182 T. H. Havelock 
by substituting in (20) the expansions 
Kpe e+ sinh € = 2) ( +7)" 1 nd, (Kpe*) sinh n€, (22) 
T 
We obtain, for n> 1, 
~ ( _4)n-1 Ff* K e-2ka —yn-l Pei K e—2Kb . ak 
ee Sate ee eee an) 
2 f( — r= 1F(— K)—0"— 1 F(x (k)} e-2xd M6 dk BS 
-[ aS nJy(pei®)E, (28) 
(ee) ==} n—1 Pe K) e72ka _ nl Fre ( — K e—2kb . aK 
1D}, = ~irb,—| ( ( 1 Dod: ( nJ,(Kpe*’) ay 
0 3 K 
eo) {( pS t) ee F(= k) —yn-1 F(k (k)} eae 9 dk - 
+f Ripe we Wie eaacar Bae ce nJ,,(Kpe ) K? (24) 
while for n = 1, C, has the additional term icpe’ and D, the additional term 
lepetd 
— Cpe”. 
The boundary condition on the contour C is that the real part of dw/dé 
should be zero for = &); this gives 
DE = —e?"60C,. (25) 
Using this in (23) and (24) we obtain an infinite set of equations for the 
coefficients b,,; these are, for n> 1, 
© H(k) dk 
0 1— e72kd K H 
(k) = {ng" F(x) J,(Kkpe~”) + (— 1)" nF *(x) J,(Kpe")} ex 
+ {(—1)"ng”F(—K) J,(xpe-) + nF*(— x) J,(epe®)} e-™ 
= nl{a"F*(—K) + (= 1g F*(w)} I, (ape) 
+{(—1)" F(—k) + F(k)} J, (kpe®)] e-24, (26) 
with a similar expression for 2b, including an additional term 
ib, = 
— gcepe + tepge—, 
and with q = e£0 
By using (19), these results may be combined into an integral equation 
for the function F(x); it is 
F(x) = Iy(xpe!)— eple® — ge) J (xpe? 
ap ae )G, + F*(v) Gy} e244 {F*(—v) Gy + F(—v) Gy} e2” dv 
0 ] —e-2vd ) 
(27) 
2 
_ (°tP*(< 2) G+ F(—») G.+ F(v) Gy + F*(v) Gy} e-24 dv 
0 ] —e-20d v 
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