Astronomical and Nautical Collections. 301 
supposing their depth and extentknown,” and their form sufficiently 
simple, ‘‘ may easily be deduced from the important theorem of 
Lagrange, by which the velocity of a wave of any kind,” when suffi- 
ciently broad, “is shown to be equal to the velocity of aheavy body, 
which has fallen through half the height of the fluid concerned : 
but in the case of a tide extending to any considerable portion of 
the surface of the globe, this velocity must be somewhat modified 
according to the comparative density of the central and the super- 
ficial parts. 
‘“* The most remarkable consequence of this analogy is the law, 
that if the simple oscillations, of which the moving body is suscep- 
tible, be more frequent than the period of the recurring force, the 
pendulum will follow its point of suspension with a direct motion; . 
but if the spontaneous vibrations be the slower, the motions will be 
inverted with respect to each other: and, with regard to the tides, 
we may infer from this mode of calculation, that supposing the 
earth to be between five and six times as dense as the sea, the 
oscillations of an open ocean can only be direct, if its depth in the 
neighbourhood of the equator be greater than fifteen or sixteen 
miles: and that if the depth be smaller than this, the tides must 
be inverted, the time of low water corresponding, in this case, to 
the transit of the luminary over the meridian. 
“‘ This distinction has not been explicitly made by Mr. Laplace, 
although he has calculated, that for a certain depth, of a few miles 
only, the tides of the open ocean must be inverted, and that for 
greater depths they will be direct: but the intricacy of his formule 
seems to render their use laborious, and perhaps liable to some in- 
accuracy; and in the application of his theory, he seems to have 
lost sight even of the possibility of an inverted tide. In narrower 
seas, which Mr, Laplace has not considered, a smaller depth will 
constitute the limit between these two species of tides; and in 
either case the approach of the depth to this limit will be favour- 
able to the magnitude of the tide.” It may also be remarked, that 
if the depth of the sea became gradually smaller in receding from 
the equator, till it vanished at the poles, its surface, as well as that 
of the earth, having the form of an oblate spheroid, the time re- 
Vou, XVII. ¥ 
