36 
PEOFESSOR A. E. H. LOVE ON THE INTEGRATION OF THE 
quantities; the functions by which it is expressed are those denoted by . . . 
These functions are defined for points on the further side by expressions which are 
not continuous up to and across the aperture; but they represent the transmitted 
radiation at any finite distance from the aperture. The actual disturbance is 
continuous up to and across the aperture. We seek accordingly to represent the 
disturbance on the nearer side by means of functions which are defined for the nearer 
side, but are not continuous up to and across the aperture, the discontinuities being 
so arranged that the displacements on the nearer side shall be continuous with those 
on the further side. In § 27 we separate the expressions of these functions into two 
parts, thus regarding the disturbance on the nearer side as consisting of two super¬ 
posed disturbances, there called A and B. The functions representing the disturbance 
A are continuous up to and across the aperture ; those representing the disturbance B 
are not ; but their discontinuities cancel exactly those of the functions , . . . The 
determination of B is in a certain sense unique. In § 28 we verify the supposition 
that B may be regarded as a system of standing waves, l)y actually determining, in 
accordance with this supposition, the functions involved in B, viz., u_, . . . , in terms 
of the functions ... In § 29 we show that the displacements, of which the func¬ 
tions . . . are the values, at certain times, and at points within the ajierture, are 
the displacements belonging to the disturbance A. The disturijance B and the trans¬ 
mitted radiation are thus determined in terms of A, and the general question of § 24 
is reduced to the determination of A. 
The components u', . . . f\ ... of A are subject to the following conditions :— 
(1.) On the nearer side they satisfy the equations of § 9 everywhere, except 
possibly at certain singular points. 
(2.) These singular points are the actual sources of the incident radiation. 
(3.) The functions u', . . . f\ . . . are continuous up to, and across, the aperture. 
(4.) At all points of the screen, not points of the aperture, they satisfy certain 
boundary conditions. 
The boundary conditions depend, to some extent, on the material of the screen ; 
and they will usually take the form that some components of electric or magnetic 
displacement vanish. The components, affected by the condition, are tliose of the 
displacement on the nearer side compounded of A and B, i.e., such quantities as 
u -|- ; but, as B falls off rapidly, with increasing distance from the ajDerture, it 
will generally be sufficient to impose the boundary condition on the components of 
A only. 
We may now give the following interpretation of the anal 3 "sis ;—The disturbance 
B, consisting of a system of standing waves, which are important in the neighbour¬ 
hood of the apertute only, can be described as the “ effect of the aperture.” The 
disturbance A can be described as “ the incident radiation, as modified by the action 
of the screen.” The result of § 29 can be stated in the form :—The transmitted 
radiation is to be calculated from the incident radiation, as modified by the action 
