Derivation of Symmetrical Gravitational Fields. 25 

 (C) with respect to x 2 and eliminating -^— 3 by means 



of (A), we see that H 2 2 ^ is independent of #1; but it 



qx 2 



cannot involve any of the variables but as l9 since H 2 and 



H 4 are functions of a?i alone, and so we conclude that it 



is a constant. 



H 2 2 ^ — 4 = a, where a is an arbitrary constant. . (C) 



Let us now write the two remaining equations Gr 2 2 = 0, 

 Gr:33 = 0. These take relatively simple forms under the 

 hypotheses just made. 



G 22 = {21, 12} + {23, 32} + {24, 42} 



• = iay, .-.i/bm 1 1 ay' 1 /ay.y 



2 3«i 2 4# 2 \d<z > 1 / 2^ 3 B^ 2 2 4# 3 2 Vdff 2 / 



— TTTT//, 1 B 2 H 3 H 2 ^H 3 , HgTT ,tt ,_ n sj^s 



primes denoting as usual differentiations with regard to 

 the argument of the function. 



G 33 = {31, 13 j- + {32, 23} + {34, 43} 



2 d^ 2 4# J \d# 1 / 2g 2 ~bx 2 2 4:g 2 g z \dx 2 ) 



f 1 /B^/BM | 1 Bf/3^4 



d 2 H 3 H 3 B 2 rJ 3 3H 8 r 1 3H 2 1 3H 4 1 



W "^H/B.^ 2 B.11 IH 2 B^'i H 4 "B^ J ~ 



...(E) 



Now «??! and ,r 2 are the only variables that can enter the 



coefficients g r = TL r 2 , so that (A) gives at once -^—^ = H 2 <//, 



where <£ is a function of x 2 alone, cj>' being its derivative ; 

 whence H 3 =H 2 (£+ t /', where f is a function of x x alone. 



Then (B) shows that ^- s — ¥ * s a function of x x alone, 



H3 O^'i 

 so that its derivative with respect to x 2 vanishes identically, 

 yielding 



4>'{fR 2 "-f"H s } =0. 



= H 



