thus A] dx' expresses in invariant form the 
first variation of the components of A* as the x'^ vary. Hence if we write 
(2.1) in the form 
A)do^lds =0, (2.3) 
we see that the first variation of the components of the tangent vector to 
a path is equal to zero. In this sense we speak of the tangents to a path as 
parallel. 
Suppose now that A^ are the components of any contravariant vector 
whatever, and consider the vectors at points of any curve C not necessarily 
a path. The components A' and coordinates x^ along C are expressible 
in terms of a parameter 5, and dx^ /ds are the components of the tangent 
to C. If these functions are such that equations (2.3) are satisfied, we say 
that the vectors A^ are parallel to one another with respect to the curve. In 
particular the tangents to a path are parallel with respect to it. Some 
time ago Professor Veblen, in discussing the covariant derivative of a 
tensor, pointed out that it should be interpreted as the system of turning 
components of the given tensor with respect to the given direction. In 
this sense (2.3) expresses the fact that the turning components of the 
vector along the curve are zero. 
In order that our definition may be such that if A^ are the components 
of parallel vectors with respect to a curve so also are (p A\ where is a 
scalar, we say that the vectors of components A^ are parallel with respect 
to a curve whose tangents have the components dx' /ds, provided that a 
scalar function <^ exists such that 
bx' J ds 
3. Equation (2.4) is satisfied independently of the curve, if 
^Vri,A«-^^A^ = o, (3.1) 
c>x> ' -bx' 
that is 
