584 
MR. G. H. DARWIN ON PROBLEMS CONNECTED 
For the sake of comparison with the approximate solution for the tides of a viscous 
spheroid, a precisely parallel process will now be carried out with regard to the fluid 
spheroid. 
We obtain a first approximation for , when inertia is neglected, by omitting v~ in 
CLL 
the denominator of (77); whence 
d* d ( 2i +1 g a > 
— ' -v z cos vtr%li ). 
dt dx\2i(i — l) a 
Substituting this approximate value in the equations of motion (73) we have 
dp . d , 2-i+l q , 
■ +— W+w . r " T \ ~ v " cos vt r*Qi 
ax dx\ 2i(i — l) a 
and two similar equations 
= 0 
• (80) 
From these equations it is obvious that the second approximation to the form of the 
tidal spheroid is found by augmenting the equilibrium tide due to the tide-generating 
potential r'Q, cos vt in the proportion 1 -f -v~ to unity. 
JL ) CO 
When i— 2 the augmenting factor is 1+^—. 
- g 
This is of course only an approximate result; the accurate value of the factor is 
l-r( 1—-g—j, and we see that the two agree if the squares and higher powers of \— 
are negdigeable. 
o o 
Now in the case of the viscous tides we found the augmenting factor to be 
l + i^oT cos 3 e. When e=0, which corresponds to the case of fluidity, the expressions 
9 
are closely alike, but we should expect that the 79 ought really to be 75. 
The explanation which lies at the bottom of this curious discrepancy will be most 
easily obtained by considering the special case of a lunar semi-diurnal tide. 
We found in Part II., equation (21), the following values for a, (3, y, 
a — W g in 2e[(8cr— br")ij+lxhy] 
Oo v 
q sin 2 e [( 8 a 2 — 5 r 3 ) x + 4.r f) 
O O 
y=d sin 2e • K'/- 
(81) 
