Electromagnetic Waves. 



145 



§ 1. A general solution of the electromagnetic equations. 



Let £, r), f denote a set of orthogonal curvilinear co- 

 ordinates, arranged so that their directions of increase 

 form a right-handed set of axes when taken in this order 

 (or in any cyclical permutation) : further let the element 

 of length in terms of these coordinates be expressed in 

 the form 



ds 2 = A 2 de + B 2 dr) 2 + C 2 d^, 



where A, B, C are supposed positive, and are in general 

 functions of f , 77, f. 



To specify the electric and magnetic forces we shall use 

 electromagnetic units; further we write X, Y, Z for the 

 components of the electric force, resolved along the directions 

 °£ ?? Vi K respectively, while a, /3, 7 will denote the corre- 

 sponding components of magnetic force. 



Then the fundamental equations of electromagnetism, 

 in a medium of dielectric constant K, and of magnetic 

 permeability /n, can be written as follows : 



(E) i 



| r (A«)-;|(C 7 ), 



c l dt 



an< 



■ ***¥<=&»*> 



r -*<>£= &< CZ) 



(M) 





(A*), 



(BY), 



°^f=| (AX) -<l (CZ) ' 



-^ 



h (BY) -h iAX} > 



where c is the velocity of radiation. 



For these equations see for instance Macdonald's ' Electric 

 Waves,' ch. vi. § 36. Or we may proceed directly as follows : — 



Apply Ampere's circuital relation to the curvilinear rectangle 

 on a surface £=const. bounded by the curves 7; = ?^, »?=i? 2 , 



s=£ 15 r=r 2 . 9 



Phil. Mag. S. 6. Vol. 38. No. 223. July 1919. L 



