1000 
av 
at aye Me a gy A ny a 
ip , 
oP 
hence we get 
‚ > 2 
ds? = v (da? + dy? 4-dz”) + (u—v) & de + ce dy + — a: ) + wdt?, 
r r r 
in which w, v,and w are functions of 7. From the form of ds? we find 
immediately for the values of the g’s the scheme 
ve vy uz 
v + —(u—v) | (2) —(u—v) 0 
7 r 7 
“! (u—v) v + a (u—v) ad (u—v) 0 
rT (le 
vz ye Ke 
ed) eN Aan Ld 0 
r r 7 
0 0 0 w 
A similar scheme holds for the ¥’s, viz. 
0 vy ne 
ga AP) 2 (P—9) pg U 
(id Ue r 
Ly y? yz 
pal) ot alae ne) hal de Ua 
UZ yz Zi 
— (p—9) me ty 4 (Py) 0 
r r r 
0 0 0 s 
In this p, g, and s are functions of 7 satisfying the relations 
pega WES rs So eee 
which is seen in the simplest way by choosing P on one of 
the axes of coordinates. 
3. In order to find the differential equations, which w, v, and w 
or, what comes to the same thing, p, g, and s satisfy, we make use 
of the thesis of the calculus of variations, which occurs in the second 
paper of EiNsreiN and Grossmann cited above, and which states that 
the first variation of [Hd is equal to 
nf (2 ys fr) de. 
x ey 
afsp Ot x Ow, 
the integration is to be performed over a region of the manifold 
(v,y, 2,0), dr is an element of that region, and the variations must 
In this 
