330 
MATHEMATICS: L. P. EI SEN HART 
Proc. N. A. v S. 
3. If we put 
5 = ay — fi 2 , A = 0'y - &y' , B * y'a - ya', C = a'§- a(3', 
(where primes indicate differentiation with respect to Xi) we find that all 
of the Christoffel quantities for the form (I) vanish except the following: 
(22) = _oS (2S\ = _ fi> (33) = _ t' /13\ _ A_ 
\\) 2' \ 1 J 2 ' (If 2'\ 2 J ~ 25' 
C?K('-»M?K-&.w-i('+»> 
Also we find 
5 » = § - i v ~ h {<"*' ~ ^ = = °- 
Equation (4) reduces to 
2a7" + 2a"T + 3^'7'- /3' 2 ) - 4 0 0" - y = 0. (6) 
The problem reduces to the integration of (3), when a, /3, y are subject 
to the condition (6). We separate the discussion into the four cases, 
when V is independent of x 2 and Xz] of either x 2 or x$\ of %\\ involves all 
three variables. Expressing the conditions of integrability of (3) we 
are led to linear partial differential equations of the first order and these 
are ultimately solved. The first and second cases are the only ones 
giving solutions ; in the course of the investigation it is shown that in both 
cases a change of variables can be made so that /3 = 0. The solutions 
of the first type are: 
2('k + 1) 2k (k + 1) k 
«-*i»+»+ i v-.*i* ,+ * +1, v , -*r *'+*+ f - (7) 
where k is any constant. Solutions of this kind have been found by 
Kasner. 4 For the second case the linear element of 5 may be written 
ds 2 = d * _|_ ad%2 2 + ( b + aa ~ 2\ dxz 2 f ( 8 ) 
b a + a a V2 \ / 
and 
V = a l/ V(* 2 ), t*.+ bp = D, (9) 
where a and b are constants. This is the solution found by Levi-Civita 5 
