176 DE. T. J. I'A. BROMWICH ON THE 



points behind the sphere. The method of this section is similar in some respects to 

 one used by Prof. MACDONALD in a later paper.* 



The formulae of 6, 7 have been delayed in publication for two reasons : in the 

 first place I wished to obtain some confirmation from direct numerical calculation. 

 This has now been carried out by Messrs. PROUDMAN, DOODSON and KENNEDY (of 

 Liverpool University).! It appears that the agreement with the formulae of 6 is 

 quite close (for KO, = 9, 10) from 9 = to 90, and for the Z-component up to about 

 120. The formulae of 7 also give good results in a cone of about 10 behind the 

 sphere (that is, from = 170 to 180). It is clear, however, that an approximation 

 suitable from 6 = 90 to 170 (for Y) and from 6 = 120 to 170 (for Z) has still to be 

 obtained. But nevertheless the present approximations proved a valuable auxiliary^ 

 in checking and testing the numerical work. 



[The paper in its original form was presented to the Society on April 13, 1916 ; 

 owing to the difficulties in regard to labour and paper during the war, I was asked to 

 condense the introductory matter of 1-3. This proved to be impossible until now, 

 on account of pressure of war-work of various kinds. In the present version 1 has 

 been re-written so as to reduce its bulk ; in 2, 3 certain formulae have been omitted 

 which were not used in the applications of 4-6. 



In re-arranging the paper it proved convenient also to number the formulae 

 differently. The decimal system has now been adopted ; here the figure before the 

 decimal point indicates the section of the paper in which the formula occurs. The 

 figures following the decimal point are to be regarded as following the same order as 

 ordinary decimal fractions. Thus (5'2l) and (5'22) fall between (5'2) and (5'3), and 

 all these formulae occur in 5. Added March 18, 1919.] 







1. A GENERAL SOLUTION OF THE FUNDAMENTAL ELECTROMAGNETIC 



EQUATIONS. 



The fundamental equations of electromagnetic waves may be written 



c 2 3 3y 3z c 2 at 3z 3x c 2 9 3o; 3y 



8 a 3Z 3Y 3)8 3X 3Z a y 3Y 3X 



__ .. _ __ __ _ . ___ .. . ^_ . _____ _ . .. _ i_ _ _. , 



M 3 ' By 3z M 3t " 8z dx M 3 " 3a; Sy 



* 'Phil. Trans. Roy. Soc.,' A, vol. 212, 1912, p. 299. The two methods are not identical; but they 

 appear to yield equivalent results in all the cases- to which they have been applied. 



t 'Phil. Trans. Roy. Soc.,' A, vol. 217, 1917, p. 279. The calculation was originally undertaken by 

 Dr. PROUDMAN in consequence of a suggestion made in my lectures of 1912; the work, however, proved 

 to be longer than had been anticipated and was completed by Messrs. DOODSON and KENNEDY. 



J See the paper last quoted, p. 292 et seq. 



Revised March 18, 1919 ; see note at the end of the introductory remarks above. 



