Vol.. 8, 1922 
MATHEMATICS: L. P. EISENHART 
237 
from (5.4) it follows that if S„ and 5'„ are Riemann spaces, we must have 
<Pji = Vv» that is (pi must be a gradient. 
Suppose now that 5„ is any space. From the preceding theorem it 
follows that we can find a space S„" whose paths are in one-to-one corre- 
spondence with S„ and such that 5"^- = 0. If S„ is a Riemann space in 
correspondence with 5„, and consequently with S„'\ it follows from (5.4) 
that the vector giving the relation between 5„" and S„' is the gradient of 
a function (p. The algebraic consistency of equations of the form (cf. 
(5.3) former paper) 
leads to more than n {n + l)/2 equations. Hence when the expressions 
for B'iki as given by equations of the form (4.2) are substituted in these 
equations of condition, we have more than n{n + l)/2 algebraic condi- 
tions upon the n{n + l)/2 second derivatives of <p. Consequently the 
functions B"ijk for 5„" are subject to conditions and hence 5„ cannot be 
arbitrary, if its paths are to be in one-to-one correspondence with the 
geodesies of a Riemann space. It follows also from these remarks that the 
final theorem of the former paper does not give the complete characteriza- 
tion of geometries whose paths can be identified with the geodesies of a 
Riemann space. 
7. Suppose that 5„ is a Riemann space. If SJ also is to be Riemannian, 
it follows from (5.4) that (pi must be the gradient of a function (p. If the 
components of fundamental quadratic tensors of 5„ and 5„' are denoted 
by gij and g'^ respectively, we must have 
which in consequence of (3.4) and (3.6) becomes 
^JJl _ .r« - n - 2£^-^ - - = 0 (7 1) 
g^.^Jk -^g^,^^, gkj^^, g^k^^J U. (/.i) 
If g and g' denote the determinants of gij and g'y, we have (Einstein, Ann. 
Phys., 49, 1916, p. 796). 
Vg bx" bx" 
Multiplying (7.1) by g'^\ summing for i and /, and making use of (7.2), 
we obtain an equation from which it follows that 
2 (n -f 1) ^ g 
where C is a constant. 
Again, if we put Aij = e'^*^ gij and indicate by Aij^ the covariant derivative 
of Aij with respect to the fundamental form of S„, we obtain 
Aijk + Ajk, + Akij = 0, 
