It may be shewn in the same way, in order that the wheels may not clog at the point B before the driving point at A comes into action, that

and as a, b, a, b, must be positive quantities, both these conditions will be fulfilled, if

which may be called the clearing equation; if the value of k, from this be substituted in the equation

for determining the radii of a friction wheel of 2n teeth; and by giving different values to n, sets of friction wheels will be found which will not clog theoretically just before or after the change of the driving teeth and such wheels will not clog at other points, unless the depth of the teeth be very great in proportion to the radii of the wheels, or the curves used for the construction of the teeth be of complicated forms.

An example is given in figure (24), where k = 0, and the clearing equation (7) becomes 2k2= 12, and equation (8) for determining the radii is therefore

Hence, for a wheel of eight teeth, which is derived from a curve of four lobes,

for a wheel of twelve teeth to turn with the former, and the teeth (or half-lobes) may be described from rules before given.

The flat sides of the teeth must be a little hollowed out to allow of the free motion of the points, but these have no connection with the rolling sides.

IV. Note on the Motion of Waves in Canals. By G. GREEN, Esq. B.A.

of Caius College.

[Read February 18, 1839.]

IN a former communication I have endeavoured to apply the ordinary Theory of Fluid Motion to determine the law of the propagation of waves in a rectangular canal, supposing the depression of the actual surface of the fluid below that of equilibrium very small compared with its depth; the depth y as well as the breadth ẞ of the canal being small compared with the length of a wave. For greater generality, ẞ and y are supposed to vary very slowly as the horizontal co-ordinate x increases, compared with the rate of the variation of, due to the same cause. These suppositions are not always satisfied in the propagation of the tidal wave, but in many other cases of propagation of what Mr Russel denominates the "Great Primary Wave," they are so, and his results will be found to agree very closely with our theoretical deductions. In fact, in my paper on the Motion of Waves, it has been shown that the height of a wave varies as

With regard to the effect of the breadth ẞ, this is expressly stated by Mr Russel (Vide Seventh Report of the British Association, p. 425), and the results given in the tables, p. 494, of the same work, seem to agree with our formula as well as could be expected, considering the object of the experiments there detailed.

In order to examine more particularly the way in which the Primary Wave is propagated, let us resume the formulæ,

where we have neglected the function f, which relates to the wave propagated in the direction of a negative.

Suppose, for greater simplicity, that ẞ and y are constant, the origin of a being taken at the point where the wave commences when t = 0. Then we may, without altering in the slightest degree the nature of our formulæ, take the values,

But for all small oscillations of a fluid, if (a, b, c) are the coordinates of any particle P in its primitive state, that of equilibrium suppose; (x, y, z) the co-ordinates of P at the end of the time t, and Φ = født when (x, y, z) are changed into (a, b, c), we have (Vide Mécanique Analytique, Tome II. p. 313.)

Applying these general expressions to the formulæ (1) we get

and consequently,

a

(a-t√gy).da = - √1⁄2 F(a-t√57).

✓ g

supposing for greater simplicity that the origin of the integral is at

a = 0.

Hence the value of x becomes

Suppose

α=

length of the wave when t = 0; then (a) = 0, except when a is between the limits 0 and a. If therefore we consider a point P before the wave has reached it,

the whole volume of the fluid which would be required to fill the hollow caused by the depression

when t = 0. Hence we get

below the surface of equilibrium

x' being the horizontal co-ordinate of P, before the wave reaches P.

Also, let x" be the value of this co-ordinate after the wave has passed completely over P, then

If

were wholly negative, or the wave were elevated above the surface of equilibrium, we should only have to write V for V, and thus

-