482 MR. G. H. DARWIN ON THE PRECESSION OF A VISCOUS SPHEROID, 
There are obviously two solutions of the equation, one of which represents that the 
earth is very nearly fluid, and the other that it is very nearly rigid. 
In the first case, that of approximate fluidity, e, e, e" are very small, and therefore 
12 4 
sin 4e=4e, sin 2e / = 2e' = 2e, sin 4e" = 4e"=4—e=-pp—z€ 
n ZT' 62 , 
Hence 
(1650 + 248-f-2-V 2 of 7-04)e = 8 
whence 
That is to say, the semi-diurnal tide only lags by the small angle 14'. But this is 
not the solution which is interesting in the case of the earth, for we know that the 
earth does not behave approximately as a fluid body. 
In the other solution, 2e and e' approach 90°, so that p is small; hence 
• , 4 np p . n , 2 np 2 p n n ■ , „ 4/2p 
sm 4e= ——sm 2e = — ■-= very nearly, and sm 4e = — , , 
p* + 4n* n’ p° + n 2 n J /d+4/2 3 
Hence we have 
412-6(£) + 128-9gf) + 7-04a^=8 
Put ~-T 
2/2 5 
so that x— cot 2e" 
then substituting 
for — its value 
27 - 32 ’ 
we have 
whence 
1320-7 
27-32 
x+7- 042 
x 3 --l655x 3 +l-2921a;--1655 = 0 
This equation has two imaginary roots, and one real one, viz. : -12858. Hence the 
desired solution is given by cot 2e"= -12858 ; and 2e ,/ = Hr—7° 20', and the corres¬ 
ponding values of 2e and e' are 2e=^77—16', and e'=^7 r — 32'. If these values for e, 
e, e" be used in the original equation (63), they will be found to satisfy it very closely ; 
and it appears that there is a true retardation of the moon of 3 //- l in a century, whilst 
the lengthening of the day would make an apparent acceleration of 7 //- l,—the difference 
of the two being the observed 4". 
With these values the semi-diurnal and diurnal ocean-tides are, according to the equi¬ 
librium theory of ocean-tides, sensibly the same as those on a rigid nucleus, whilst the 
fortnightly tide is reduced to sin 2e" or *992 of its theoretical amount; and the tune 
7T 
of high tide is accelerated by - — —, or 6T hours in advance of its theoretical time/''" 
vt J L J L 
* In the abstract of this paper (Proc. Roy. Soc., No. 191, 1878) the height and lag* of the bodily tide 
were accidentally given instead of the height and acceleration of the ocean tide. 
