one who succeeds to a share of an inheritance divided among several persons. The corpuscular forces, on which the mechanical properties of the aggregates of matter depend, have been in some measure considered, as far as they relate to solids, in the articles Bridge and Carpentry. There are however other modifications of these forces, which are principally exemplified in the Cohesion of Fluids, and which afford us a series of phenomena, highly interesting to the mathematician, on account of the difficulty of investigating their laws, and of considerable importance to the natural philosopher, from the variety of forms in which they present themselves to his observation.

Sect. I.—Fundamental Properties of the Cohesion of a Single Fluid.

The three states of elastic fluidity, liquidity, and solidity, in all of which the greater number of simple bodies are capable of being exhibited at different temperatures, are not uncommonly conceived to depend on the different actions of heat only, giving a repulsive force to the particles of gases, and simply detaching those of liquids from that cohesion with the neighbouring particles which is supposed to constitute solidity. But these ideas, however universal, may be easily shown to be totally erroneous; and it will readily be found, that the immediate effect of heat alone is by no means adequate to the explanation of either of the changes of form in question.

There can never be rest without an equilibrium of force; and if two particles of matter attract each other, and yet remain without motion, it must be because there exists also a repulsive force, equal, at the given distance, to the attractive force. If we imagined the atoms of matter to be impenetrable spheres, only resisting when their surfaces came into actual contact, it would follow, that the degree of repulsive force exerted at the same distance must be capable of infinite variation, so as to counterbalance every possible modification of the attractive force that could operate between the particles. In this there would be no mathematical absurdity, and it may sometimes even be convenient to admit the hypothesis as an approximation; but we know, from physical considerations, that the actual fact is otherwise. The particles of matter are by no means incompressible; the repulsion varies indeed very rapidly when they approach near to each other; but the distance of the particles and the density of the substance must inevitably vary, in some finite degree, from the effect of every force that tends to produce either compression or expansion.

In elastic fluids, the law of the repulsive force of the particles is perfectly ascertained; and it has been shown to vary very accurately in the inverse ratio of their mutual distances. It is natural to inquire whether this repulsive force, continued according to the same law, would be capable of affording the resistance exhibited by the same bodies in a liquid or solid form, and holding the cohesive force in equilibrium; but in order to answer this question, it would be necessary to determine the law of the variation of the cohesive force with the variation of the density. Now if this force extended to all particles within a given distance of each other, whatever the density might be, the number of particles similarly situated within the sphere of action being as the density, and each one of this number being attracted by an equal number, the whole cohesion urging any two particles to approach each other would obviously, as Laplace has observed, be as the square of the density; but since this cohesive force would increase with the increase of density accompanying compression, more rapidly than any repulsive force like that of elastic fluids, there could never be an equilibrium between forces thus constituted: for, as Newton has justly remarked, the force of repulsion must be supposed to affect the particles immediately contiguous to each other only, their number not increasing with the density. Nor is there any reason to infer, from the phenomena of cohesion, that this force extends to a given minute distance, rather than to a given number of particles, as that of repulsion appears to do. It would indeed be possible to assign a law for the variation of cohesion, which would reduce the repulsion of liquids and of elastic fluids to the action of the same force, without any other modification than that which depends on the mutual distance of the particles; but this law is in itself so improbable that it cannot be considered as affording an admissible explanation of the phenomena; for it would be required that the force of cohesion should diminish, instead of increasing, with every increase of density, and with a rapidity nineteen times as great as the repulsion increased. For the height of the modulus of elasticity of all kinds of gaseous substances remaining unaltered by pressure, that of steam would still be only one twentieth as high as the modulus of elasticity of water, even if the steam were compressed by 1200 atmospheres; and the resistance to any minute change of dimensions would be twenty times as great in water as in steam of equal density, and the variation of the repulsion would be in the same proportion. It is therefore simplest to suppose the repulsion itself to be also twenty times as great, and the cohesion little or not at all altered by the effect of a slight compression or extension; and we shall have no difficulty in imagining this abrupt change in the magnitude of the repulsive force to depend on an increase of the number of particles to which it extends; supposing that when cohesion begins to affect them, this number becomes four or five times as great as before, and that it is not further increased by a greater increase of density; although, like the distance to which the force of cohesion itself extends, it may be liable to some modification from the effects of a change of temperature. Thus it is probable that the number of particles co-operating, both in repulsion and in cohesion, is diminished by the effect of heat; for the diminution of the elasticity of a spring is much more than proportional to the expansion of its substance, although the primitive repulsive force of the single particles may very possibly be as much augmented by an elevation of temperature in this case as in that of an elastic fluid; the cohesive powers of liquids are also diminished by heat, and indeed in a considerably greater degree than the stiffness of springs, although there can be no doubt that there is a considerable analogy in these changes. However this may be, it appears that the force of cohesion cannot be supposed to vary much with the density, and it is therefore allowable to consider it as constant, at all distances, as far as its action extends; while that of repulsion, though it may operate in some degree at distances somewhat greater, may still be considered, on account of its greater intensity at smaller distances, as equivalent to a resistance terminating at a more minute interval than that to which the action of cohesion extends.

The distance at which cohesion commences between the particles of gaseous fluids appears to depend entirely on the temperature, and for any one fluid it is generally reduced to one half by an elevation of about 100° of Fah- In whatever way the particles are caused to approach nearer than this distance to each other, they become subject to the action of this force, and rush together with violence, and with a great extrication of heat, until the increased repulsion affords a sufficient resistance to the cohesion, and the gas is converted into a liquid. Superficial observers have sometimes imagined that liquids possessed little or no cohesion; and it has generally been supposed that their cohesive powers are far inferior to those of solids. But that all liquids are more or less cohesive, is sufficiently shown by their remaining attached, in small portions, to every substance capable of coming into intimate contact with them, in opposition to the effect of gravitation, or of any other force; and the cohesion of mercury is still more fully exemplified by the well-known experiment of a column, standing at a height much exceeding that of the barometer, when it has been brought, by strong agitation or otherwise, into perfect contact with the summit of the tube, and is then raised into a vertical position; the summit of the tube supporting, or rather suspending, the upper parts, and each stratum the stratum immediately below it, with a force determined by the excess of its height above that of the column equivalent to the atmospherical pressure. The perfect equality of the cohesion of a given substance in the states of solidity and liquidity appears, however, only to have been asserted in very modern times; and the assertion has only been confirmed by a single observation of the sound produced by a piece of ice, compared with the elasticity exhibited in Canton's experiments on the compressibility of water; the results demonstrating that the resistance is either accurately or very nearly equal in both cases.

The real criterion of solidity is the lateral adhesion, which prevents that change of internal arrangement, by which a fluid can alter its external dimensions without any sensible difference in the mutual distances of its particles taken collectively, and consequently without any sensible resistance from the force of cohesion. It is probable that this lateral adhesion depends upon some symmetrical arrangement of the constituent parts of the substance, while fluidity requires a total independence of these particles, and an irregularity of situation, affording a facility of sliding over each other with little or no friction. The symmetry of arrangement, when continued uniformly to a sensible extent, is readily discoverable by the appearance of crystallization; but there are several reasons for supposing it to exist, though with perpetual interruptions, in more uniform masses, or in amorphous solids. It is obvious that the lateral adhesion, confining the particles so as to prevent their sliding away, performs an office like that of the tube of a barometer to which the mercury adheres, or like that of the vessels employed by Canton and Zimmerman for confining water which is compressed; and enables the cohesive and repulsive powers of the substances to be exhibited in their full extent. Nor can we obtain any direct estimate of these powers, from the slight cohesion exhibited, in some circumstances, by liquids in contact with the surface of a solid which is gradually raised, and carries with it a certain portion of the liquid; an experiment which had been often made, with a view of determining the mutual attractions of solids and fluids, but which was first correctly explained, as Laplace observes, by our countryman Dr Thomas Young, from its analogy with the phenomena of capillary tubes.

There are, however, still some difficulties in deducing these phenomena from the elementary actions of the forces concerned, whatever suppositions we may make respecting their primitive nature. The intermediate general principle of a hydrostatic force or pressure, proportional to the curvature of the surface, had been employed long ago by Segner, and had been considered by him as the Cohesion result of corpuscular powers extending to an insensible distance only. But Segner's reasoning on this point is by no means conclusive, and he has very unaccountably committed a great error, in neglecting the consideration of the effects of a double curvature. There is also an oversight in some of the steps of the demonstration attempted by Dr Young in his Lectures, which has been pointed out by an anonymous writer in Nicholson's Journal; and Mr Laplace's final equation for determining the angle of contact of a solid and a liquid, which Dr Young had first shown to be constant, has been considered as completely inaccurate, and as involving an impossibility so manifest as to destroy all confidence in the theory from which it was deduced. A demonstration which appears to be less exceptionable was lately published in the Philosophical Magazine; and it may serve, with some further illustrations, for the present purpose.

It is only necessary to consider the actions of such of the particles of the liquid as are situated at a distance from the surface shorter than that to which the cohesive force extends; for all those which are more internal must be urged equally in all directions by the actions of the surrounding particles. Now it will readily be perceived, that the first or outermost stratum of particles will cohere very weakly with the stratum below it, having only its own attraction to bind it down; that the second will be urged by a force nearly twice as great; and that the cohesion will gradually augment by increments continually diminishing, until we arrive at the depth of the whole interval to which the force extends; and below this it will remain constant, the number of particles within the given distance not undergoing any further change. It has been observed by Mr Laplace, that this partial diminution of the density of the surface is likely to be concerned in facilitating the process of evaporation; and it has been cursorily suggested in another quarter, that the polarisation of light by oblique reflection may be in some measure influenced by thisgradation of density. But its more immediate effect must be to produce that uniform tension of the surface which constitutes so important a principle in the phenomena of capillary action; for since the cohesion in the direction of the surface is the undiminished result of the attractions of the whole number of particles constituting the stratum, acting as they would do in any other part of the substance, it follows that a small cubical portion of the liquid, situated in any part of the space which we are considering, will be pressed laterally by the whole force of cohesion, but above and below by that part only which is derived from the action of the strata above it, so that this minute portion must necessarily tend to extend itself upwards and downwards, and to thicken the superficial film, and at the same time to become thinner in the direction of the surface, and to shorten it in all its dimensions, unless this alteration be prevented by some equivalent tension acting in a contrary direction; and this tension must be always the same in the same liquid, whatever its form may be, the thickness of the whole stratum being always extremely minute in comparison with any sensible radius of curvature.

Upon these grounds we may proceed to determine the actual magnitude of the contractile force derived from a given cohesion extending to a given distance. Supposing the corpuscular attraction equable throughout the whole sphere of its action, the aggregate cohesion of the successive parts of the stratum will be represented by the ordinates of a parabolic curve; for at any distance \( x \) from the surface, the whole interval being \( a \), the fluxion of the force will be as \( dx (a - x) \), since a number of particles proportional to \( dx \) will be drawn downwards by a number proportional to \( a \), and upwards by a number proportional to \( x \), and