110 PEOF. A. E. H. LOVE ON THE TRANSMISSION OF 



should tend to zero for large positive values of z, the positive sense of the axis of 2 

 being directed from the conductor to the dielectric, and that II' should tend to zero 

 for large negative values of z ; and further, that at great distances from the axis of z, 



111 an( i n' should tend to forms which represent waves travelling outwards from 

 that axis. 



The problem in a slightly different form was solved by SoMMERFELD.( 6 ) He took 

 the doublet to be situated on the plane boundary, so that II', as well as II, has a 

 singularity on this surface ; and he obtained an exact solution in terms of definite 

 integrals involving BESSEL'S functions, and devised methods of evaluating the integrals 

 approximately. His main result is an approximate expression for IIj at a point close 

 to the boundary, and at a distance p from the doublet, in the form 



(12) 



This expression gives a valid approximation if kp is not small and not too great ; 

 but, as p increases, the value of II,/!!,, does not increase so rapidly as this formula 

 indicates, and for very great values of p it tends to zero. Within the region of 

 validity of the formula the absolute value of IIj diminishes according to the inverse 

 square root of the distance ; and the effect represented by II 1( being a wave affected by 

 cylindrical divergence, is described as a " surface wave." This surface wave is the effect 

 of resistance ; and it appears that, owing to resistance, the electric and magnetic forces 

 diminish less rapidly witli increasing distance than they would if there were perfect 

 conduction. SOMMERFELD (") and ('") maintained that this effect of resistance would 

 probably be intensified by curvature of the surface, and might thus counteract the 

 tendency of the signals to become enfeebled owing to curvature. With a wave-length 

 of 5 km. and the conductivity of sea-water (a- = 10~ n ) the region of validity of the 

 formula would include distances of 1000 to 10,000 km., and in this region the ratio 

 II | /| II would increase regularly from 1'003 to 1'032. 



The value of I I^II,, depends upon the resistance and the wave-length as well as upon 

 the distance p. It varies directly as the square root of the resistance and inversely 

 as the wave-length. The formula (12) indicates that in the region within which it 

 gives a good approximation, increased resistance is favourable to long-distance trans- 

 mission, increased wave-length unfavourable. These results are directly opposed to 

 those which were noted in 3, as derived from the study of the limiting case in 

 which the waves are treated as plane. On the other hand, the range of values of p 

 within which the formula is valid becomes narrower as the resistance increases or the 

 wave-length diminishes. The optical theory of shadows shows that increased wave- 

 length must be favourable to long-distance transmission, and we see that we cannot 

 hope to obtain any equivalent result from the solution of the plane problem. It is, 

 however, quite feasible that resistance also should be favourable within a restricted 

 range. The only way to settle the question is to solve the problem of the spherical 

 conductor, supposed imperfectly conducting. 



