232 
MR. S. S. HOUGH OH THE APPLICATION OF HARMONIC 
By a comparison of' the 2nd and 3rd columns it will be noticed that in most cases 
the roots of the frequency-equation are given at once with a fair degree of accuracy 
by simply solving the equations = 0, and that this approximation improves the 
greater n becomes. In the fifth column I have given the periods of oscillation for an 
ocean of the same depth when the rotation is annulled, calculated by means of the 
formula 
_ n in + 1) lujn 
where w now denotes a constant such that ir/w = 12 hours. It will be seen that the 
approximation obtained by omitting the rotation continually improves with increasing 
values of n, but in no case will it lead to as accurate a result as the formula 
1 
{2n - 1)(2 7 i + 3) 
For instance, taking the case hgjAoy^a' = n = 8 , the error introduced by using 
the first formula amounts to about 14 per cent., whereas the second form gives the 
frequency with an error less than one per cent, of its true value. 
8. Unsymmetrical Types. 
An exactly similar method of treatment is applicable to the types which are 
represented by a series of harmonics of odd order ; the period-equation for these types 
is given by 
II ^;_2 + 3 — 0 
where n now denotes an odd integer and K„ denote respectively the continued 
fractions 
_ 1 _ _ 1 _ _ 1 
{2n -f 1) (271 -f 3)~ (2ft -1- 5) {2n — 3) (27i — 1)- {2n -f 1) 3.5- 
1 1 
(2ft - 3) (2ft - l)-(2ft + 1 ) (2ft -f l)(2ft + 3)-(2ft -|- 5) 
L„ — — . . . ad inf. 
Treating this case in the same manner as the last, I have found the first six roots 
and the corresponding periods of oscillation for the four depths employed as follows :— 
