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Now if y = + 3,

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

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

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If moreover, x – {, , = 0 be the equation of the upper surface, - d: da da dt

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

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provided as above we neglect (disturbance)'. Vol. VI. PART III. 3 N

Again, the condition (2) gives by equating separately the coefficients of powers and products of y and x,

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If now by means of (a), (b'), (c) we eliminate p" p, from (2), there results

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For this we shall suppose 3 and y functions of a which vary very slowly, so that if written in their proper form we should have

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Hence if we allow ourselves to omit quantities of the order wo, and assume, to satisfy (4),

q}, + Af(t + X), where A is a function of a of the same kind as 3 and y, we have, omitting (#) 3.

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dA d.V.
i. i.i./.

d’op, d X\* 2, do Y
da:* = 4 (#) f + 4 +...f4 2

Substituting these in (4), and still neglecting quantities of the order wo,

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Hence if we neglect the superfluous constant kvg, the general integral of (4) is, ("." A = 3-3 y-1),

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and the actual velocity of the fluid particles in the direction of the axis of ar, is

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neglecting quantities which are of the order (w) compared with those retained.

If the initial values of K and u are given, we may then determine j" and F', and we thus see that a single wave, like a pulse of sound, divides into two, propagated in opposite directions. Considering, therefore, only that which proceeds in the direction of a positive, we have

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Suppose now the value of F(a) = 0, except from a = a to a = a + a, and 8a to be the corresponding length of the wave, we have

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Lastly, for any particular phase of the wave, we have

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(8). do = V gy, is the velocity with which the wave, or more strictly speaking the particular phase in question progresses. From (5), (6), (7), and (8) we see that if 3 represent the variable breadth of the canal and y its depth, K = height of the wave oc 3-4 y-4, u = actual velocity of the fluid particles or 6-3 y-3,

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XXIV. On the Theory of the Equilibrium of Bodies in Contact. By the Rev. H. Moseley, M.A., of St. John's College, Professor of Natural Philosophy and Astronomy in King's College, London.

[Read May 15, 1837.]

IN a paper on the Theory of the Arch, read before the Cambridge Philosophical Society in October 1833, and published in the fifth Volume of their Transactions, I have discussed the conditions of the equilibrium of a system of bodies in contact, on a principle referring it to the direction in respect to the surfaces of contact of a certain line, given in terms of the magnitudes and directions of the forces which compose the equilibrium. The condition that no one portion of the system shall turn on the edge of its surface of contact with another, being determined by the condition that the point at which the line leaves the surface of any one of the contiguous bodies, to enter the adjacent body, shall be within the boundary of the common surface of contact of the two; and the condition that no two contiguous bodies shall slip upon one another, by the condition that the direction in which this line intersects their common surface shall lie within a certain angle, which I have called the “limiting angle of resistance,” and which is dependent for its magnitude on the circumstances of the friction of the two surfaces upon one another.

If a surface be imagined to intersect the system, and continually to change its position, and, if necessary, its form so as to coincide, in order, with all the surfaces of contact, and if, in each position, the resultant be taken, in respect to those forces which are impressed upon one of the

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