26 
MR. DARWIN ON THE BODILY TIDES OF VISCOUS AND SEMI-ELASTIC 
A= 
B=- 
G 
i CCT Sill 6 
m (y 2 — gli) 2 (a 2 co 2 — gli) 
F-E cj a(geose-fyE) 
m(V 2 — gli) 2 (ft 2 ® 2 — gli) 
In the case of such seas as exist in the earth, the tide-wave travels faster than the 
free-wave, so that cr<A 2 is greater -than gli ; and the denominators of A and B are 
positive. 
We have then— 
( a?ar—gh) 
~ cos e—r (/E^ sin — ~ sin e cos j* 
But the present object is to find the motion of the wave-surface relatively to the 
bottom of the canal, for this will give the tide relatively to the dry land. Now the 
height of the wave relatively to the bottom is 
PQ=h — E cos \m(x—vt)-\-e]—7) 
— h—h 
And 
r/£_ 1 
dx a "co" —gli 
dx 
T '—i \ T 
- cos e — jg E j cos -J- sin e sin 
4 
Hence reverting to the sphere, and putting a for a-\-h, we get as the equation to 
the relative spheroid of which the wave-surface in the equatorial canal forms part— 
h sin 2 6 fr . . 2„ r , . , 
r—a— - 2 - 0 ——-j -cos 2{(p—(i)t) — jghj cos [2(</> — cot)-\-e] J 
But according to the equilibrium theory, if Y has the same form as above, viz.— 
T V 
g + -g[-) E sin 3 6 cos[2(^>—<y^) + e]-{-q( - ] sin 3 6 cos 2(<f> — cot) 
and if r—a-\-u be the equation to the tidal spheroid, we have, as in Part I., 
sin 2 or t , 3„ r . 
U = — -j - COS 2 ((f)— cot) -j-; t/E cos — cot) -f e] 
and the equation to the relative tidal spheroid is 
r=a-\- u — <r 
, sill 2 OC T . . . 2 „ \ . 
= «+ - cos 2(<f>— cot) — _yE cos [2(<£— cot) + 
9 L 2 0 
Now in either the case of the dynamical theory or of the equilibrium theory, if E be 
put equal to zero, we get the equations to the tidal spheroid on a rigid nucleus. A 
comparison, then, of the above equations shows at once that both the reduction of tide 
and the acceleration of phase are the same in one theory as in the other. Bat where the 
