17(3 
DR. T. J. I’A. BROMWICH ON THE 
points behind the sphere. The method of this section is similar in some respects to 
one used hy 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. Proddman, Doodson and Kennedy (of 
Liverpool University).t It appears that the agreement with the formulae of § 6 is 
quite close (for kci = 9, lO ) from 0 = 0° to 90°, and for the Z-component up to about 
120°. The forimdae of § 7 also give good results in a cone of about 10° behind the 
sphere (that is, from (9 = 170° to 180°). It is clear, however, that an approximation 
suitable from 0 = 90° to 170° (for Y) and from 0 = 120° to 170° (for Z) has still to be 
obtained. But nevertheless the present approximations proved a valuable auxiliaryl 
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 (5’21) 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 
(E) 
(M) 
Kax 
dy 
CL* 
KaY 
da 
Kaz 
a/3 
da 
c" dt 
dz 
c" dt 
dz 
dx 
a^ 
dx 
dy 
da. 
dZ 
aY 
a/3 
a_x 
_dZ^ 
ay 
dji 
_dX 
dy 
a^ ’ 
dz 
dx 
dx 
dy 
* ‘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. 
X See the paper last quoted, p. 292 et seq. 
§ Revised March 18, 1919 ; see note at the end of the introductory remarks above. 
