Vol. 8, 1922 
MATHEMATICS: 0. VEBLEN 
349 
in which we are defining the functions ip^^ by the equations 
^i = rj, -A', (3.2) 
The functions thus defined are the components of a tensor which is con- 
travariant with respect to i and co variant with respect to a and /3. This 
is because a transformation to new - independent variables z^, z^, . . ., z^, 
changes the F's according to the formula 
This tensor defines a covariant vector by means of the formula 
^» = ;^ii< (3.4) 
By eliminating 0 from the equations (3.1) we obtain 
Remembering that the values of dx^/dt, dx^/dt, . . ., dx"/dt are arbitrary it is 
easy to infer from (3.5) that 
= ■ (3-6) 
where is 1 or 0 according as = aov i 9^ a. 
When (3.6) is substituted in (3.1) there results 
2 f = e. (3.7) 
4. Conversely it can be shown that if we start with the differential 
equation (1.1) and any covariant vector <^„, we can find a second differen- 
tial equation of the type (1.2) which defines the same set of paths as (1.1). 
The first step is to define the functions (p^^ by means of equations (3.6). 
Since b\ is a tensor and (p^ is a covariant vector it follows that is a ten- 
sor of the third order. We then define Aj^ by means of the equations 
(3.2) and write the differential equations (1.2) which are by definition 
identical with 
d'^x ^ ^ dx^ dx^ _ dx"^ dx' , . 
dt' ~^ ^ ~dt 'df '^"^-li ~di ^^-^^ 
If we compare (4.1) with (2.5) we see that (4.1) is obtainable from(l. 1) by 
means of the substitution 
e * +h{x) (4-2) 
where k {x) is an arbitrary function of x^, x'^, . . 
