EQUATIONS OP PROPAGATION OP ELECTRIC WAVES, 
aud remember that 2 and z’ are constants in the integration ; we find 
1 r-'^ u — 2 ' . 2 . 
=r — (J(f) (]}■ ^ (cos Q + /cr sin Q) + ^ (sin'Z' — ki- cos xjj) 
Jrvr J 0 .2 — 2' r' X- r" 
1 
/cr- 
■(2 - P)- 
+ sin^ (/) ( I 
(2 - P)= 
{(3 — K'l '-) sin ip — SKVCosxp] 
rK [z — z' . 2 . 
= L dr ■ ,/ (cos ip + sin xp) -|-^ (sin ip — k)- cos \p) 
}, _ -f T“ KV^ 
~ oTihl ^ ^ I [(3 — Kh-')sinxp — SKrcosxp] 
iKl- 
This is immediately integrahle, and we find''" 
u 
cos K {z — Gl -j- 11) -j- 
/(R 
I — j sin K [z — ct II) 
+ cos K (2 — ci) ; .... (56); 
and, when R is very great, this is approximately equal to 
cos K {z — ci) — .f cos /c ( 2 ' — cf + R). (57)- 
Thus the value of u for the primary wave is I'eproduced l)y the secondary waves 
sent out from the parts of the plane, which are not at a very great distance. In like 
manner we shoidd find for y the value 
K sin k(z — Qt) 
+ 
1 + 
R 
sin K [z' — ct + R) 
[ 
/cR \ 
(2 - Pfi ] 
IP 
cos K ( 2 ' — ct -fi R) 
giving, when R is great, the a])proximate value 
— K sin K (2 — C?) + J /c sin k {z — Ct + 11);.(1^3); 
and, as before, the value fbi' tlie primary wave is re})roduced by the secondary waves 
sent out from the parts of the plane, which are not at a very great distance. Both 
for n and for g, the distant parts of the plane contribute something finite to the 
disturbance ; just as, in the ordinary elementaiy theory, there may remain a ])ortion 
of a Huygens’ zone uncompensated ; such portions are always disregarded.! 
The difticulty here considered arises entirely from our having applied to an infinite 
plane, formulse, which were obtained on tlie express supposition that the surface, to 
whicli they are applied, lies entirely within a finite distance of tlie jioint, at which 
* Equation (.56) determines the intensit}^ at a point on the axis, of light diffracted through a circular 
aperture, the incident light being parallel, and the ordinary optical rule being assumed to hold (§§ 24, 30). 
t Cf. Basset, ‘ Physical Optics,’ p, 46, or Lord Rayleigh, ‘ Wave Theory of Light,’ p. 429. 
