1903 - 4 .] Prof. Chrystal on Mathematical Theory of Seiches. 331 
It appears at once from (9) that the real positive roots of 
C (c, 1) = 0 are 
The roots of (S(c , 1) = 0 and @(c, 1) = 0 are neither commensurable 
nor so easily found. A somewhat laborious arithmetical calcula- 
tion, in which I have been kindly assisted by Dr Burgess and Mr 
E. M. Horsburgh, has given for the smallest positive root of (£(c,l) 
= 0c = 2-77...., and for the corresponding root of @(c, 1) = 0 
c = 12-34 
It should also be observed that, when c has one of the values 
(11), C(c, v) reduces to an integral function of v ; and the same 
happens to S(c, v) when c has one of the values (12). 
If we assume <r(v) = h( 1 + v 2 /a 2 ), the equation for P is 
which, if we put w = v/a , and take 
c = n 2 a 2 jgh , 
reduces to either (5) or (6). Hence A(x)£ can he expressed in 
terms of the seiche functions ; and £ is given by 
In the case where the breadth of the lake is constant and the 
cross section rectangular, hut the depth variable, say h(x) — 
h{ 1 -x 2 /a 2 ), we can replace the variable v by x. The constants h 
and a are then linear magnitudes (whose meanings are obvious) 
instead of a volume and an area as in the general case. It will be 
observed, therefore, that all the general features of the phenomena 
of seiches are to be found in this more special case, regarding which 
we now give some particulars. 
1.2, 3.4, 5.6, ilk 2, 12, 30, . . (11) 
and of S(e, 1) = 0 
c=2.3 ; 4.5, 6.7, i.e. 6, 20, 42, . . (12) 
1 du 
{= - 
a dro 
