144 Dr. T. J. I'a. Bromwicli on 



is derived from § 1 ; and it is shown that this solution? 

 coincides with a Cartesian form given by Prof. A. E. H. 

 Love *. In particular in § 4 the field of an oscillating 

 electron is. determined and the rate of radiation of energy ; 

 it is found that the analysis is much simpler and easier 

 to grasp in terms of spherical polar coordinates than in the 

 usual Cartesian forms. 



A further restriction is introduced in § 5, by supposing 

 the waves to be simple harmonic with respect to the time ; 

 the solutions ^o found (in spherical polar coordinates) are 

 substantially the same as those given by Profs. J. W.- 

 Nicholson t and G. Mie J. 



it is shown in § 6 that these spherical polar solutions are 

 actually equivalent to the Cartesian solutions originally 

 given by Prof. H. Lamb §. 



Finally. in § 7. it is proved that the solution of § 1 also 

 includes certain general solutions given (for problems with 

 an axis of symmetry) by Hertz |] and FitzGerald II ; and 

 further that it leads to Lord Rayleigh's solution ** for waves 

 travelling along two long parallel conducting wires. 



The solution of § 3 (in spherical polar coordinates) was 

 originally worked out in 1899 ; and has been found also by 

 Prof. H. M. Macdonald. It was published as a question in 

 Part II. of the Mathematical Tripos, 1910; and has formed 

 the basis of a paper on the scattering of plane waves by 

 spheres, communicated to the Eoyal Society in 1916. 



It may be convenient to explain here the system of 

 numbering the formulae adopted in this paper: the principal 

 point to be borne in mind is that the decimal notation is 

 followed. The figure before the decimal point indicates 

 the section of the paper; and those following the point 

 ar supposed to be arranged in order of magnitude as 

 decimal fractions. For instance, 



3-4, 3-48, 3-491, 3-5, 3*51 



all refer to the third section (§3) and follow in the order 

 indicated. 



* Phil. Trans. A, vol. 197. p. 10 (1901). 

 t Phil. Mag. ser. 6, vol. xiii. p. 259 (l907). 

 X Annalen der Physik, ser. 4, vol. xxv. p. 382 (1908). 

 § Proc. Lond. Math. Soc. ser. 1, vol. xiii. p. 51 (1881). 

 11 < Electric Waves* (English edition), p. 140. 

 11 'Scientific Writings,' p. 122. 



** Phil. Mag. ser. 5, vol. xliv. p. 199(1897); 'Scientific Papers,' 

 vol. iv. p. 327. 



