24 Prof. F. J). Murnaghan on the 



Writing out G n = {12, 21} + {13, 31} + {14, 41} with 

 the simplification #i=l, we obtain 



fys d 



a?, 



__L/^iV 



and on introducing the H r and equating this to zero 

 there results 



JL?>!H» i ;*!I? i ^h 4 _ ^ 



H 2 B^i 2 H 3 B^ H 4 d^ 2 * ' W 



The next in simplicity is 



G 44 = {41, 14} + {42, 24} + {43, 34}. 



Remembering that g 4 is a function of x x alone and that none 

 of the coefficients involve x if we have on equating this 

 to zero 



1 'd?jh_±_ /B,<7 4 y , J_5^i ^ 2 j 1 d^Bffs _q 



2 a^i 2 4^ \a^i/ "*" 4^ <3<^2 cU'i %3 d-% 9#1 

 or 



4 d^i 2 4 d#i 1 H 2 a^i H 3 B^ S " 

 which is immediately integrable with respect to # 1? yielding 



H 2 H 3 ^ — - independent of ^ (C) 



Now the two-dimensional spreads x± = const. Xi= const, 

 are what may be termed geodesic spheres. On one of 

 these let us suppose that «r 3 is a " longitude " coordinate 

 whilst x 2 is a latitude coordinate. As a demand of 

 symmetry we say that the expression for (ds) 2 on this 

 sphere cannot have its coefficients g 2 and gs involving the 

 longitude coordinate # 3 , whilst the arc differential along 

 a "meridian" curve ds = \/g 2 dx 2 cannot depend on the 

 latitude x 2 . Thus g 2 is a function of x\ alone, whilst g 3 is 

 a function of x 1 and x 2 at most. Just as the non-appearance 

 of the time-coordinate x± in the coefficients made G u = 0, 

 G 24 = 0, G 34 = 0, so now the absence of x% from the co- 

 efficients gives in addition Gis = 0, G2 3 = 0. Differentiating 



