Electromagnetic Waves. 143 



An explanatory figure is scarcely necessary. The proof 

 just given is based upon the lemma : All angles ft subtended 

 by the arc ad of the "central" angle aOd and having their 

 vertices on k are equal to \aOd. The proof of this lemma, a 

 generalization of the familiar theorem on peripheral angles, 

 offers no difficulties, and may be left to the reader as a simple 

 exercise in vector algebra. The generalization of the above 

 construction to non-coplanar a', a is again recommended to 

 the reader. 



B. Any vector R = OR being given, construct upon a giveii 

 ray T s the vector S = OS, ivhose tensor is equal to that of R, 

 in symhols, 



|B| = |BJ, 



•or in (generalized) familiar language, construct a segment 

 OS congruent to OR. 



The vector r = Or being the unit of R, and s=Os the unit 

 of the required S, draw the join sr up to its terminus T f . 

 Then T'R will cut 0T S in the end-point S of the required 

 vector. 



Thus also we can easily draw a conic /e R , homologous to /c, 

 through any given point R. In fact, all rays centred at the 

 point just called 7" will cut the r-line and the s-line in 

 points M, iV, such that ( ON\ = \ 0M\, or in other words, 

 in points M, N lying on the same conic homologous with k. 

 The extension to non-coinciding origins or centres offers no 

 difficulty. 



London, March 12th, 1919, 

 Hesearch Dept., Adam Hilger, Ltd. 



X. Electromagnetic Waves. 

 By T. J. I'a. Bromwich, Sc.B., FM.S* 



Introductory Summary. 



f pHE following pages contain a general solution of the 

 JL electromagnetic equations of wave-propagation : in § 1 

 these formulae are expressed in terms of general orthogonal 

 coordinates, and in § 2 it is proved that these formulae are 

 the most general possible — at any rate for spherical polar 

 coordinates. 



It naturally follows that the solution of § 1 should include 

 all known types of solution ; and this fact is confirmed bv 

 •detailed examination. In § 3 the spherical polar solution 



* Communicated by the Author. 



