THEORY Of 1 TIDES). 
225 
Severn, it assumes almost the appearance of a steep bank, 
(SeeNich. Journ. vol. XVIII, 1807, p. 118.) 
Scholium 1. Asa continuation of the conclusions which have Different 
been deduced from the analogy of the oscillations of the fluids Nation* ^ 
with the vibrations of the pendulums, we may obtain similar 
results from the immediate consideration of the progress of the 
tide in an open ocean following the luminary like a widely ex- 
tended wave. If the actual height of the tide be called q , its 
virtual height, with respect to the form which would afford a 
momentary equilibrium, will be q + h, and the propelling force 
will be proportional to this height : now the natural velocity 
of the wave, m </ (Jd),is generated by its exposure to a force 
proportional to its simple height for a time proportional to that 
which it occupies in passing over its own breadth h, that is for 
the time 
and if it be exposed to a force greater or 
le s in the ratio of 0 + k to 0 , for the time -• , where n is the 
n 
velocity of rotation, its velocity will become (\d) 
q n 
m ^(\d) 
.... r,™ 
nq 
m?. \d * AtA , which must be equal tow, and 
n^q + and putting ip ~ rn^.-r, tq = d (q+k) and 
q — AAA , and d — precisely as already demonstrated j 
r—d q+h 
r being the depth, affording a velocity equal to the velocity of 
rotation. And supposing, in the next place, such a wave to 
be liable to a resistance proportional to the velocity, we may 
illustrate the effect of the resistance by comparing it with that of 
another wave, which we may call its representative, combined 
with the original wave, and altering the inclination of its surface 
to the horizon. Thus, if the ordinates of the curve A (Pl.V. fig. 1) 
represent the elevation and the horizontal velocity of the tide, 
those of the curve B will exhibit the force derived immediately 
from the resistance, which will be the same as would be pro- 
duced by the combination of the wave G with the original wave, 
or as if the momentary position of the virtual horizon were 
altered from its natural state to the form V, which is the re- 
verse 
