WITH THE TIDES OF A VISCOUS SPHEROID. 
575 
Then since tan V \ , therefore ~~-=- cot e, and (58) becomes 
zgaw 
l9u a 
V ,(a . , 79 S 3 ~ , .1 
p—a- cot e( b cos vt — 5 - + —cos e b cos (vt — e) ]. 
But the radial surface velocity is equal to y f , and therefore so that 
dt 
cot e.cr=a 7 cot e^S cos - cos eS cos (vt — e)) . . . (59) 
Then if we divide cr into two parts, cr', a", to satisfy the two terms on the right 
respectively, we have 
a' S , 
= cos e • - cos (vt—e), 
a 9 v y 
which is the first approximation over again, and 
Therefore 
<r" S 79 ^ 
-= cos e- - -zrzrz — cos e cos (vt — 2e). 
a Q 150 q v ' 
a S 1 , N , 79 ^ t o \ 
-= cose--*! cos (vt— e) + —cos e cos (vt — 2e) 
a a [ v l°0 S v 
. . ( 60 ) 
This gives the second approximation to the form of the tidal spheroid. We see that 
the inertia generates a second small tide which lags twice as much as the primary one. 
Although this expression is more nearly correct than subsequent ones, it will be well 
to group both these tides together and to obtain a single expression for cr. 
Let 
tan y= 
-r4%— sin e cos e 
lo°,j 
Then 
i + rVV^r cos 2 e 
cr S C OS 6 n a V o \ it \ 
=-I + i 5 o _ cos e cos \vt — e— y) 
ci g cos y\ 15 0 g 1 v 
• • ( 61 ) 
This shows that the tide lags by (e-fy), and is in height ^ 6 (1 -j-i~ 5 %~ cos ' 2 e 'j of the 
equilibrium tide of a perfectly fluid spheroid. 
By the method employed it is postulated that - is a small fraction, because the 
9 
MDCCCLXXIX. 
