ON THE HYDRODYNAMICAL THEORY OF SEICHES, 631 
The boundary conditions are then as follows :— 
A,C(q,, 1) + BS(e,, 1)=0; 
A, = Ap, B,/a, = Bo/a ; 
AgO(C , Wy) — ByS(¢y , Wy) = AgE(cg , V3) + BsS (cy , Ws), 
— A,O'(e,, Wo) + B,S'(¢ » W,) = ae { A,O'(cg , 03) + BaS (eg , 05) ; ; 
3 
A,= Ay’, B,/a,= By /a,! ; 
— Ay’ (6,', 9’) + Ba'S(cq’ , Wg) = Ag G(cg , Wy’) — By’ E(c,’ , We) » 
gO (cy, ty’) + ByS(cy’, 10) = 2, | — Ay C(ey’, 124) + BYS'(ey st) } ; 
3 
Ay’ =A,’, B,'/a,' = By'/ag ; 
Ay Ce’, 1)- B/(¢, 1)=0. 
m these, since C(c, , W.)S'(Cy , Wo) —C’(ey , We)S(Cy, W) = 1, we derive 
dA O(c, 1) + a B,S(e,, 1)=0, 
A,= | Sey. My) ECs, 5) + 28(6, 5 Wa) (Cg, Ws) \ As 
3 
a { S'(Cy , Wy) SG (Cg » Ws) + “28(c Wy) S (Cg , Ws) \ B,, 
3 
=)AA,+ pB, (say). 
By= | O(c 1) E(C 09) + 2C (6a, 25) (ey, 9) | Ay 
3 
1 ; O'(Cy Wy) G(cg Ws) + “2.0 (C9, Wy) (Cg 5 Ws) } B, 
3 
=vA, +B, (say). 
x A,0(¢,, 1) {AA, + uBs} +.a,S(c, 1){vA; + pB,} = 0, 
{a,AC(c,, 1) + a,vS(c, RAS 
+ {aqpCO(c, , 1) + aypS(c,, 1}B,=0. 
: dy Ao’ O(c)’, 1) - %'B,/S(e', 1) =0 
; 4 A= \ S'(co' , Ws )E(Cg , ws’) + “28(cy) 1 Wy )B'(ca’ » Wg’) } A, 
3 
a = { S'(c,", ws)S(cq , Wy’) + “280, pt (Cy axthy:) \ Baas 
3 
=)'A,' —y'B,’ (say). : 
li { C'(eo' , 5’) O(c’ , ws’, ) + “2,0 (6, wo’ )G'(cz, Ws) \ As’ 
3 
1 { O'(e,' , We’) S(Cy, Ws) + “2,0(¢9 1 Wy )S'(cs’ , wy) } B,’ 
3 
= = —vA,' +B,’ (say). 
Gem DNAs = WB hoa s(e,1)(—v'A, # BPO 
{a d'C(e,’ 1) + ay'v'S(c,', 1)} Ag’ = {atg'p'C(cy’, 1) + a,'p'S(e,’, 1)}B,' =0. 
ince A,’=A., B,’=a,'B,/a,, the last equation may be written 
; {a d'C(e,’ , 1) + a,'V'S(c,’, 1) } Ag — a {a9'u'C(cy’, 1) + a,'p'S(e,', 1)} B, = 0. 
_ TRANS. ROY. SOC. EDIN., VOL. XLI. PART ITI. (NO. 25). 93 
