ON THE HYDRODYNAMICAL THEORY OF SEICHES,. 619 
§ 25. We have now, of course, 
Ce Nate oe ee Cis 
S(c, 1)=(1- 51-35) ey. P ad « ; . (22); 
Gp (SN a ietieie Sadek tim eiliniiw «2s (289s 
Se,‘ =1(1 +55 \1+75) eee Hiiobegnd ; . (24). 
We shall have frequent occasion to use C(c, 1) and S(c, 1); and it is convenient 
for purposes of calculation to express them in terms of the gamma function. 
We have 
c\7, 4n?+6n+2-c 
Oe =(1-5) enya) 
a c\yyimt4(3 + a) } {n+ 3(3 - 2)} 
= ==) (n+4)(n+1) ; 
where a= ,/(4c+ 1). 
Since #(3 +a)+4(3—a)=4+1, it follows by a well-known theorem * that 
ce) = (1-$) 4. arena) 4) 
Since P(4)=77, P(1)=1, and a’ = 4c +1, we get finally 
C@)1) = n= ==) 2 _) ; » G5 
and in exactly the same way 
Se. 1) = he a iS = =) : I e6). 
Tt follows that 
K¢e, 1) = ar(? - ae ; re ; ae) ; ' ~— Oy 
a formula which has been much used by Mr WeEppERBURN and myself in the numerical 
calculation of seiche periods. If we recollect that 
rez) 100475), 
n(n) =e 5), 
we can put (27) into the form 
K(c, 1) = —, cos (a = \r : i (C = : ‘ es 28)e. 
which is useful for determining the sign of K(c, 1). 
* See Whittaker’s Modern Analysis (1902), § 96. 
