T2 REPORT—1846. 
canal on the earth be supposed circular. In all cases the disturbing force 
will give rise to an oscillation in the water having the same period as the force 
itself. This oscillation is called by Mr. Airy’a forced wave. It will be suf- 
ficient here to mention some of the results of this theory as applied to the 
case of the earth. 
In all cases the expression for the tidal elevation contains as a denominator 
the difference of the squares of two velocities, one the velocity of propagation 
of a free wave along the canal, the other the velocity with which a particular 
phase of the disturbing force travels along the canal, or, which is the same, 
the velocity of propagation of the forced wave. Hence the height of the 
tides will not depend simply on the magnitude of the disturbing force, but 
also on its period. Thus the mass of the moon cannot be inferred directly 
from the comparison of spring and neap tides, since the heights of the solar 
and lunar tides are affected by the different motions of the sun and moon in 
right ascension, and consequently in hour-angle. When the canal under 
consideration is equatorial the diurnal tide vanishes. The height of high water 
is the same at all points of the canal, and there is either high or low water at 
the point of the canal nearest to the attracting body, according as the depth 
of the water is greater or less than that for which a free wave would be pro- 
pagated with the same velocity as the forced wave. In the general case there 
is both a diurnal and a semidiurnal tide, and the height of high water, as well 
as the interval between the transit of the attracting body over the meridian 
of the place considered and the time of high water, is different at different 
points of the canal. When the canal is a great circle passing through the 
poles, the tide-wave is a stationary wave. When the coefficient of the dis- 
turbing force is supposed to vary slowly, in consequence of the change in 
declination, &c. of the disturbing body, it is found that the greatest tide oc- 
curs on the day on which the disturbing force is the greatest. 
The preceding results have been obtained on the supposition of the absence 
of all friction; but Mr. Airy also takes friction into consideration. He sup- 
poses it to be represented by a horizontal force, acting uniformly from top to 
bottom of the water, and varying as the first power of the horizontal velocity. 
Of course this supposition is not exact: still there can be no doubt that 
it represents generally the effect of friction. When friction is taken into 
account, the denominator of the expressions for the tidal elevation is essen- 
tially positive, so that the motion can never become infinite. In the case of 
a uniform tidal river stopped by a barrier, the high water is no longer simul- 
taneous at all points, but the phase of high water always travels up the river. 
But of all the results obtained by considering friction, the most important 
appears to be, that when the slow variation of the disturbing force is taken 
into account, the greatest tide, instead of happening on the day when the 
disturbing force is greatest, will happen later by a certain time, p,. More- 
over, in calculating the tides, we must use, not the relative positions of the sun 
and moon for the instant for which the tide is calculated, but their relative 
positions for a time earlier by the same interval p, as in the preceding case. 
The expression for p, depends both on the depth of the canal and on the 
period of the tide, and therefore its value for the diurnal tide cannot be 
inferred from its value for the semidiurnal. It appears also that the phase of 
the tide is accelerated by friction. 
The mechanical theory of the tides of course belongs to hydrodynamics ; 
but I do not conceive that the consideration of the reduction and discussion 
of tidal observations falls within the province of this report. 
Before leaving the investigations of Mr. Airy, I would call attention to a 
method which he sometimes employs very happily in giving a general expla- 
* 
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