ON THE HYDRODYNAMICAL THEORY OF SEICHES, 617 
GENERAL SOLUTION FOR A PARABOLIC LONGITUDINAL SECTION. 
INTRODUCTION OF THE SEICHE FUNCTIONS. 
§ 24. Consider the equation 
d?P c : 
a hae” . . . . . (i3)e 
uming y=+a,v+ugv2+ .... , we find in the usual way 
| ¢4+1.2a,=0, ca, + 2.3a,=0, 
' (c-1.2A)a, = 3.4a,=0, (BBN -=0, 
| (c ae 3)(n— 2)N}a as Naa, =. 
nee we have 
more c(e — 1.20) 4 e(¢ — 1.2A)(¢ - 3.4A) \ 
- ee At Wasps esa 
2.3 SDB \\ie= 46 
$B yel vf oe o coe ek ; as 
ere A and B are arbitrary constants. The series in the brackets are obviously 
ent if |v|<1/,/|A|; and divergent if ;v|>1/,/|A|. They are also convergent 
+1/,/; for the general term, of say the first series, may be written 
euo(Q- ga) (oe) eernen 
and, since the infinite product 
Ee 
3 convergent, wv, is ultimately of the same order as 1/n’*. 
aa 
If we introduce the notation 
Pesce c(¢ — 1.2A) 4 
C(c,A,v)=1 i” bear ant ER 
Sane ew? c(e = 2.3) 4 en ; 
ee) Jeo} 1 oat 23x45 
e that the functions C and S have certain properties in common with the circular 
ons. For example, we have 
Cie,A,-v)= Cle,rA,v); 
S(c,A, -v)= —S(e,A, rv); 
C(c,r,0)=1, S(ce,A, 0) =0. 
C(e, 0, v)=cos( fev), S(c,0,v)=sin ( ,/cv) ; 
that the cosine and sine are particular cases of the functions just defined. 
