162 



Dr. T. J. Fa. Bromwich 



The form (6-42) agrees with (5*3) in consequence of the relation 

 (6'41) already found ; this would be anticipated from the general 

 theorem of § 2, that the radial component of force suffices to 

 determine the other components. Similarly we deal with the 

 0-component of electric force. 



§ 7. Comparison of the general formula* of § 1 with 

 other known solutions. 



Before concluding it may be of interest to note that the 

 formulae (1*6) and (1"7) include certain other known solutions 

 of very general character. 



The first group of solutions is obtained by using cylindrical 

 polar coordinates p, cp, z, such that 



takii 



ds 2 = dp 2 + p 2 dd> 2 + dz\ 



A = 1, B = p, C = 1. 



We make, however, a cyclical interchange of coordinates 

 so that z corresponds to the f of our general formulae of § 1*; 

 then the solutions (1*6) can be written in the forms: — 



Y = 

 Z = 



'dp'dz' 

 1 d 2 U 



p'dt'dz' 

 ?> 2 U /xK 2 U 



COL 



= 1JWK3U\ 



p d<t>\c -dt r 



B^V c 



cy =-. 0, 



-dz 2 c 2 -dt 2 ' 



where U is any solution of the equation 



(7-1) 



S it 2 ~ p'dpV dp/ V^ 2 



B 2 U 



(7-2; 



Similarly the solutions (1*7) lead to the forms : — 



B 2 V 



x = 



P~d<l>\c ~dt 



Z =0, 



c/3 = 



dpcu' 

 l d 2 V 



P ci«3z' 

 B 2 v ^Kyy 



}.(7-3) 



C7= S7 2 " 



3t> 



where V is also a solution of equation (7 "2). 



It may be noted that here equation (7 2) reduces to the 



* It is therefore necessary to see that C=l, and that A/B is inde- 

 pendent of a : both of these conditions are satisfied here. 



