m 



.28 Prof. F. D. Murnaghan on the 



where C is a constant, or 



2(H 2 -«/3) = 0+^. 

 Writing a{3 = m, C = 2, we obtain 



tt , m2 tt /-, ™>\ 2 



■R 2 -m = r + Ir , or H 2 = ^l + -J; 



whence TT /_, ?n\ 2 



H 1 rfHg _ 1 BH 2 dr 1 / _ m^\ J- _ 1 ^~2^ 



9 



and then 



' + 2r' 



which is essentially the result of the paper quoted. 



It may be interesting, in conclusion, to say a few words 

 about the case where <j> ! = 0, so that H 3 is a function of x 1 

 alone, as are all the other coefficients. This will be 

 interesting in connexion with certain axial - symmetric 

 solutions of G rs = if the coordinate lines x x are interpreted 

 as geodesies though not necessarily through a fixed point in 

 the sub-space of three dimensions ^ 4 = const. G12 is now 

 automatically zero for the same reason as are all the other 

 components G r „ r^=s ; (B) yields 



H 2 + H 2 + H 4 U ' 



where primes denote as before differentiations with respect 

 to the only variable occurring, a x . 

 From (C) we have 



H 2 H 3 H 4 / = constant — c 4 (say), 



where the constant c 4 may be zero (taking care of the case 

 11/ = tacitly omitted on division across by H 4 ' before 

 ntegration). 



