] 2 
MR. DARWIN ON THE BODILY TIDES OF VISCOUS AND SEMI-ELASTIC 
x 1 <j^ 1 + *7sin 2 a exp (— Kt) j- +y 2 +z 2 -j^l + ^y(l— sin 2 a exp( — Kt 
( &77b 
+ 5 m sin a cos a xz exp (— Kt) = a~i 1 + — 
Let a be the inclination of the principal axis at this time to the axis of z, then 
sin 2 a exp (—id ) 
tan 2a 
1—2 sin 2 a exp (—id) 
If a be small, as it was in the case I considered in my former paper, then 
a' = a exp( — Kt) and yv = — Ka - 
Therefore the velocity of approach of the principal axis to the axis of rotation varies 
as the angle between them, which is the law assumed. 
2waa 
Also k= , so that k (the v of my former paper) varies inversely as the coefficient 
of viscosity,—as was also assumed. 
5. Bodily tides in a viscous earth * 
The only case of interest in which S; of equation (11) is a function of the time, is 
where it is a surface harmonic of the second order, and is periodic in time; for this 
will give the solution of the tidal problem. Since, moreover, we are only interested 
in the case where the motion has attained a permanently periodic character, the 
exponential terms in the solution of (11) may be set aside. 
Let S c = Scos and in accordance with Thomson’s notation,! let 
|= 5 , and 
r - U ,= V ; and therefore "q- =“• 
5 iva 2 19 v r 
Then putting i —2 in (11), and omitting the suffix of cr for brevity, we have 
da 
dt 
+ L—-S cos (vt+ij) . 
(14) 
It is evident that cr must be of the form A cos (v£ + B), and therefore 
A{ — vX sin (^+B)-f-g cos (v£+B)}=aS cos ( vt-\-r )) 
* In certain cases tFe forces do not form a rigorously equilibrating system, but there is a very small 
couple tending to turn the earth. The effects of this unbalanced couple, which varies as the square 
3 
of 2 pr> will be considered in a succeeding paper on the “ Precession of a Viscous Spheroid.” (Read 
before the Royal Society, December 19th, 1878.) 
f ‘Nat. Phil.’, § 840, eq. (27). 
