6 REPORT—1846. 
If the surface of the fluid be cut by a vertical plane perpendicular to the 
ridges of the waves, the section of the surface will be the curve of sines. 
Each particle of the fluid moves round and round in an ellipse, whose major 
axis is horizontal. The particle is in its highest position when the crest of 
the wave is passing over it, and is then moving in the direction of propaga- 
tion of the wave; it is in its lowest position when the hollow of the wave is 
passing over it, and is then moving in a direction contrary to the direction 
of propagation. At the bottom of the fluid the ellipse is reduced to a right 
line, along which the particle oscillates. When the length of waves is very 
small compared with the depth of the fluid, the motion at the bottom is in- 
sensible, and all the expressions will be sensibly the same as if the depth were 
infinite. On this supposition the expression for ¢ reduces itself to ie 
T 
The ellipses in which the particles move are replaced by circles, and the 
motion in each circle is uniform. The motion decreases with extreme rapid- 
ity as the point considered is further removed from the surface; in fact, 
the coefficients of the horizontal and vertical velocity contain as a factor the 
exponential e~”Y, where y is the depth of the particle considered below the 
surface. When the depth of the fluid is finite, the Jaw of the horizontal and 
vertical displacements of the particles is the same as when the depth is infi- 
nite. When the length of the waves is very great compared with the depth 
of the fluid, the horizontal motion of different particles in the same vertical 
line is sensibly the same. The expression for ¢ reduces itself to ./gh, the 
same as would have been obtained directly from the theory of long waves. 
The whole theory is given very fully in the treatise of Mr. Airy*. The 
nature of the motion of the individual particles, as deduced from a rigorous 
theory, was taken notice of, I believe for the first time, by Mr. Green+, who 
has considered the case in which the depth is infinite. 
The oscillatory waves just considered are those which are propagated uni- 
formly in fluid of which the depth is everywhere the same. When this con- 
dition is not satisfied, as for instance when the waves are propagated in a 
canal whose section is not rectangular, it is desirable to know how the velo- 
city of propagation and the form of the waves are modified by this cireum- 
stance. There is one such case in which a solution has been obtained. In 
a paper read before the Royal Society of Edinburgh in January 1841, Prof. 
Kelland has arrived at a solution of the problem in the case of a triangular 
canal whose sides are inclined at an angle of 45° to the vertical, or of a canal 
with one side vertical and one side inclined at an angle of 45°, in which the 
motion will of course be the same as in one half of the complete canalj. The 
velocity of propagation is given by the formula(B.), which applies to a rectan- 
gular canal, or to waves propagated without lateral limitation, provided we 
take for h half the greatest depth in the triangular canal, and for A, or *, a 
quantity less than the length of the waves in the triangular canal in the ratio 
of 1 to “2. As to the form of the waves, a section of the surface made by 
a vertical plane parallel to the edges of the canal is the curve of sines; a 
section made by a vertical plane perpendicular to the former is the common 
catenary, with its vertex in the plane of the middle of the canal (supposed 
complete), and its concavity turned upwards or downwards according as the 
section is taken where the fluid is elevated or where it is depressed. Thus 
* Tides and Waves, art. 160, &c. 
+ Transactions of the Cambridge Philosophical Society, vol. vii. p. 95. 
+ Transactions of the Royal Society of Edinburgh, vol. xv. p. 121. 
