Vol. 8, 1922 MA THEM A TICS: EISENHART A ND VEBLEN 
23 
(Hij) - Hpi Tfk - Mpj Tfk = 0. (6.5) 
Hence the vector is the gradient of a scalar function, and can be put 
in the form 
<Pk= - — (6.4) 
Substituting this value of (p^ in the right member of (6.2) and the explicit 
expression for g^jk (cf. 2.4) in the left member, we obtain 
bx^ 
Consequently the functions Xgy satisfy (2.4) and give a Riemann geometry. 
It is interesting to note that the geometry of paths obtained from (6.2) 
without imposing the condition (6.4) is the geometry used by Weyl as 
the basis for a combined electromagnetic and gravitational theory. For 
(6.2) is equivalent to 
n = l(~f+T^i'-^S)+\ fe« + Sjk 'Pi - Sij <Pk)- (6.6) 
2 \ox^ ox ox / 2 
7. Let us now assume that (5.3) are algebraically consistent, and that 
all of their solutions satisfy (5,5). Let g^lj , g^fj be a com- 
plete set of solutions. The general solution is expressible in the form 
gij — <P gij ^ ^ gij ^ ■ ■ ■ ■ n- (p gij ' U-i; 
Differentiating (5.3) covariantly, we get (6.1), and consequently we must 
have 
The vectors X^"'^'* {a, ^ = 1, . . . .p) must be such that the functions 
gjji satisfy (6.3). On substituting (7.2) in (6.3) we find 
+ 2 (^*'"'' - ) =0. (7.3) 
7 = 1 
When we express the condition that the functions g^j given by (7.1) 
shall satisfy the conditions gijk = 0, we find in consequence of (7.2) that 
the functions (^^^^"^ must satisfy the equations 
(a) ^ 
+ 2^^'^^^^'"^ = 0. (7.4) 
/3 = i 
In consequence of (7.3) this system of equations is completely integrable,. 
and hence there exists a set of <^'s which by means of (7.1) determine a 
system of g's which yield a Riemann geometry. Hence we have the 
theorem: In order that the geometry of paths shall he a Riemann geometry 
it is sufficient that the F's he such that the equations (j.j) he algebraically 
consistent, and that all of their solutions satisfy (5.5). 
