18 
MR. DARWIN ON THE BODILY TIDES OF VISCOUS AND SEMI-ELASTIC 
Now these equations have exactly the same form as those for the motion of a viscous 
/I d\~ l 
fluid, save that the coefficient of viscosity v is replaced by n (q+y) • We may there¬ 
fore at once pass to the differential equation (11) which gives the form of the surface 
of the spheroid at any time. 
Substituting, therefore, in (11) for -, ~ W+ j-\ we get 
1 + 
gwa 
da-, 
gwa 
(T, = 
i(2i + l) 
2(-i +1) 3 +1 n ]dt ' 2(f+l) 3 + l Tit ‘ 2(i —l)[2(i +1) 3 + 1] n 
ioct i+x [ 1 
This equation admits of solution just in the same way that equation (11) was solved ; 
but I shall confine myself to the case of the tidal problem, where i— 2 and 
S 2 =S cos ( vt-\-rj ). In this special case the equation becomes 
2 gwa\da 2giva 5 wed 
19?i Jdt 19/it°” 19 n 
- cos (vt-{-7]) — v sin ( vt-\-rj) 
S. 
19 n 1 'z,q 
And if we put —' +1 tan xp=vt, and ft=V 
1 Snnnn f£ H 
2 gwa 
This may be written 
% 
5a 
da , k vak ^ 
— H— cr =—:-rS COS (vt + rj + xp). 
dt t g sin -xjr v ‘ r/ 
In the solution appropriate to the tidal problem, we may omit the exponential term, 
vt 
and assume cr= A cos (ttf + B). Then if we put tany=y 
k 
da , k Av . , . \ 
— + -cr=-— cos m-4-B-f-v). 
dt n t sm y v n 1 
Wheiw ’ u follows that B=t?+i|/—- y, and 
A = a h smx = a C ° SX 
2 sin yp 2 cos y l r ’ 
so that 
«S cos 
or- 
2 cos yp 
~ COS (vt-{-r)-\-xp — y). 
Hence the bodily tide of the elastico-viscous spheroid is equal to the equilibrium tide 
COS V 
of a fluid spheroid multiplied by — and high tide is retarded by y — 
The formula for tan y may be expressed in a somewhat more convenient form ; we 
have tan xp=v\ and therefore tan y=tan xp- 
l^nvt 
2gioa 
