26 Prof. F. D. Murnaghan on the 



Hence either <£ is a constant, so that all the coefficients- 

 depend on x 1 alone, or /B 2 '— -/' H 2 is a constant giving 



f = constant Xn 2 ^5-9. 



J H 2 



We shall discuss in some detail the case when /=0, 



as this leads immediately to the Einstein-Schwarzschild 



form. We have „ „ , 



J±3 = Jti 2 <p, 



and on substituting this in (D) we get 



,H/ + </>" 

 H 4 4> 



Since the first three terms involve x h alone and the last 



x 2 alone, they must separately be constants which cancel each 



other. If the constants are zero, we derive <£' = constant. 



Otherwise, by at most a mere linear transformation on the 



variable % 2 alone (which is unessential to the argument, 



since it merely changes the origin of measurement and 



the size of the unit in which it is measured) we may 



write 



(ft = sin .ty (h) t 



tn 



On substituting ¥— = —1 in the expression just obtained 



H 2 H/' + H 2 ' 2 + H 2 H 2 '4v L + " = 0. 





 from (D) we find 



H,H/' + H s H a '{^ + ^} = l,. • • (D') 



and (E) yields exactly the same equation connecting H s 

 and H 4 . On writing H 3 = H 2 sin# 2 , (B) becomes 



2^H-| = (B') 



Differentiating (C) and eliminating H2H4" we obtain 



H 2 "H 4 = H'H4', 

 which yields on integration 



HV = /3H 4 , where /S is an arbitrary constant. . (C") 

 .From (C") we have 



H 2 " = /SH 4 '=g_ .... by (C);. 



,. W> = 2 ( 7 -0 



where 7 is an arbitrary constant. 



