206 Proceedings of the Eoyal Society of Edinburgh. [Sess. 
2D = c; . ^^1^1 _ 
2E = c - ^ 
2a 2c 
2F = 6,, - , 
" 2a 26 
The expressions can be much simplified b}^ changing the variables. 
Substitute 
a = — , h— - , c= — 
d - 82 
and, noting that a is (by hypothesis) a function of alone, introduce a new 
variable X-^' such that 
adx^ ■■= dx^ ; 
also let /3/= , y/ = J^ , and so on. Then 
' ^ dx^ 
A = 
B^/5Sft'8/, 
C = ySy/S/, 
- 7(722 + /5ft Vi'). 
E= -a2yyj/, 
F= 
M = ay(y./-^/3/). 
For the Einstein equations 
= B 22 = 0 , Bgg^O, B^^ = 0, M = 0 
we can substitute the equivalent set 
A B C ^ 
— + -y + — = 0, 
a 0 c 
Va-Vd-^^ 
7-^-0 
ah cd ' 
M = 0, 
obtained by resolving them linearly so as to express A, D, E, F in terms of 
B and C. 
