ANALYSIS TO THE DYNAMICAL THEORY OF THE TIDES. 
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U sin X “ ^ cos x> 
__r . 1 df 
'Iwcifjb ^ \/(l — fj?) dcf) 
+ COS X v/(l - 
1 Bt/t 
2 ( 0 //, ds 
Hence, if dxjf/ds = 0, there will be no flow across the element d.s. It follows that 
the function xjj may be regarded as a stream-function, the paths of the particles of 
water always coinciding with the lines 
xjj = const. 
But from the equation of continuity we have, on putting = 0, and replacing 
U, Y by their values in terms of xjj, 
_ d j k Bt/t I d f k Bi|r I 
d/x \ /j, d(f) \ dcf) \ /x dfx } ’ 
or 
B / \ d)fr B / \ Bi/r 
B/i \ /X j 3</> d(f>\ /X / d/x ’ 
the general solution of which is 
^=f(hliO, . .( 37 ), 
where f {hj/x) denotes an arbitrary function of hj/x. 
It follows that the stream-lines i// = const coincide in direction with the lines 
hj/x = const 
(38). 
Thus, if the depth be a function of the latitude alone, the stream-lines must necessarily 
coincide with the parallels of latitude, and the only forms of steady motion possible 
are those in which the water has no latitudinal velocity. In the more general case, 
the stream-lines of the possible steady motions are given by the equation (38), and 
from this equation they might at once be traced out on a chart if we had a sufficient 
knowledge of the depth of the ocean in different parts. In particular, whatever be 
the law of depth, the equator will be one of the free stream-lines corresponding to an 
infinite value of hj/x, while the shores will also be stream-lines corresponding to zero 
values of this expression. An infinite number of stream-lines will converge towards 
those points where the coast-line intersects the equator, and it is only by passing 
through one of these points that a particle of water could pass from the northern to 
the southern hemisphere, or vice versd. As however the velocities at these particular 
points tend to become infinite, the equations which we have used which involve the 
neglect of the squares of the velocities will not be applicable to the region immediately 
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