202 
Mr Coolcson, The Oscillations of a Fluid 
in which k must satisfy (9) and m is given by (8). There are an 
infinite number of roots (9) for all values of n from n = 0 to n=oo : 
so that the complete expression for cf> is a doubly infinite series of 
terms of the form (10). 
For shortness put 
Bn (*r) = Jn (*r) - 7 n (kt), 
n = <x> 
thus (f) = 2 2* A n>K B n (/cr) . sin nO . cosh k (z + h) cos mt. 
n=0 
Instead of the single trigonometrical factor A cos mt sin n 6 in 
the typical term, we might put 
(A cos mt + B sin mt) sin n 6 + (0 cos mt + D sin mt) cos n 6 . 
Let now 77 denote the elevation of the free surface at any 
moment above the mean level : then 
* = B)_ = (IL„ to the ° rder re< ^ uired - 
and since the liquid was originally at rest, 77 must be zero when 
t = 0, so that we require sin mt and not cos mt in the expression 
for (p. The typical term is therefore 
sm 
</> = A n B n (/cr) nO . cosh k (z + h) . sin mt. 
cos 
Hence 77 = tcA n B n (w) n 6 . sinh /ch . sin mt, 
cos 
and 77 = — - A n B n (/cr) S1R n 6 . sinh /ch . cos mt (10), 
m cos 
a possible form of the initial free surface is defined by putting 
t = 0 in this expression for 77. 
By superposition of two fundamental modes of the same period 
but in different phases, we obtain a solution 
77 = A n B n (/cr) . sinh /ch . cos (nO ±mt+ e), 
which represents a system of waves travelling unchanged round 
the origin with angular velocity m/n in the negative or positive 
direction of 6 . 
We may write (10) in the form 
sm 
77 = A n B n (/cr) n 6 . cos mt, 
