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sequently ensues, the warmer parts ascend and are continually being replaced by the descent of colder. The effect of this circulation is to make the gradation of mean temperature from the lower to the upper strata less rapid than it would otherwise be, and so far as this cause

7 - - - - d also operates, # will be positive. Hence we may assume # to be

some positive function of x, and as we have no means of determining

à priori the form of this function, we will assume that

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* ITA (* ~ ā)** 3GIE * 5(IIE) supposing 6 = 0 when x = 0.

The empirical formula which Atkinson gives in his Memoir, for expressing the relation between the altitude (h) in English feet above

any place on the Earth's surface, and the depression of temperature (n) in degrees of Fahrenheit at that elevation, is the following:

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The third term amounts to about 1" for an elevation of three miles, and may, within the heights to which observations can extend, be neg

lected in comparison of the others. Then finding the numerical values of A and B by comparing the two expressions for 6, we shall obtain,

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giving a positive value to #. It also enables us to estimate to what

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

equation (5) involving # is about one-fourth the other term. The

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 formulae p = pressure, p = density, and p is such a function of a, y, x 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 equa

tion of one of these surfaces, the corresponding condition is [Lagrange, Mec. Anal. Tom. II, p. 303. (I.)],

o – 44 d'A 124, 144, = d/ t do dy als “” Hence _ dA dA dip , dA dip , dA dip (A) 0 = 77 * da, da; + dy dy + dx d. (when A = 0).

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

uda-H valy + wal: is an exact differential.

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

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

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