ANALYSIS TO THE DYNAMICAL THEORY OE THE TIDES. 
217 
in comparison with 
c (Li + 
But we have seen that dU/dn, 0V/3n' are of the order U/a, while 
d fc^ {u~ + jj?) 
'dll' (_ \/ (1 + v“) 
are of the order c'vJa. 
Hence 
h gE {CVC + E) n/(1 - E) U}, h i. 
C^(p^ + V 
l\/ (1 — \/ (1 + r"') 
are both of the order 
U 
r h . 
O-'Pn 
The aj^proximations will therefore be admissible, provided h/a is a small quantity, 
that is, provided that the depth of the ocean is small in comparison with the radius 
of the solid earth, a hypothesis as to the validity of which there can be no doubt. 
Beturning now to equations (14), and solving for U, Y we find 
U(X2 - 4w-cos’= 
V(X2 - Tco-^cos^^) == - 
A \/ (1 ~ A<-") ^ 1 -o) co.s 9 
a dfx, « y/ (1 — fj?) 
0\p 
2(0 cos 0 
y/( 1 — /u-'b 
a 
'/X d^Jr 
ft y/ (1 — /A-) dcf) 
But we have rigorously 
cos 0 = jX 
^ (i'o~ + 1) 
v/ (v + 
and therefore, with errors of the order of the ellipticity, we may replace cos 0 by p. 
Hence, finally, we obtain the values of U, V with errors of the order hja, e compared 
with their true values in the form 
i\ y / (1 — 1 2uj/x Syjr "^1 
ft(X^ — 4«"p^) 0p ft y/ (1 — p^) (Y — 4ft)-p" d(p 1 
|» • • (15). 
2(ofjb y/ (1 — P^) I 
ft (X“ — 4(w®p') 0p ft y/ 11 — p") (^" ~ 4w"/u,“) 0<^ J 
2 F 
MDCCCXCVII.—A. 
