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XV. A General Solution of the Electromagnetic Relations. 

 By J. W. Nicholson, D.Sc, B.A.; Isaac Newton Student 

 and Scholar of Trinity College, Cambridge *. 



THE only general mode of solution of the differential 

 equations of wave propagation in electromagnetic 

 theory, which has led to a result directly useful for definit 

 physical problems, has been that of Prof. Lamb f- This 

 method is concerned with the determination of a vector 

 (u, v, w) simultaneously satisfying 



(Y 2 + k 2 )(u,v,w) = 



and * + |* + |H =0 ..... (1) 



ox oy o- 



(w, v, w) being Cartesian components. The general expres- 

 sion found for this vector is somewhat cumbrous, and though 

 well adapted for the solution of many problems, it frequently 

 leads to tedious work. In this paper, an alternative solution 

 will be developed by a different process. It is of a form 

 more suited to certain classes of physical problems connected 

 with spheres. Although it is not really distinct from that of 

 Prof. Lamb, the equivalence of the two is not easy to prove, 

 involving some laborious applications of reduction formulae. 

 The circuital relations, and not the equation of wave motion, 

 form the starting point of the investigation. Instead of the 

 Cartesian components of the vectors, we treat the components 

 along the three main spherical polar directions at any point. 



Let (js 9 y, z), (a, b, c) be the components of electric and 

 magnetic force along the directions of (r, 0, <£) increasing at 

 any point. We may suppose the magnetic induction and 

 force to be identical, without loss of generality. The space 

 elements in the three principal directions are 

 (dr, rdOj r sin 6 d<o). 



Since the work done in taking a unit magnetic pole round 

 any circuit, divided by 4-7r, is the total current crossing that 

 circuit, we have 



1 (a dr + br dO + cr sin 6 dco) 



= 4-7T 1 1 (ur 2 sin 6 dO dco + vr sin 6 dr dco + cor d6 dr) 



= V2 7 1 1 ( Xi ' 2 sm @ d6 dco+yr sin dco dr + zr dO dr) 



under the conditions supposed. 



* Communicated bv the Author. 



t Proc. Lond. Math. Soc. xiii. p. 51 (1882). 



