2 
PROFESSOR A. E. II. LOVE ON THE INTEGRATION OF THE 
element of the normal to S. Kirchhoff’s Integral can be shown to include Poisson’s 
by taking, for S, a sphere, of radius Gt, with its centre at the point. In the case of 
sound, or for any scalar disturbance, Kirchhoff’s integral is directly interpretable in 
terms of ijnagined sources of disturbance situated on the surface S ; for all the 
cpiantities that occur can be interpreted in terms of condensations and velocities. It 
miglit tlms he regai’derl as providing an exact equivalent of Huygens’ principle,* if 
tlie disturbance involved were of a scalar character. Its application to light is open 
to criticism (see § 8. infra). 
Besides satisfying the partial differential equation = c" V the components 
of a vector quantity, propagated by transverse waves, are also subject to the circuital 
condition ; and the problem of integrating the system of equations is, accordingly, 
not tlie same as tlie problem of integrating the single equation satisfied by the several 
components. Sir G. Stokes! has attacked the more general problem, by extending 
and transforming Poisson’s solution of the single equation. He has shown that the 
components of the disturbance at any point 0, and at any time can be expressed 
as the sums of two parts, one depending on initial values on a sphere of radius Gt, 
with its centre at the point 0, and the other depending on initial values in space 
outside this sphere.j The latter part is relatively unimportant when, as in the 
a])j)licatious made l)y Sir G. Stokes, the radius of the sphere is great, compared with 
the wave-length of tlie disturbance; the former part has precisely the character 
required for representing transverse vector disturbances, and it admits of transforma¬ 
tion to a form in wliich it expresses the radiation received at the point O as due to 
secondary waves sent out from surfaces other than spheres with their centres at 0. 
The transformation to a plane wave front was given in the paper above quoted, and 
the results, which were deduced from this form of the Integrals of the system of 
eipiations, have liad a very important hearing on the development of the tlieory. 
3. The object of this })aper is to present an investigation of a new system of 
integrals of the system of equations that govern the propagation of transverse vector 
disturbances, and to exemplify the use that can be made of such integrals. The 
components of tlie vectoi'S that constitute the disturliance ought to he expressible, as 
in Kirciihoff’s solution, in terms of surface values on an arbitrary surface ; the 
elements of the integrals ought to be quantities characteristic of transverse vector 
disturliances, as in Sir G. Stokes’s solution; and the results ought to admit of inter¬ 
pretation in terms of sources of disturbance of definite types, as Kirchhoff’s result 
does when applied to souiifl waves. It is shown that the method developed by 
Kirchhoff can lie adapted to tlie system of equations in such a way as to lead to 
* It is so regarded l»y KuiGiniOFF {Joe. rit.), l)y PoiNCAlui, ‘ Theorie math, do la Lumiere,’ vol. 2 (Paris, 
1892), ch. 7, ami by Dkiuie, ‘ Lehrlnich d. Optik.’ (Leipz., 1900), p. 167. 
t “Oil the Dynamical Theory of Dift’raction,” ‘Camli. Phil. Soc. Trans.,’ vol. 9 (1849), ‘Papers,’ vol. 2, 
p. 241. 
I Sec equations (29) and (30) of the paper above quoted, ‘ Papers,’ vol. 2, p. 268. 
