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lected in comparison of the others. Then finding the numerical values of A and B by comparing the two expressions for e, we shall obtain,

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This result accords with the preceding theoretical considerations in

de giving a positive value to It also enables us to estimate to what dx amount the variation of the atmospheric temperature with the height above the Earth's surface is affected by causes distinct from that of variation of density. It appears, that for small altitudes the term in

de equation (5) involving is about one-fourth the other term. The dx formula of Atkinson from which these inferences are made, is strictly applicable only to the lower parts of the atmosphere where the grand aerial currents prevail, beyond which the law of the decrement of temperature probably undergoes some variation.

I have thus endeavoured to advance in the theoretical part of this problem, as far as the present state of our knowledge appears to admit, and to give as much exactness as possible to the mathematical reasoning. With respect to the latter, the course pursued in this paper may lay some claims to originality, but the fundamental principles regarding the atmosphere are not essentially different from those advanced by Dalton and Ivory in their writings on this subject.

XXIII. On the Motion of Waves in a variable Canal of small Depth and Width. BY GEORGE GREEN, ESQ. B.A. of Caius College.

[Read May 15, 1837.]

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 x, are,

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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, ≈ and t, that the velocities of the fluid particles parallel to the three axes are

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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.)],

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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 x, are so small that may be expanded in a rapidly convergent series in powers of y and ≈, so that

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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, x,) p will evidently contain none but even powers of y, and we shall have

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

y - ẞ, = 0,
Bx

since the other will then be satisfied by the exclusion of the odd powers of y from p.

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and (c) becomes, since is of the order of the disturbance,

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