634 PROFESSOR CHRYSTAL 
Also, if the dash on L(¢,z) denote differentiation with respect to z, 
= Le, 2) =S8(e, 1)C(c, 1 — 22) - Cle, 1)s(e, 1 — 22) 
OF il s (55) ’ 
ae S(e, 1)C'(e, w) — C(e, 1)8'(c, w) 
we may also notice the particular relations 
Legs; 1; . Ueda) =Ge, ; . (aig 
§ 42. 
THE L-SoLutTiIon FoR A CoMPLETE PARABOLIC LAKE 
is obviously 
EN(1 — w?) = AL(c , z) sin n(t — 7) ; 
== Le, z) sin n(t-7). | (51), 
- a 
The values of c are the roots of the period equation 
. L(c, 1) 
that is to say, : . 8 
=v(v+1); 
and 
T, =27a/ ,/{cv(v + l)gh}, 
as before. 
The values of z corresponding to the nodes of the »-nodal seiche are the roots of 
the equation 
Lites =Or, te 2 . : . 6am 
Tue L-SoLuTIon FoR A BIPARABOLIC LAKE. 
§ 43. For the part between w=0 and w= +1, 
ER(1 — w?) = (c,2) sin n(t —7) ; 
ua (60) ; 
Ae mele sin 2(t — 7) ; 
2 
where z=3}(1—-w). 
For the part between w’=0 and w’= —-1, 
A(1 —w'2) = 1 (c', 2’) sin n(t— 7) ; 
Eh( \= ae a c’, z’) sin n(t — 7) | | | aa 
A Pio ANC i 
(€ - Jae (2 sinn(t=ns| 
where 2 =4(1+w’). 
The period equation is now 
aL(c, Fy Wey ale, sled) Oe — 
where 
c=na2/gh, c =n'a'2/gh. : : . , (63) ; 
and 
T, =2x/n, = 2a! /(c,gh) = 2a’! J(c', gh) . ; . (Coa 
