620 PROFESSOR CHRYSTAL 
§ 26. By Srurm’s Oscillation theorem, applied to the solutions of the differential 
equation (13), we see that for any real value of v not exceeding unity the equations 
C(c,v)=0, S(es0)—0, ‘ : : : (29). 
C(e,o)=0, Cle, =0, ‘ ‘ : : (30) , 
have each an infinite number of real roots, and that the roots of each equation of either 
pair separate the roots of the other equation of the same pair. 
In particular, the roots of 
Ces 1) — Ob eare.: cs 24 eee £ 
of 
S(¢; l)=S0) are: “eS 223) 400), we eo ; 
unfortunately the roots of 
G(c,1)=0, G(c, 1)=0 
are not commensurable ; and, owing to the slow convergence of the series involved, 
they are very difficult to calculate directly. By a very laborious calculation, I find :— 
for the smallest root of @(c,1)=0, c=2°'77... .; and for the smallest root of 
6(¢,1)=0, c=12°34.... As these figures agree with the approximations given 
by Dr Hato in the paper on the seiche functions above referred to, and with calcula- 
tions which Dr Burasss, Professor Gipson, and Mr HorssurcH have been kind 
enough to make for me, probably they are correct. It would, however, be hopeless to 
calculate the higher roots, or even these two to greater accuracy, by direct use of the 
series as it stands in (17) and (18). 
SEICHES IN A CONCAVE SYMMETRIC COMPLETE PARABOLIC LAKE. 
h(x) =hx(1—2a"/a’). 
A a 5 oO a A 
Hiren 2: 
§ 27. The equation for determining P is, by § 22, 
o2P n2 
des LIES 2S 1) 
Ga? * gh(1 — 22 /a2) 
; 
or, if w—Z/a, 
GINO Tors 
ape pS 
( aya ohir, o, 
say, 
a2P ; 
(1 — et) teP=0, : : : . : (31), 
where 
c=n?u2/gh , : ; j : ; (32). 
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