332 MATHEMATICS: L. P. EISENHART Proc. N. A. S. 
-and 
V = ylaa- 1 ^ - 1 ' 
where a is a constant. This is the longitudinal solution obtained by 
Levi-Civita. 6 
For V independent of x 2 two cases arise, according as j3 can be made 
equal to zero by a transformation of coordinates, or not. 
When /3 = o, the functions a and 7 must satisfy the three equations 
«-+K*)^-H)£ + 70"H 
■>■ ■+(' 4)^ + a 7 T +2 ( 1+ *') = •>■ < 12 > 
2 «V-^! + (. + i)^ + -(^ + i + i) = «, 
where k is any constant. For k = 1, 
t2 i a 2 a a , /Va + Va — 7 2 ^ Tr * 3 . — , 1oN 
77 +4t =4a, - = &( ) V = e 3 Vay (13) 
7 \ 7 / 
where a and b are constants. Consequently, 
ds 2 = 4(a ~ 7 ^ dy 2 + adtff - 2 a x 2 dx 2 d£ 3 + ^a*| + 7^*3- (14) 
When k dp lj we solve the second of (12) for a' '/ a and substitute in 
the third of (12). The first integral of the resulting equation is 
(t' + vy* + 4fe 7 y » 
r n+f = a T 2 ' (15) 
[(l + fe 2 ) V 7 '2 + 4fe 7 + 6 7 'J + 
where o is an arbitrary constant and 6 = ="= (1 — k) Vl 4- fe 2 . 
Then we have 
2 
J" 1 = 7 * [(1 + & 2 ) V 7 ' 2 + 4^ T + 6 7'] ! 
V = e kx > <y[-\ 2 K 
& 
(16) 
When |8 4= 0, we have 
y = 5 1/4 *- 2 *\ 5 = a T -^ (17) 
with the f o llowing conditions upon a, /3 and 7 : 
a"- 
/ s/2 2 
a: 6 a 0 „ . a 
— - 14— = 0, 
45 2 5 2 5 
0'a - fia' = kb l/ \ (18) 
b' 2 + 25 (a' 7' - iS' 2 ) + 24 a 5 = 0, 
where /V denotes an arbitrary constant. 
