224 
MR. S. S. HOUGH ON THE APPLICATION OE HARMONIC 
§ (). The Period-Equation for the Free Oscillations. 
Returning to the case of uniform depth, we see that the equations (23) divide 
themselves into two groups, in one of which only even suffixes and in the other of 
which only odd suffixes are involved. We therefore conclude that the types of 
oscillation divide themselves into two classes, in the former of which the height of the 
surface-waves will be expressible entirely by harmonics of even order, and in the 
latter by harmonics of odd order alone. An exactly similar treatment is applicable 
to each of these classes; we shall therefore select for discussion the former set, 
contenting ourselves as regards the latter with merely stating results. 
Denote by L„ the expression 
f - 1 , _ 'i _^ 
n {71 + 1) (2n - 1) {2n + 3) 4wV“ 
(25). 
Then, putting all the y’s equal to zero, the types of free oscillation will be deter¬ 
mined by the equations 
7.9 
-aL, + .-;, = o 
C., 
5.7 ^ ' 11.13 
— -f — 0 
(26). 
{2n - 3) (271 - 1 ) 
'-'n+2 
{271 +3) {2)1 -p 5) 
r: = 0 
J 
At first sight it might appear that whatever be the value of X these equations will 
serve to determine C 4 , Cg,. . . in succession in terms of C 2 , whereas we know that this 
should only he possible for certain determinate values of X corresponding to the 
different periods of free oscillation. The manner in which these values of X are to be 
determined involves arguments similar to those used by Kelvin* in justification of 
the procedure of Laplace with reference to the forced oscillations after it had been 
attacked by AiRvt and Ferrel|. 
From equations (26), we obtain by actual solution 
C 4 ^ _ 
7.9 
7.9.11.13 
= C 2 
5.7’ 
i 
I 
1 
* ‘ Phil. Alag.’, 1875. Cf. also au analogous problem treated by Niven, ‘ Phil. Trans.,’ 1880, Part I., 
p. 133 et seq. 
t “ Tides and Waves,” § 3. 
t “Tidal Researches ” (U. S. Coast Survey 1873). 
