Again, the condition (2) gives by equating separately the coefficients of powers and products of y and x, If now by means of (a), (b'), (c') we eliminate p′′, from (2′), there results For this we shall suppose ẞ and y functions of x which vary very slowly, so that if written in their proper form we should have Hence if we allow ourselves to omit quantities of the order w2, and assume, to satisfy (4), where A is a function of x of the same kind as ẞ and y, we have, Substituting these in (4), and still neglecting quantities of the order w2, equating now separately the coefficients off" and f", we get Hence if we neglect the superfluous constant k√g, the general integral of (4) is, (·.· A= ẞ-1y ̄+), and the actual velocity of the fluid particles in the direction of the axis neglecting quantities which are of the order (w) compared with those retained. If the initial values of and u are given, we may then determine f 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 Suppose now the value of F(x) = 0, except from x = a to x = a + a, and dx to be the corresponding length of the wave, we have Lastly, for any particular phase of the wave, we have 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 ẞ represent the variable breadth of the canal and its depth, and = γ height of the wave - ẞ-y-, u = actual velocity of the fluid particles ∞ B-y-i, dx = length of the wave ∞ y, dx dt XXIV. On the Theory of the Equilibrium of Bodies in Contact. By the [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 two parts into which this surface divides the system; then the locus of the consecutive intersections of these resultants is that curved line to which I have assigned the properties of equilibrium described in the preceding page. I wish now to correct this definition. To the properties assigned to this line it is necessary that at each of the points where it intersects contiguous surfaces of the component masses, the whole pressure upon those surfaces should be supposed to be applied. Now, according to the definition given of it, this supposition is not, except under certain circumstances, admissible. The resultant of the pressures upon each surface of contact is necessarily at some point or other a tangent to the locus of the intersections of the resultants, but it may be, and except in particular cases, will be, a tangent to it at a point other than that in which this line intersects the surface of contact itself. The point where the resultant intersects the dividing surface to which it corresponds, is that element in the theory on which the condition, “that one portion of the system shall not turn over upon the boundary of its surface of contact with the adjacent portion," depends. I propose, therefore, in the following paper, to determine the line which is the locus of intersections of the consecutive resultants, with the corresponding imaginary surfaces of division, these surfaces being, here, supposed to be planes. This line I shall call the LINE OF RESISTANCE, including as it does the points of application of the resultants of all the resistances of the surfaces of contact. The direction in which the resultant intersects two common surfaces of contact, is that on which the condition, "that these surfaces shall not slip upon one another," depends; moreover this direction is a tangent to the line which is the locus of the intersections of the consecutive resultants, drawn from the point where the line of resistance cuts the surface of contact. The determination of this line is therefore also an important feature in the theory. I propose that it should retain the name before given to it of the LINE OF PRESSURE. |