454 MR. G. H. DARWIN ON THE PRECESSION OF A VISCOUS SPHEROID 
Semi-diurnal. 
Diurnal. 
Fortnightly. 
Tide . 
Slow 
(2n -2/2). 
Sidereal 
(2 n). 
Fast 
(2a+2/2). 
Slow 
(n -2/2). 
Sidereal 
(n). 
Fast 
(n + 2/2). 
(2/2). 
Height . 
E, 
E 
E, 
E\ 
E 
E', 2 
E" 
Lag . . . 
2e 1 
2e 
2e 3 
/ 
e i 
e 
/ 
€ 2 
2e" 
The E’s are proper fractions, and the e’s are angles. 
Let r=a-\- cr be the equation to the surface of the spheroid as tidally distorted, a 
being the radius of the mean sphere,—for we may put out of account the permanent 
equatorial protuberance due to rotation, and to the non-periodic term of V. 
It is a well known result that, if ivr ~S cos {yt-\-rj) be a tide-generating potential, 
estimated per unit volume of a homogeneous perfectly fluid spheroid of density w. 
(S being of the second order of surface harmonics), then the equilibrium tide due to this 
5 o? 2 o 
potential is given by cr— —S cos {yt + 17 ). If we write this result may be 
Z CJ DCC 
• <X S , , 
written -=- cos (vt-hri). 
a g \ ' n 
Now consider a typical term—say one part of the slow semi-diurnal term—of the 
tide-generating potential, as found in (3) : it was 
—wEt\p^ sin 3 6 cos 2</> cos 2 (n — S2). 
The equilibrium value of the corresponding tide is found by putting - equal to this 
expression divided by tvr~Q. 
Then if we suppose that there is a frictional resistance to the tidal motion, the tide 
will lag and be reduced in height, and according to the preceding definitions the 
corresponding tide of our spheroid is expressed by 
° = — T E l \p 4! sin 3 6 cos 2(j) cos \2(n—ft,) — 2e{] 
Cl 
All the other tides may be treated in the same way, by introducing the proper E’s 
and e’s. 
Thus if we write 
^> e — E ]L cos (2 n — 2/2 — 2e,) + Ap 3 q 3 cos (2n— 2e)-\-E. : cos (2n+2/2 — 2e 2 ) 
Tq = E\2p z qcos (n — 2/2 — e\) — E'2pq( p~ — g 3 ) cos (n — e') — E'. 2 2qxj 3 cos (n + 2/2 — eb) 
X e = E"3pY cos (2/2 - 2e") 
( 8 ) 
and if in the same symbols accented sines replace cosines, then, by comparison with (5), 
we see that 
