since in the formula (8), r ought to be made equal to zero at the end of the process.

By conceiving the auxiliary variable u to vanish, it will become clear from what has been advanced in the preceding number, that the values of the function V within circnlar planes and spheres are only particular cases of the more general one (49), which answer to s=2 and 8=3 respectively. We have thus by combining the expressions (48) and (49), the means of determining V when the density p' is given, and vice versa; and the present method of resolving these problems seems more simple if possible than that contained in the articles (4) and (5) of my former paper.

ON THE MOTION OF WAVES

IN A VARIABLE CANAL OF SMALL DEPTH

AND WIDTH*.

* From the Transactions of the Cambridge Philosophical Society, 1838. [Read May 15, 1837.]

ON THE MOTION OF WAVES IN A VARIABLE CANAL OF SMALL DEPTH AND WIDTH.

THE equations and conditions necessary for determining the motions of fluids in every case in which it is possible to subject them to Analysis, have been long known, and will be found in the First Edition of the Mec. Anal. of Lagrange. Yet the difficulty of integrating them is such, that many of the most important questions relative to this subject seem quite beyond the present powers of Analysis. There is, however, one particular case which admits of a very simple solution. The case in question is that of an indefinitely extended canal of small breadth and depth, both of which may vary very slowly, but in other respects quite arbitrarily. This has been treated of in the following paper, and as the results obtained possess considerable simplicity, perhaps they may not be altogether unworthy the Society's notice.

The general equations of motion of a non-elastic fluid acted on by gravity (g) in the direction of the axis z, are,

supposing the disturbance so small that the squares and higher powers of the velocities &c. may be neglected. In the above formulæ p= pressure, p= density, and is such a function of x, y, z and t, that the velocities of the fluid particles parallel to the three axes are

To the equations (1) and (2) it is requisite to add the conditions relative to the exterior surfaces of the fluid, and if A=0 be the equation of one of these surfaces, the corresponding condition is [Lagrange, Mec. Anal. Tom. II. p. 303. (I.)],

The equations (1) and (2) with the condition (A) applied to each of the exterior surfaces of the fluid will suffice to determine in every case the small oscillations of a non-elastic fluid, or at least in those where

is an exact differential.

udx + vdy + wdz

In what follows, however, we shall confine ourselves to the consideration of the motion of a non-elastic fluid, when two of the dimensions, viz. those parallel to y and z, are so small that may be expanded in a rapidly convergent series in powers of y and a, so that

Then if we take the surface of the fluid in equilibrium as the plane of (x, y), and suppose the sides of the rectangular canal symmetrical with respect to the plane (x, z), 4 will evidently contain none but even powers of y, and we shall have

represent the equation of the two sides of the canal, we need only satisfy one of them as

y-B2 = 0,