Vol.. 8, 1922 MATHEMATICS: L. P. EISENHART 25 
where g^^ is the cofactor of g^s in the determinant of the g's divided by 
the determinant. Hence from (4) we have 
^nk= - RprHK, (6) 
k 
where 
Rtr = " g'"Rp,.rs, (7) 
and consequently is the contracted curvature tensor. From (4) it follows 
that ri^h = o, and therefore 
= S f,^ - - RpXK (8) 
k 
is the sum of the Riemann curvatures determined by ih) and each of 
n — \ directions orthogonal to {h) . Ricci calls the mean curvature 
of Vn for the direction {h) at the point. Thus Ricci not only obtained 
the fundamental contracted tensor in 1904, but gave a geometrical inter- 
pretation of it. 
In general as the vector {h) is changed the value of p/^ varies. Since 
the components X| are bound by the equations gpr^tK, = 1 in order to 
find the directions giving the maximum and minimum values of pf^, 
we equate to zero the derivatives with respect to {r = l, n) of 
Rpr^h^h /n\ 
This gives 
(Rpr + PhgprM = 0. (10) 
Hence the maximum and minimum values of p^ are the roots of the 
equation 
\Rpr + pgprl = 0, (11) 
and the direction for each p is given by the corresponding n equations 
(10) for r=l, n. Following Ricci we call these n directions the 
principal directions of the Vn. It is readily shown that if the roots of 
(11) are distinct, the n corresponding directions are mutually orthogonal. 
If X;j|^ denote the covariant components of Qi), and we multiply (10) 
by and sum for h, we have 
Rpr = — 2 Ph^h\p^h\r^ ' 
h 
which is the form given by Ricci as characteristic of principal directions. 
In order that the principal directions be completety indeterminate at 
every point, it is necessary that the coefficients of the X's in (10) be zero; 
that is 
Rpr = <Pgpr, . (12) 
where c? is a scalar. When this is satisfied, we have from (9) that p^^ is 
the same for all directions. Consequently the space may be thought of 
as homogeneous, and (12) is the necessary and sufficient condition for 
such homogeneity. 
