342 T. H. Havelock. 
in which 
F, = — {(k — ky sec? 0) P, — pQ, sec 6} D 
F, = — {(k — ky sec? 6) Q, + pP, secb} D 
F; = — {(« — ky sec? 6) Qy — uP, sec 0} D 
F, = — {(k — k, sec? 8) Py + wQy sec 6} D 
D = ky sec? O/z« {(k — Ky sec? 0)? + u? sec? 0}, (12) 
and the quantities P, Q are given in terms of the source distribution by 
a | ae—! cos (kh cos 0) cos (kk sin 0) dS 
Py= | oe sin (kh cos 9) sin (kk sin 0) dS 
Q,= | oe “sin (kh cos 8) cos (kk sin 8) dS 
OQ, = | ce—*! cos (kh cos @) sin (kk sin 9) dS. (13) 
Similarly from (10), 0¢/0z is obtained in the same integral form as in (11), 
with quantities G instead of F given by the same expressions as in (12) but 
with 
D=k/r{(k — ky sec? 6)? + py? sec? 6}. 
The expressions for the surface values of ¢ and 0¢/éz are now in a form to 
which we may apply a theorem derived from the Fourier integral theorem in 
two variables ; namely, we have, with the above notation 
@@ awe ce ll a) GG 2G. GL Gi\ede (4) 
az | , 1 
Using (8), this reduces readily to 
Cs (o) 2 2 2 2 
18 = doshan che, ayn sec 6 a0 | Heese Honan ede Oho) gy 
0 A) o (kK — Kp sec? 0)? + u* sec? 0 
= 16mic,20 ( (P2-+ P,?-+ Q2-+ Q,2) sec 0 d8, (15) 
0 
where in (15) the quantities P and Q have the values given by (13) when « has 
been replaced by x, sec? 0. 
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