32 
PKOFESSOR A. E. H. LOVE OX TflE IXTEORATION OF THE 
for B u = y._ = Q.osKCt + sin «: C2 , '] 
I.(70). 
t = /_ = //' cos K Qt -f- Jj' sin K ct , 
« f . • ^ 
We have now to express the conditions of continuity of magnetic and electric 
displacement at the aperture. We su})pose that . . . are formed for a point 
P, {z > z), and that u_, . . . are formed for a point P', [z < z), and we take any 
point p) in the aperture. The functions . . . are continuous in the neighhour- 
hood of and have definite values at (}. We form the dhference 
and allow P and P' to approach Q hy any pathsf"' the tangents to which at Q do not 
lie in the plane of the aperture. Then the conditions of continuity are 
lim[.,(P) 
hm -! t\ (P) 
The functions ii_, . . . satisfy the general e(piations of § !) at all points on the 
nearer side (; < 2 '), and are free from singularities in this region ; these conditions, 
with the conditions of continnity (71), suffice to determine the functions in question, 
•in terms of the functioiis <7j, . . . introduced in § 25. (One such determination will 
he worked out presently ; here it is important to observe that it is effectively unique. 
The conditions (71) recp.iire, in fact, that tlie discontinuities of u_, . . . -should be 
arranged so as to cancel exactly those of ?«+, . . . Now let us s\q)pose that two 
sets of functions u. . ., and //_ Au_, . . . have l)een f nmd, Ixffh of which oliey 
all the conditions imposed iq)on the hmctions u_, . . . ; their differences \u_, . . . 
have no discontinuities at the aperture, and no singularities 011 the nearer side 
(s < 2 '); thus the disturbance rejjresented hy Azi_, . . . belongs to the disturbance 
* The path of F lies, of course, entirely in the region i and that of F' in the region 
— (P^)} = III (Q) COS KCt nj (Q) sin kCI , "j 
.... 1^ 
— ./_ (P')] = /,' (Q) COS KCt + /h (Q) sin k ct , [ 
