33 
EQUATIONS OF PROPAGATION OP ELECTRIC WAVES. 
A, and not to the disturbance B. The relation between A and B is similar to that 
between the “complementary function” and a “particular integral” of a linear 
differential equation, with a right-hand member; the difference between two particular 
integrals is part of the complementary function.* 
Perhaj)s the simplest way of building up the functions u_, . . . is to act upon a 
hint, derived from a study of Helmholtz’s theory of acoustical resonators, f We 
may, in fact, attempt to satisfy the conditions, by regarding the disturbance B as 
consisting of a system of standing waves ; and we find that the method thus 
suggested is successful. We shall proceed on the assumption that the displacements, 
represented by u_, , constitute a system of standing waves.] 
28. Having regard to the proposed plan of passing to a limit when the point 
[x, y, z) is brought to coincidence with {x', y', z'), we see that especial importance 
attaches to the limiting values of such expressions as 
cos K {ct — r) — KV sin k (ct — r), 
which, in § 26, have uniformly been placed in { } ; and it appears that all these 
limiting values are numerical multiples of cos k Ct. This remark indicates that the 
discontinuities of the terms sin k Ct, . . . are independent of those of the terms 
cos K ct, . . . Again, when the expressions are, as above, replaced by their 
limiting values, it appears that every term in . . . might be interpreted as a 
differential coefficient, either of the first order, or of the second order, of the potential 
of a distribution of surface density on the area within the aperture. Now it is 
known| that, the charged surface coinciding with the plane 2 = const., first differential 
coefficients of the potential with respect to x and y are continuous in crossing the 
surface, and the first differential coefficient with respect to 2 has a definite discon¬ 
tinuity ; further, it is known that all second differential coefficients are continuous in 
crossing the surface, except the two that are formed by differentiating with respect 
to 2 once and either x or y once. These considerations guide us to a proper choice of 
displacements in the standing waves represented by w_, . . . ; for example, in the 
second line of the expression for Attu^, the factors 
K 
vx~ 
must he retained, and multij^lied by a function of r and t, of which the limit at ?’ = 0 
is cos K ct. But we should not arrive at a proper choice by replacing the expressions 
in { } by their limiting values ; for the system of displacements thus arrived at would 
not satisfy the fundamental differential equations. This consideration suggests that 
* Forsyth, ‘Treatise on Differential Equations,’ ch. 3. 
t See the memoir already quoted, particularly equation (29c), Helmholtz, ‘Wiss. Abh.,’ vol. 1, 
p. 377. 
J Cf. Poincare, ‘ Theorie du Potentiel Newtonien,’ ch. 3. 
VOL. cxcvir.—-A. r 
