ANALYSIS TO THE DYNAMICAL THEORY OF THE TIDES. 
213 
vibrations small. If we now suppose that the depth of the water is small, so that 
Yo — Yi is a small quantity, the above equation admits of considerable simplification. 
Using square brackets to denote values at the mean surface y = Jq, we have, by 
Taylor’s Theorem, 
whence 
U/Mo = [U/V^g] + (y - Yo) 
p(u//a,) A = [U/Aj!3](r, - r.) - i(x„ - y,)’[|;(u//a 3 ) 
+ ... 
Now, if Jq — y-^ be small as supposed above,even though ^(U/ZioAg) is finite, we 
may omit all the terms on the right except the first. This amounts to supposing 
that the horizontal velocity is sensibly uniform throughout the depth, not on account 
of the small value of its rate of variation, but on account of the small distance 
through which this variation can take effect, a supposition which is not inconsistent 
with the results of § 1. Hence, on neglecting small terms of the order (yg “ Yi)^’ 
we have 
f(U/Vi3)f?y = [U//a3](r„-y,), 
* Yi 
and, in like manner, 
f(V,V,3),/y = [Y//,3A,](y„-y,). 
• yi 
Let h denote the depth of the ocean at the point (a, /3). Then, provided h be 
small in comparison with the radii of curvature of normal sections of the surfaces 
fx — const, ^ = const, y = const, we may put 
7o ~ 7i 
[/^sl 
= /q 
with errors of the order of the square of the ratio of li to these radii of curvature ; 
and therefore 
['■(U/Wis) dy = h [U//l.], 
* Vi 
f"(V/V,3)c;y = A[V/A,]. 
* Yi 
Substituting these values in (10), we find 
a? 
0^ 
— — 
a /iu\ a_ nil 
0« \ lu j a/3 \ /q /_ 
.( 11 ). 
where we have now used bars to denote surface values. 
* Tlie standard of conqiaiHson is considered in the next section. 
