Vol. 7, 1921 
MATHEMATICS: L. P. EISENHART 
331 
and called the quadrantal solution. 
4. For the linear element (II) the expressions for the Christoffel symbols 
which are different from zero are: 
{?}--£•{?} -K-'*-4 
{?} - a( - - * + «') {?} - 2 i(W + + 4 
{?}=-i(«-4{?}=" 
(23) = a(ax 2 -(3) (33) = _ (ax 2 - ff) 2 
\ 3 J 5 '( 3 j 6 
The expressions for the functions Bik are 
Sl3= " 4( 2C, * 2 + S )' 
2 
B 22 =L,B 2 s =M -x 2 L, B 3 z = Lx 2 2 -2Mx 2 + N, 
where 
^-£+s(-v-'K 
2 45 26\ / 5 
The condition (4) is 
-^-^a' T '- 0' 2 ^ + 4a = 0. (10) 
/ 2 
7 
We consider the system of equations (3) as in § 3, taking up separately 
the five cases where V is independent of x 2 and x z ; of x 2 ; of X\ ; of x z ; involves 
all three vaiiables. Only the first two cases lead to solutions. 
For the first case we have the linear element of 5 
do? 
ds 2 = 77 rr \ + a dx 2 — 2 ax 2 dx 2 dx 3 + a {x 2 + 1) dx 3 2 , (11) 
4(a a - 2 — a) 
