EQUATIONS OF PROPAGATION OF ELECTRIC WAVES. 
81 
which occur in these expressions, represent fnnctioiis, v/hich are continuous, and have 
definite values, at all points, that do not lie in the plane 2 : = 2 .', and within the aper¬ 
ture. At points within the aperture, the functions, defined by these integrals, present 
discontinuities of one or other of three following kinds'^" :— (a.) The integral, obtained 
by replacing (x, y, z) by ix' , ?/', z), is convergent, and is different from the limit 
obtained by bringing [x, y, z) up to coincidence with (x', y', z') through values, for 
which 2 ; > z', or through values, for whicli 2 < 2 ;'; these two limits are finite and 
definite, and they are not the same. The term of (63), containing d)~^/dz, is an 
example of this peculiarity, (h.) The integral, obtained by replacing [x, y, 2 ) by 
{x', y\ z), is not convergent ; but the limit obtained by bringing (a’, y, 2 ) up to 
coincidence with (.if, y', 2 '), on either side, is finite and definite, and these limits are 
the same. The term of (65), containing di~^jdx, is an example of this peculiarity, 
(c.) The integral obtained by replacing [x, y, z) by (.»', y', z) is not convergent, nor is 
any definite limit obtained by bringing (x, y, 2) up to coincidence with [x , y', z) on 
either side; hut the diffei'ence of the two values, obtained liy taking (x, y, 2 ) at two 
points, near the aperture, and on opposite sides of it, can be made less than any 
assigned quantity by sufficiently diminishing the distance between the })oints. The 
term of (63), containing is an example of this peculiarity. 
The discontinuity of the expressions for . . . arises from the representation oi 
tlie disturbance on the further side (2 > z) as due to imagined sources in the 
aperture ; there are not really any sources in the aperture, hut the disturbance 
on the further side (2 > 2 ') is continuous with the disturbance on tlie nearer side 
(2 < z). To restore continuity, it is most convenient to regard the disturbance 
on the nearer side as consisting of two superposed disturbances, denoted by A and 
B. The disturbance A is represented by functions, which are continuous in a region 
of space, containing all tlie points vuthin the apei'ture, and within a finite distance 
on either side of it ; these functions have no singularities on the nearer side, 
except the actual sources of the incident radiation. The disturbance B is repre¬ 
sented by fimctions which have no singularities on the nearer side, but have 
discontinuities at the ajierture, and these discontinuities may be of any of the kinds 
presented by the expressions for . . . We shall denote the magnetic and 
electric displacements, that belong to the disturbance B, by . . . , /L, • • . , and 
those that belong to the disturbance A by u' , . . . , f\ . . . The displacements, 
that belong to the disturbances A and B, satisfy the general equations of § 9. We 
shall take them to be given by the following equations :— 
for A, u = u' = ii^ cos K Qt + uj sin k of, ] 
t — f = fi cos KOt // sin K ct , 
.... ' J 
* Cf. Poincare, ‘ Theoric du Potentiel Newtonien ’ (Paris 1899), ch. 3. 
• (fi'J); 
