PHILOSOPHICAL TRANSACTIONS. 
I. The Integration of the Equations of Propagation of Electric Waves. 
By A. E. H. Love, F.R.S., Sedleian Professor of Natural Philosophy in the 
University of Oxford. 
Received December 29, 1900,—Read February 7, 1901. 
1. Ix the older forms of the Undulatory Theory of Light, the propagation of the waves 
was traced by means of Huygens’ principle; each element of a wave front was 
regarded as becoming a source of disturbance from which secondary waves are 
emitted. The principle is indefinite, inasmuch as the nature and intensity of the 
sources of secondary waves are unrestricted, save by the conditions that the secondary 
waves must combine in advance so as to give rise to the disturbance actually pro¬ 
pagated, and must interfere in rear so as to give rise to no disturbance. That these 
conditions are insufficient, for the comjDlete determination of the nature and intensity 
of the sources in question, is proved by observing that different writers, proceeding by 
different methods, have arrived at different expressions for “ the law of disturbance 
in secondary waves,” all these expressions satisfying the imposed conditions.^ 
In. the more modern forms of the theory, the propagation of the waves is traced by 
means of a system of partial differential equations. This system has the same form, 
whether we regard the luminiferous medium as similar in its mode of action to an 
elastic solid, transmitting transverse waves, or regard light as an electromagnetic 
disturbance obeying the fundamental equations of the electric field. In both cases it 
appears that all the components of the vector quantities which represent the dis¬ 
turbance satisfy a partial differential equation of the form d'l/dd = C' V ~(j). 
2. This equation is the same as occurs in the Theory of Sound. It has been 
integrated in two ways. Poisson f expressed the value of <^, at any point, at time t, 
in terms of the initial values of <f) and d(f)/dt on a sphere of radius Ct, with its centre 
at the point. KiRCHHorr| obtained a more general integral, in which the value of </> 
at any point is expressed in terms of the values of (f), ^4>jdv and d<hft on a closed 
surface S, separating the point from the singularities of the function (f), dv being the 
* Lord Rayleigh, “tVave Theory of Light,” ‘ Eiicycl. Brit.,’ 9th ed., vol. 24, pp. 429, 453. 
t ‘Mem. de I’lnstitut,’ vol. 3 (1820), p. 121. Cf. Lord Rayleigh, ‘Theory of Sound,’ vol. 2, ch. 14. 
X ‘Wied. Ann.,’ vol. 18 (1883), p. 663 ; or ‘ Vorlesungen ii. math. Optik’ (Leipz. 1891), pp. 23 et seq. 
The result is given explicitly in equation (2) of § 7, infra. 
VOL. CXCVII.—A. 287 
B 
17.7.1901. 
