660 DR J. HALM 
The Seiche-polynomials C,,~*(w) are therefore represented by the coetticients of tl 
ascending powers in the series 
1e ; 
(1 -— 2hw + h?)t = Dez) 
0 
where 
Bane Sh ee CSO ce epee 
(Ole ( v) =, ( 1) pa Ge) dw 24 qd 2) \ 
Aa ia =) aa | 
SG ess eee a ee (22) 
The following relations between contiguous C-functions may be mentioned (see 
Wauittaker, /.c., p. 236). 
tC nr a(t0) — Cona(ta) = Cat) 
FE) = 20s) (23) 
; Cn" (ew) = 00 'i(«e) =" 20, (0) 
nC,,¢(w) =(n — 1 + 2a)wCr_,(w) - fas — w)Cn_3(w). 
In the case a = 0 the functions C,°(w) are of course zero, since 
(1 — 2hw+h?)?=1 
: 
But it can be shown that the limiting values of ee (lim a=0) represent the 
coefficients in the expansion w 
4 log (1 — 2hw +h?), 
and we derive the following relation : 
oo e Rer-t h 4 d 
log /1—2hsinz+h? = Da -1)" ona] SB (2n- 1)z- yy 008 2nz| (24) 
n=1 os 
w= sinz. 
The relations (21) are of particular interest in the Seiche-theory because they lead | 
to elegant expressions for the horizontal and vertical displacements € and ¢ If we 
write the Seiche-equation referring to a parabolic concave lake 
d?P 
Spay Efi 
(1 wi) 3 + n(n INO) 
we have, usine Professor CHRYSTAL’S notations 
’ 5 > 
uf ‘ 2A Ges { e } 2A a” a) 
= Ke 19h See ae ay i ee ee eS — y;2\P-1 = — ap2\n—l 
(oe) ee) sin 7,t h(1 — w?) dw? \ a) hn(n—1) dw” { (ae | 
cists 2) he BAP (ap 
(=1)2.4..... Qn—2) Ry aE (1 -— w?) ; 
