ANALYSIS TO THE DYNAxMICAL THEORY OF THE TIDES. 
239 
and this equation, as we have already seen, is the equation which determines the 
periods of free oscillation. 
It is usual in Tidal Theory to express the height of the forced tides in terms 
of the height of the corresponding “ equilibrium-tides.” If we denote the height of 
the equilibrium-tide arising from the disturbing potential in question by (p.) 
we see, on omitting all the terms on the left of (23) which depend on inertia, 
and replacing G,, by that 
= y,//./ 4a»'a' ; 
and therefore 
= gMn- 
If then denote the height of the equilibrium-tide, we have 
-b (2n-I) (2/i-3) (2^.-5) {'ln-7) 
+ {2ii —l) (2« — 3) II,,_3P,;_2d-P;, 
+ (2ri-p3) (2«-|-5) + + 
+ (2?i-p3) (2n-p5) {2n-\-7) (2n-|-9) K;; + 3 K„_,.^P^; 
+ ... 
The most important practical application of the above theory is the case where the 
disturbing potential involves only a single harmonic term of the second order, and 
the period of the disturbance is long compared with the period of rotation. Thus 
taking = *00133, hgjiuPcir' = 1/40, which corresponds to the case of lunar- 
fortnightly tides in an ocean of depth 7260 feet, we hnd 
Lo = - *09349, Ps = - '03105, 
= - *04745, = — *02886, 
L,; = — *03561, = — *02764. 
whence, neglecting we obtain in succession 
log Kj3 = /i4*T2, log Kjq = /z4*432, log Kg = '/i4*815p 
log Kg = w3'3055, log Kj^, = w3*9992. 
Thus 
— K^, = — *08351, 
log (C 4 /C 2 ) = wT* 7986, log (CyC^i) = wl*4608, log (Cg/Cg) = nl*2216, 
log (Cio/Cs) = ’^1'033, log (Cio/Cio) = 1 ^ 2 * 88 , 
jn 
fo (H«-2 ”t Kh + 2 — Pi) Oii 
