Derivation of Symmetrical Gravitational Fields. 29 



(D) gives on division by H 2 H 2 ' and integration 



H 2 'H 3 H 4 = con stant = c 2 , 



where as before c 2 may be zero. 



(E) yields similarly 



H3 H 2 H 4 = c 3 . 

 These equations may be written symmetrically 



JV = JV = Ml = L_ 



C4H4 c 2 H 2 C3H3 H 2 £[ 3 H 4 



it being understood that if c 4 , for example, is zero we omit 

 this member of the equations and write instead 



H/ = or H 4 = constant. 



It is apparently advisable to write Z 2 = logH 2 etc., and 

 we have 



I A. to tQ I 



c± c 2 c 3 H 2 H 3 H 4 



1 

 g^g^V-ffe' + W 



Ci Ccy C 3 



Suppose, for example, c 2 =fcO and eliminate Z 3 and / 4 and 

 obtain 



V o 2 c 2 J 



c 2 c 2 



whence 



_,= (_— -j(* 1 + const.). 



Choosing the origin of measurement of x v so that this^ 

 additive constant is zero, we obtain 



C2 + C3 + C4. %i 



Co 



.*. log H9 = loo- x x + const. 



6 - C2 + C 3 + Ci ° 



c 2 C3 



or H 2 = C 2 a? 1 C2+C3+C4 ; H 3 = C 3 ^i C2+C3+C4 ; 



C4 



H 4 = C 4 a7 1 ^ +c 3+ c - t . 

 (There is an exceptional case, c 2 + c 3 + c 4 = 0. Here 

 l 2 " = 0, y = const. = c 2 k (say). 



