1. When a solid body is partially plunged in a fluid, the level surface near it is disturbed, and the fluid is observed either to ascend or descend, so as to form a ring round the part immersed. If a tube of glass be inserted in a vessel containing water, the liquid will rise in a concave ring, both on the outside and the inside; and if the tube be small enough, the cylinder of water within it will be elevated above the general level, and the elevation will be greater nearly in the same proportion that the bore is less. On the other hand, if the tube be plunged in mercury, the fluid in contact with the glass will be depressed, forming a hollow ring with the convexity upward; and when the diameter is very small, the cylinder of mercury in the inside will sink below the level on the outside. In all these appearances the physical cause is the same, and it has received the name of Capillary Action, because its effects are most remarkable in the case of tubes with extremely minute diameters.

No part of natural philosophy has been the subject of a greater variety of researches than capillary action. It has been viewed in almost every possible light, and it would be difficult to suggest a new principle that has not been proposed by some philosopher in order to account for the observed appearances. One advantage has resulted from repeated discussion; for by this means the true cause of the phenomena is no longer doubtful, although there is still considerable difference of opinion with regard to the manner in which the effects are produced. It is now universally allowed, that the suspension of fluids in capillary tubes is to be ascribed to the attraction observed to take place between the elementary particles of which bodies are composed. We shall not stop to detail the different experiments which prove the reality of this attractive force, and we shall at once assume that the two following facts, which are the fundamental principles of this theory, are fully established; namely, that glass and other solid bodies attract the particles of fluids with which they are in contact, and that the particles of fluids attract one another. Admitting these two kinds of attraction, it remains to investigate the consequences that flow from them.

2. Corpuscular attraction acts with great intensity in law of contact, or at the nearest distances, but it decreases very rapidly as the distance increases, and, on the whole, is confined within a very small range. Clairaut supposed that the sides of a capillary tube extend their action to the central parts of the contained cylinder of fluid. But in this opinion he is singular. All other philosophers confine the sphere of attraction within much narrower limits. They suppose that the corpuscular force has produced its full effect, and has become evanescent, at a distance so small that it cannot be appreciated by the senses. But from this we are not to conclude that a particle attracts those only which are quite contiguous to it; its action, although confined within a sphere of a very small radius, nevertheless extends to some distance, and reaches to the particles beyond the nearest.

As corpuscular attraction extends its influence to a distance, it must vary, within the sphere of its action, according to some law, which is unknown, and in all probability will never be discovered. But a knowledge of this law is not necessary to explain the capillary phenomena; for these are caused by the accumulated action of the force in its whole range, and are independent of the intermediate variations of intensity which it may undergo. In this respect capillary action resembles the attraction by which transparent bodies refract the rays of light. In both cases, what we observe is the total effect of the attractive energy, which may remain the same, although the intermediate degrees of intensity be infinitely varied.

3. Conceive a fluid mass (Plate CXLV. fig. 1), the particles of which attract one another, but which is subjected to the action of no other forces, not even to that of gravity; and let an imaginary surface be traced through the fluid, having at every part a depth equal to the utmost range of the corpuscular force. Then a particle placed within the imaginary surface may be considered as occupying the centre of a sphere of the fluid, described with a radius equal to the greatest distance to which attraction reaches; whence it is manifest that the particle will be urged with equal forces in all opposite directions. If the particle be placed between the boundaries of the superficial stratum or film, the sphere of which it is the centre will extend above the fluid's surface; and, on account of the defect of matter, the particle will be less attracted outward than inward. Let N be a particle so situated, and suppose that n is another particle as much elevated above the fluid's surface as N is immersed below it; and trace the surface PQ in the fluid as far below N as that particle itself is below the outer boundary of the fluid mass. Then the particle N will be in equilibrium with regard to the attraction of all the fluid above PQ; but it will be urged inward by the force with which it is attracted by the fluid below PQ; and as the particles at N and n are similarly situated with regard to the whole fluid mass, and the part of it below the surface PQ, it is manifest that the attraction of the whole mass upon the particle at n is equal to the force which urges the particle at N inward. From this it follows, that all particles placed in a stratum which is everywhere at the same depth below the fluid's surface, are drawn inward with the same force, equal to that with which the whole mass attracts a particle placed at an equal height above the fluid's surface.

If now we conceive a canal passing through the interior of the fluid, and terminating both ways in the surface, it follows, from what has been said, that the attraction of the whole mass upon the superficial drops placed at the two orifices will propagate equal pressures in opposite directions through the canal. In order to estimate the force of compression, we may denote by K the pressure inward, caused by the attraction of the whole fluid upon a square inch of the superficial film; then a portion of the fluid within the canal will be compressed by the equal forces, K, acting in opposite directions. This is true of all portions of the fluid within the superficial stratum; between the boundaries of that stratum the compressive force is less, being always of the same intensity at the same depth, but decreasing rapidly in approaching the surface, where it is evanescent.

We may now conceive a fluid mass, whatever be its figure, to consist of a central part, surrounded by an indefinite number of thin beds or strata, placed at equal depths below the surface; and it will follow, from what has been proved, that the compression is constant in all the central part; and likewise that it is uniformly of the same intensity throughout every superficial stratum, varying from one stratum to another, and decreasing very rapidly near the surface. Such a body of fluid will therefore be in equilibrium whatever be its figure; in other words, the corpuscular attraction will oppose no resistance to a change of figure in the fluid, nor obstruct, in any degree, the perfect mobility of the particles among one another.

It must be observed, however, that the conclusion just obtained is exact only when we confine our attention to the direct action of the attractive forces, as is done in the theory of the figure of the earth. But there is another effect caused by the direct attraction of the particles of a fluid; to be afterwards considered, which takes place only at the surface, and from which this consequence results, that a body of fluid subjected to no forces but the attraction of its own particles will no longer be indifferent to any figure, but will arrange itself in a perfect sphere.

A change in the temperature of a fluid mass will produce an alteration in the cohesive force; but it appears very difficult, if not impossible, to estimate, in any satisfactory manner, the effect arising from this cause.

A variation of temperature will affect the attraction of the particles of a fluid by the change of density which it induces. When two portions of a fluid attract one another, if we conceive one of them to have its density changed, while that of the other remains unaltered, it is evident that their cohesion will be proportional to the number of particles of the first portion placed within the sphere of action of the second; that is, it will vary in the direct proportion of the density. Again, if we now suppose the density of the second portion to vary, the attractive force will on this account also suffer a proportional change. Wherefore, when both portions undergo an equal change of temperature, their cohesion will vary as the square of the density.

Again, the variations in the mutual distances of the particles of a fluid, caused by changes of temperature, must bear a finite proportion to the range of the corpuscular force; and, on this account, a change in the fluid's cohesion will take place, depending upon the law that attraction follows in regard to the distance. At a given temperature, and under a given pressure, the particles are separated from one another to a certain distance, at which there is an equilibrium between the attractive force which impels them to one another, and the repulsive power attending the action of heat. In these circumstances, the actual cohesion is due to that part only of the whole corpuscular force which is exerted upon the particles placed beyond the limit of approach allowed by the given degree of temperature. The cohesion, too, is diminished not only by the decreased intensity of the attractive force, but also by the increased repulsion of heat. Our ignorance of the laws that regulate the action of these forces makes it impossible to subject to calculation the effect of a change of temperature; but, when we consider that corpuscular attraction decreases very rapidly as the distance increases, it is extremely probable that the cohesion of a fluid undergoes much greater changes from this cause than from the variations of density.

But capillary action arises from the cohesion between the particles of a fluid, and the attraction that takes place between them and the solid bodies with which they are brought into contact. Experiments show that these forces continue to act so long as the state of fluidity endures; their action is constant under the same temperature; and they are affected in degree only by the variations of heat. In the further prosecution of this inquiry we shall there- Capillary fore throw out of view the effect of temperature, and shall confine our attention to develope the consequences of corpuscular attraction.

4. The attraction of a solid body on every particle of a fluid within the sphere of its action, is a force perpendicular to the surface of the solid. This is manifest from the homogeneity of the solid when its surface is a plane; for, on account of the uniform arrangement of the parts, there is no reason why the attractive force should decline from the perpendicular to one side rather than to another. And when the solid is bounded by a curve of any kind, we may still consider the extremely small part of the surface which acts on a particle, as coinciding with the tangent plane; whence we may conclude that, in all cases, the attraction on every particle is perpendicular to the surface of the attracting body. The same thing is true in the action of transparent bodies on light; for, if the motion of a ray be decomposed into two parts, one parallel to the refracting surface, and the other perpendicular to it, the observed law of refraction implies that the velocity of the first part will remain unchanged, while the velocity of the other part will be increased or diminished by the refracting force.

If a smooth plate of glass be laid horizontally upon the surface of water, it is found that the glass will adhere to the water. The adhesion is not produced by the pressure of the atmosphere, for the fact is equally true in the vacuum of an air-pump. There is, therefore, evidently an attraction between the glass and the water, acting perpendicularly to the plate, and causing it to adhere to the water.

If the plate, instead of being laid horizontally upon the water, be immersed vertically in it, the part below the surface will exert the same attractions as it did in the former position. Every particle of the fluid within the sphere of action of the glass will be drawn perpendicularly towards it, and a thin coating of the fluid will attach itself to all the immersed surface of the plate.

5. Although the attractive force exerted by a solid body on a fluid is confined to insensible distances, it must still be considered as penetrating in some degree into the fluid mass. The thin film on which it acts retains possession of all the properties of a fluid. The particles of water in contact with the glass press upon its surface; and the particles farther off press upon those nearer; and the whole film is in a state of compression. But it is a distinguishing property of a fluid, arising from the perfect mobility of its particles, that a pressure in one direction will cause an equal pressure in all directions; and hence we must infer that the thin film of water, at the same time that it is compressed by the direct attraction of the glass, will likewise press laterally, or will make an effort to spread itself towards every side on the surface of the plate. If the film, instead of being attracted by the plate, were pressed against its surface by a weight, the lateral pressure, estimated on a given superficial space, would be the same with the direct pressure. But as the strata at different distances from the plate are attracted in unequal degrees, the whole lateral force can be found only by summing up the lateral pressures arising from the attraction upon each stratum.

Let AB (fig. 2) be a plate of glass on which there stands an upright vessel or tube, containing water; and let GH be a thin section or elementary part of the water within the tube, parallel to the glass, and so near it as to be attracted by it. Suppose that \( w \) denotes the area of the section, \( a \) its distance from the plate, and \( da \) its thickness; and let \( \Psi(a) \) represent the attraction of the whole matter of the plate upon a single particle of water placed at the distance \( a \). Then the density being constant and equal to unit, the attractive force of the plate upon the thin elementary section will be equal to

\[ \Psi(a) \times wda, \]

and hence the attraction of the plate upon all the water in the tube will be equal to the integral

\[ w \times \int \Psi(a) \cdot da, \]

generated while \( a \) increases from 0 to be infinitely great.

The expression \( \int \Psi(a) \cdot da \), which we may denote by \( K' \), is therefore the force with which the attraction of the glass causes the fluid to press upon a square inch of the plate, or it is the measure of that force. If the particles of the fluid were attracted by the matter of the plate with an intensity equal to their own cohesive force, then \( K' \) would be equal to \( K \), that is, it would be equal to the force with which an indefinite mass of the fluid causes the superficial stratum to press inward.

In the inside periphery of the tube, assume any determinate length \( ab \), equal to \( \lambda \), and let the lines \( ac, bd \), be drawn in the interior surface at right angles to \( ab \). The area of the space \( abde \) is equal to \( \lambda \times a \); and because fluids press equally in all directions, the attraction which urges the elementary section towards the plate \( AB \) will cause the fluid below the section to press upon the space \( abde \) with a force which is to the attractive force urging the section downward, as \( \lambda \times a \) to the area of the section. Hence the pressure on the space \( abde \), caused by the attraction of the glass on the elementary section \( GH \), is equal to

\[ \lambda \times \Psi(a) \cdot ada. \]

This expression would evidently denote the pressure upon the surface \( abde \), if the fluid below the section were impelled towards the plate by a piston exactly fitted to the orifice of the tube. But there is no difference between the action of such a piston and that of the thin elementary section when urged by attraction with equal force in the same direction. The total force acting laterally in the length \( \lambda \), is therefore equal to the fluent

\[ \lambda \times \int \Psi(a) \cdot ada, \]

generated while \( a \) increases from 0 to be infinitely great.

Hence, if we put \( H' = \int \Psi(a) \cdot ada \); then \( H' \) will be the measure of the lateral force in the length equal to unit.

It is obvious that the direct attraction between two portions of a fluid, as well as that between a solid and a fluid, is attended with a lateral pressure. If we denote by \( H \) what \( H' \) becomes when the matter of the plate attracts the fluid with the same intensity that the fluid attracts its own particles, then \( H \) will be the measure of the lateral force arising from the direct attraction of the fluid, and it will have the same relation to \( K \) that \( H' \) has to \( K' \).

The lateral force is always very small when compared with the direct pressure. For the function \( \Psi(a) \) has a conceivable value only when \( a \) is so small as to be imperceptible to the senses; in such circumstances the product \( \Psi(a) \times a \) is very small when compared with \( \Psi(a) \); and consequently, \( H' = \int \Psi(a) \cdot ada \) is considerable in respect of \( K' = \int \Psi(a) \cdot da \).

The smallness of the lateral in comparison of the direct pressure arises from this, that every elementary part of the latter is estimated on the same finite area, while the simultaneous element of the former is confined to a space incomparably less. These two pressures resemble the power in the hydrostatic paradox, and the effect which it produces. In both cases we have a small pressure applied to a surface extremely minute, in equilibrium with a great pressure distributed over a comparatively large area.

When a piece of glass is partially plunged in water in a vertical direction, the thin film which is attracted by the immersed surface endeavours to spread itself on the glass with an effort more or less in proportion to the compressive force. Below the surface of the water the lateral actions of the parts in contact mutually counteract one another; but at the surface the expansive force meets with no opposition. The film will therefore be pushed above the general level; and as it acts by cohesion on the contiguous fluid, it will draw up a portion of it, and form a ring surrounding the immersed part of the glass. The small fluid mass on which the glass exerts its attraction performs the office of a machine, which changes a horizontal force into one having a vertical direction. In the mechanical properties of a fluid we thus have a principle adequate to account for what we observe in capillary action. But although the general view here given of the cause of the capillary phenomena is so far satisfactory, a great deal of discussion is still necessary, in order to deduce from it a clear explanation of the laws observed in the appearances that take place in different circumstances.

The idea of accounting for capillary action by means of the lateral force produced by the direct attraction of a solid body upon a fluid, is due to Professor Leslie, a philosopher to whom physical science is indebted for more than one discovery. It is developed and applied, to explain some of the principal phenomena, in a short dissertation published in 1802 in the Philosophical Magazine. This dissertation is written with the same ability that characterizes all the productions of the author; and nothing more was necessary than to pursue the observation he had made, in order to obtain a complete theory of this branch of natural philosophy. It happens that, in this instance, the views of the philosopher are confirmed by the most abstruse and refined mathematical investigation.

The formula found by Laplace for the attractive force of a fluid bounded by a curve surface, consists of two parts, one of which is the same for all surfaces, and the other varies with the curvature in each particular case. The first of these terms is the attractive force of an indefinite mass of the fluid bounded by a plane. The other term, which depends upon the curvature, is composed of a constant quantity multiplied into half the sum of the reciprocals of the radii of the circles, which have the same curvature with any two sections of the curve surface made by planes, perpendicular to one another and to the curve surface; and, on examination, this constant quantity will be found to coincide with the measure of the lateral tendency of the fluid caused by the direct action of the first force. Thus it appears that the two quantities which enter into the formula of Laplace are no other than the measures of the two kinds of force which we have been considering: the one denoting the direct pressure caused by the attraction of a fluid mass bounded by a plane, and the other signifying the derivative force acting laterally, which is a necessary consequence of the direct pressure.

In a subsequent part of this article, what has now been advanced will be proved, by deducing the formula of Laplace in a direct and satisfactory manner from the two kinds of force, with the consideration of which we have been occupied.

6. Imagine a large vessel DGHF (fig. 3), which contains a fluid subjected to no forces but gravity and the attraction of its own particles, and consequently having its surface DF horizontal; let AB represent a rectangular plate partially plunged in the fluid which it attracts; and supposing the surface of the fluid to remain level, let it be proposed to investigate the force with which the attraction of the plate tends to disturb the equilibrium of the fluid.

Suppose a horizontal plane, df, to be traced in the fluid, at a depth equal to the range of the corpuscular force; then this plane will separate all the superficial strata, in which the pressure is variable, from the rest of the mass. Below the plane df, the fluid particles cohere with the same force in every part, and they are likewise attracted with equal intensity by all the points of the plate with which they are in contact; above the same plane the attractive force of the plate remains unchanged, but the pressure of the fluid in the different strata is variable, gradually becoming less and less as we approach the surface. It will therefore be proper, first to examine what tendency the part of the plate below the plane df has to disturb the equilibrium; and, secondly, to consider the effect of the plate's attraction upon the superficial film or stratum.

If the matter of the plate have the same attraction for the particles of the fluid that they have for one another, we may consider the plate as a body of the fluid that has congealed without any other change; in which case it is evident that, below the superficial stratum, the cohesive force of the fluid particles will be equal to their adhesion to the plate, and the action of the solid matter will nowise disturb the equilibrium of the fluid in the vessel.

If the plate be supposed to have no attraction for the fluid, a canal having one end in the surface of the fluid, and the other end on the plate, will be similar to a canal terminating both ways in the fluid's surface. It will be in equilibrium by the mutual attractions of the particles within it, and will exert no pressure whatever upon the plate.

If the solid matter attract the particles of the fluid, but with less intensity than they attract one another, there will be an adhesion of the fluid to the plate in proportion to the attractive force. In this case we may distinguish the attraction between the fluid particles into two parts, one of which is equal to and in equilibrium with the attraction of solid matter; while the other part, which is over and above what balances the attraction of the solid matter, is in equilibrium by the mutual action of the particles upon one another.

The solid matter acts immediately upon a thin portion of the fluid in contact with it; that portion attracts another contiguous portion; and in this manner the attraction of the plate reaches to any distance in the fluid mass. But from this it is manifest that the whole of a force greater than the mutual attraction of the particles cannot be propagated to a distance. Part of it must remain confined to the sphere of immediate action. Hence, if the plate attract the particles of the fluid with greater intensity than they attract one another, a part only of the attraction of the solid matter will balance the whole attraction of the fluid; and the remaining part will not penetrate beyond the range of the corpuscular force, but will act only upon a thin film of the fluid in contact with the plate. In this case, therefore, the plate's attraction produces a force which is not absorbed by the fluid. As this force compresses the thin film on which it acts upon the plate's surface, it will be attended with a lateral pressure, or an effort of the film to spread itself on all sides; and it may at first be thought that this lateral tendency, by acting upon the superficial stratum, will disturb the equilibrium. But it will immediately occur, that the effort which the edge of the film adhering to the plate below the plane df makes to raise up the superficial stratum, is counteracted by the opposite effort of the fluid situated immediately above the plane df. Thus, in every relation that can subsist between the attractive powers of the plate and the Capillary fluid, that part of the solid which is immersed below the superficial stratum has no tendency to disturb the equilibrium of the fluid in the vessel.

Some philosophers account for capillary action by means of attractions between the plate and the fluid, which are supposed to take place partly at the surface of the fluid, and partly at the bottom of the plate. Laplace, in particular, has grounded his second or more popular theory entirely on such attractions. He observes, that the part of the plate's vertical plane immersed in the water attracts the fluid in contact with it as much upward as downward, and therefore has no effect in causing either an elevation or a depression; but the part above the water attracts a thin film in contact with the plate upward; and the whole vertical side of the plate likewise attracts in the same direction the fluid below it, and situated in its prolongation. According to Laplace, it is the united effect of these two attractions which supports the suspended ring. The whole of this reasoning appears to us gratuitous. No part of the fluid is attracted by the solid matter in a vertical direction, but in a direction perpendicular to the plate's surface. The immersed part of the solid presents a continuous surface to the fluid, attracting it with the same intensity at every point; whereas Laplace neglects the action of the plate's horizontal boundary, and seems to suppose that the attractive energy of the solid matter resides only in the vertical sides. We have endeavoured to prove that the thin film or coating of fluid which covers the part of the solid immersed below the superficial stratum, is everywhere in a state of equilibrium and of equal compression by the attractions which act upon it. There is, therefore, no force produced at the bottom of the plate by the attraction between the solid matter and the particles of the fluid which can contribute to support the weight of the ring raised above the level.

We proceed now to consider the action of the plate upon the superficial stratum. Trace a canal at right angles to the plate, of the same depth with the superficial film, and having its horizontal width equal to unit, and continue the canal till it terminate in a vertical plane PS, parallel to the plate. Let n be the small portion of the canal within the sphere of the plate's attraction, and suppose the canal to be divided in its whole length into the parallelopipeds m, m, m, &c., each equal to n. It is plain that the attractions of the fluid below the canal, and on the two sides of it, have no tendency to impel it in any direction, nor to impede the motion of the fluid along it. The canal is also in equilibrium with regard to gravity, since by the hypothesis it is horizontal. The rectangular wedge of fluid beyond the plane PS will attract the small parallelopiped contiguous to it with a force proportional to $\frac{1}{2}K$, because K denotes the attractive force of two rectangular wedges, sect. 5; and the same parallelopiped will also be attracted with an equal force in the opposite direction by the one next to it. In like manner, every parallelopiped in the canal is attracted with equal forces by those contiguous to it on opposite sides, except the one in contact with the plate, which is attracted in the direction of the canal with the force $\frac{1}{2}K$, and towards the plate with the force K', depending upon the intensity of the plate's attraction for the fluid.

Now, if K' be just equal to $\frac{1}{2}K$, which will happen when the intensity of the plate's attraction is half that of the fluid, the parallelopiped n will be situated, with regard to the forces that act upon it, similarly to the others in the canal; in this case, therefore, the insertion of the plate will not disturb the equilibrium of the fluid, the surface of which will remain horizontal.

If K' be greater than $\frac{1}{2}K$, it may be resolved into two parts, $\frac{1}{2}K + (K' - \frac{1}{2}K)$, of which one will counterbalance the opposite force, and reduce the canal to equilibrium; and the other part, $K' = \frac{1}{2}K$, will act only upon the parallelopiped n, and will compress it upon the surface of the plate. The compression will produce a lateral force proportional to $H' - \frac{1}{2}H$, which urges the small fluid mass to spread itself towards every side; and as this force is unopposed vertically upward, the equilibrium of the fluid will be disturbed; the superficial film will ascend all round the plate, and, by means of the force of cohesion, will carry with it a portion of the fluid till the suspended weight is sufficient to counterbalance the force acting upward.

When K' is less than $\frac{1}{2}K$, the parallelopiped in contact with the plate will be more attracted in the direction of the canal than towards the plate. When this happens the fluid is depressed below the level by capillary action; but we shall leave this case to be afterwards considered, and at present confine our attention to the former case, when the fluid is elevated above the level.

7. When the immersion of the plate causes an elevation, the fluid will assume the form of a concave ring, as the ring KLM (fig. 4). If we suppose a superficial canal divided into parallelopipeds as before, we may prove by like reasoning that the attraction of the solid matter has no tendency to disturb the equilibrium of the fluid except by the lateral force which it communicates to the small parallelopiped in contact with it. And since the attractive force of the plate upon the particles of the fluid depends only upon their perpendicular distance from its surface, it readily follows that the lateral force will undergo no variation, but will remain constantly equal to $H' - \frac{1}{2}H$, both during the rising of the ring, and when it has attained the greatest elevation. The reciprocal attraction of all the fluid in the vessel likewise produces pressures that are propagated inward from the surface of the fluid, and from the sides and bottom of the vessel, sect. 3; but these forces cannot be in equilibrium with the ring and the disturbing force arising from the plate's attraction. For the former forces have no tendency to move the centre of gravity of the whole mass, whereas the latter tend to produce motion in that point, each in its own direction. In the case of equilibrium, therefore, the vertical force arising from the plate's attraction must be equal to the weight of the suspended ring; or, which is the same thing, $H' - \frac{1}{2}H$ will express the weight of a portion of the ring in every unit of the horizontal extent.

The vertical force produced by the attraction of the solid matter begins to act the instant the fluid comes into contact with the solid; it first causes the ring to rise, and then keeps it suspended.

If $m^2$ denote the area of a section of the ring made by a vertical plane perpendicular to the surface of the plate, then $m^2$, or $m^2 \times 1$, will be the volume of a portion of the fluid equal in weight to $H' - \frac{1}{2}H$.

If two parallel plates, AB and CD (fig. 5), very near one another, have their lower ends immersed in a fluid, it is observed that the fluid will rise between them above the natural level. Conceive a superficial canal extending between the plates in a direction at right angles to their surfaces, having its depth equal to the greatest range of the corpuscular force, and its horizontal width equal to unit; then all the fluid below the canal will be in equilibrium with respect to the attractive forces that act upon it, and therefore the suspended weight must be supported by the action of the two plates upon the canal. Of the forces which act upon the canal, we may neglect the attraction of the fluid below it, which causes the particles in the inside to press perpendicularly on the bottom. At each end it is attracted by the plates with a force equal to K', or $\frac{1}{2}K + (K' - \frac{1}{2}K)$, and, at the vertical sides between the plates, by the fluid on the outside with a force equal \( \frac{1}{2} K \). Wherefore, when the canal is reduced to equilibrium by equalizing the pressure upon its sides, there will remain at each end an excess of force equal to \( K - \frac{1}{2} K \), which compresses the fluid upon the plates; and the compressive force is necessarily accompanied with a lateral pressure equal to \( H' - \frac{1}{2} H \), which tends upwards and supports the weight of the fluid suspended below the canal.

Hence the weight elevated between the plates, in the horizontal length \( \lambda \), is equal to \( 2(H' - \frac{1}{2} H) \times \lambda \); and, since \( m^2 \times 1 \) is the volume corresponding to the weight \( (H' - \frac{1}{2} H) \times 1 \), the volume corresponding to the weight \( 2(H' - \frac{1}{2} H) \times \lambda \) will be equal to \( 2m^2 \times \lambda \). Let \( D \) denote the distance of the plates, and \( Q \) the least height of the curve surface between them above the natural level; then, if we conceive a horizontal plane touching the curve surface at its lowest point, the whole fluids between the plates in the length \( \lambda \) will be composed of a small curved portion in the shape of a meniscus, and a parallelopiped equal in volume to \( \lambda \times D \times Q \). Now, when the plates are very near one another, and the elevation is considerable in comparison of their distance, the meniscus will be so small that the parallelopiped alone may be reckoned equal to the whole volume of the fluid. Hence, if we equate the two expressions of the same bulk, we shall get

\[ D \times Q = 2m^2; \]

which proves that the elevations of a fluid between plates of the same matter are reciprocally proportional to the distances of the plates; and this agrees with observation.

When a capillary tube, or one with a bore less than one tenth of an inch, is partly plunged in a fluid, the fluid will rise within the tube above the level on the outside. Let \( AB \) and \( CD \) (fig. 5) represent the sides of such a tube, \( MHN \) the curve surface of the elevated column, having below it an imaginary surface at a depth equal to the range of the corpuscular force, and conceive two planes intersecting one another in the axis of the tube at any angle, then all the fluid below the superficial stratum will be in equilibrium with regard to the attractions to which it is subjected; and the triangular portion of that stratum, bounded by the inside of the tube, and the two planes intersecting in the axis, would likewise be in equilibrium, if the pressures upon all its vertical sides were equal. But the side in contact with the tube is attracted with a force equal to \( K \), or \( \frac{1}{2} K + (K - \frac{1}{2} K) \); and each of the other two sides is attracted with a force equal to \( \frac{1}{2} K \); therefore, when the equilibrium of the attracting forces is provided for, there will remain an unbalanced pressure, proportional to \( K - \frac{1}{2} K \), upon the inside of the tube; and this direct compressive force is accompanied with a lateral tendency, proportional to \( H' - \frac{1}{2} H \), which is directed upward, and sustains the elevated fluid between the two intersecting planes.

If \( \pi \) denote the circumference of a circle that has its radius equal to unit, and \( r \) the radius of the capillary tube, then \( (H' - \frac{1}{2} H) \times r \pi \) will be the weight of the elevated column of fluid within the tube, and \( m^2 \times r \pi \) will be its bulk. Conceive a plane which touches the curve surface of the column at its lowest point, and let \( q \) be the height of that point above the level on the outside of the tube, then the elevated column will consist of a cylinder equal to \( \frac{1}{2} r^2 \pi q \), and a small meniscus above the cylinder; so that, in very small tubes, the cylinder may be taken for the whole bulk of the column; wherefore, by equating the two expressions of the same bulk, we get

\[ \frac{1}{2} r \times q = m^2; \]

which proves that, in small tubes of the same matter, the elevations are reciprocally proportional to the radii or diameters of the tubes.

And because \( m^2 \) is the same in all cases, when plates and tubes of the same matter act on the same fluid, if we equate the values of it taken from the last expression, and from the expression formerly obtained for two plates, we shall get

\[ H \times Q = r \times q; \]

and this shows that a fluid will rise between two plates, to the same height it would do in a tube of the same matter having its radius equal to the distance of the plates.

The deductions that have now been drawn from the Theory of principle of a corpuscular attraction evanescent at all sensible distances, are equivalent to the account of capillary action founded on the hypothesis of Dr Jurin. Whatever may be thought of the physical principle advanced by this philosopher, it must be allowed that his theory agrees well with observation; and it cannot be denied that he has, with great sagacity, inferred from his experiments the true place in which the capillary force resides. But it is impossible to accede to his opinion, that, when a capillary tube of glass is immersed in water, the water within the tube is attracted upward by a narrow ring of glass immediately above the surface of the liquid. If the glass attract the water, the attraction must be perpendicular to the surface of the glass; the force acting on the fluid cannot be vertical, it must be horizontal; and if we would reason strictly, the proper inference must be, that an attraction between the glass and the water is alone insufficient to account for capillary action. In order to explain the phenomena, it is necessary to attend to the remark of Professor Leslie, founded on the properties essential to fluidity, namely, that a fluid cannot be attracted horizontally by a solid body, without having a vertical force communicated to it. It is certainly not a little surprising, that an observation made in 1802, so well calculated to remove all the difficulties of the theory, should have passed entirely unnoticed, although, since that period, the subject has engaged the attention of the first philosophers of the age.

In what goes before, it has been shown that, in many cases, the height to which a fluid will rise may be found with considerable exactness, by comparing the bulk as determined by the magnitude of the capillary force with the same bulk deduced from the figure which the displaced fluid is constrained to assume; but a rigorous investigation of all the circumstances attending the capillary phenomena requires further, that we know the nature of the curve assumed by that part of the fluid's surface which is free to obey the impulse of all the forces that act upon it; and it is to this branch of the subject that we are now to proceed.

8. Resuming the first and simplest case of a single plate immersed in a fluid, which rises upon its surface in a concave ring, let a vertical plane \( PL \) (fig. 4), parallel to the plate, and at a distance from its surface greater than the range of the corpuscular force, be drawn to intersect the curve; then the part of the ring cut off, being without the sphere of the plate's attraction, must be supported by the force with which it is attracted by the fluid between the plate and the plane. Now, all the fluid below the superficial stratum is in equilibrium with regard to the corpuscular forces to which it is subjected; and hence it is the attraction of the fluid between the plate and the plane upon the superficial stratum which supports the part of the ring below the plane, in the same manner that the attraction of the plate upon the same stratum supports the whole ring. All the fluid in the vessel being supposed in equilibrium, we may conceive that the portion of it between the plate and the plane is converted into a solid without any other change of its properties; then if we consider that part of the superficial canal which lies between the vertical plane and the level surface of the fluid, Capillary Action.

The upper end of it will be pressed against the imaginary solid by the attraction of an obtuse-angled wedge of the fluid, while the pressure upon all the other vertical sides is only equal to the attraction of a right-angled wedge; and the difference of these forces remaining unbalanced, generates the force which tends upward, and supports the weight of the part of the ring situated below the point of its action.

It is now necessary to determine the attractive force of a portion of a fluid, in the shape of a wedge, contained in any proposed angle. Suppose that a fluid mass bounded by the plane AB (fig. 6) is divided by the plane PQ; and let it be required to find the force with which the attraction of the particles contained in each of the wedges APQ and BPQ will cause a small drop placed at P to press upon the plane AB. Draw PN and PG to bisect the angles APQ, BPQ; let the line PH, perpendicular to the plane AB, represent the force K, or the pressure of the drop caused by the attraction of all the fluid below the plane, sect. 3; and draw HN and HG perpendicular to PN and PG. It is manifest that the attraction of all the particles in the wedge APQ is a force in the direction PN; and, in like manner, the attraction of the particles in the wedge BPQ is a force in the direction PG. Wherefore, since PH, the united effect of both attractions, is resolved into the forces PN and PG, it follows that PN will represent the attraction of the obtuse-angled wedge upon the drop, and PG that of the acute-angled wedge. Draw NO and GL perpendicular to PH; then PO is the part of the force PN acting at right angles to the plane AB, and PL is the like part of the force PG. Draw NG, and let φ denote the angle HPQ, or the difference of each of the angles APQ and BPQ from a right angle. Then NG and PH are equal and bisect one another. Also, the angle PHG = BPG, each being the complement of HPG. Wherefore GCP = 2PHG = 2BPG = BPQ; and, taking the complements of the equal angles, CGL = CNO = HPQ = φ. Now GC = PH = K; hence CL = CO = K sin. φ; therefore PO = K + K sin. φ, and PL = K - K sin. φ. Thus the pressure of the drop upon the plane AB, caused by the attraction of the obtuse-angled wedge, is equal to K + K sin. φ; and that caused by the attraction of the acute-angled wedge is equal to K - K sin. φ.

Returning now to the canal below the vertical plane PL (fig. 4), and the level surface of the fluid, let θ denote the inclination of the curve at L to the horizon; the canal would be in equilibrium with respect to the corpuscular forces that act upon it, if the attractions upon all its vertical sides were equal. But, according to what has just been investigated, the upper end is attracted by the fluid beyond the vertical plane PL, with a force equal to K + K sin. θ; and the attraction upon each of the remaining sides is only equal to K; wherefore there is an excess of attraction equal to K sin. θ, which causes the drop of liquid at the upper end of the canal to press upon the fluid above it, and which will be attended with a lateral force, equal to H sin. θ, acting upward and sustaining the part of the ring cut off by the vertical plane.

Let β × l denote the volume of a portion of the fluid equal in weight to H. Then H sin. θ will be the weight, and β sin. θ the bulk, of the partial ring cut off by the plane PL in the horizontal extent equal to unit. Let y denote the vertical ordinate of a point in the curve, formed by the intersection of the ring, with a vertical plane perpendicular to the plate; and let x be the corresponding horizontal ordinate, or the distance of y from the plate. Then the area of the curve below the point L is equal to ydx, the fluent vanishing with y; and the volume of the partial ring in the horizontal extent equal to unit, is equal to 1 × ydx. Hence if we put z = sin. θ, and equate the two expressions of the same bulk, we shall get these equations, which are sufficient to determine the nature of the curve, viz.

\[ \beta z = \int ydx \\ \frac{dy}{dz} = \frac{z}{\sqrt{1-z^2}} \]

the negative signs must be used, because z and y both decrease when x increases.

From the first of these equations we get

\[ -ydx = \beta dz \]

and, if this be multiplied into the second equation, there will result

\[ ydy = \frac{\beta^2 dz}{\sqrt{1-z^2}} \]

that is, since \( z = \sin. \theta \), \( ydy = \beta^2 d\theta \); whence \( y^2 = 2\beta^2 (1-\cos. \theta) = 4\beta^2 \sin. \frac{1}{2}\theta \); and

\[ y = 2\beta \sin. \frac{1}{2}\theta \]

Again, \( dx = \beta \cos. \theta d\theta \); therefore

\[ -x = \beta \log. \tan. \frac{1}{2}\theta - 4\beta \sin. \frac{1}{2}\theta \log. \tan. \frac{1}{2}\theta - 4\beta \sin. \frac{1}{2}\theta \]

\( t \) being the value of \( \theta \) when \( x = 0 \). Therefore

\[ x = \beta \log. \tan. \frac{1}{2}\theta - 4\beta \left( \sin. \frac{1}{2}\theta - \sin. \frac{1}{2}\theta \right) \]

The value of the ordinate shows that \( x \) increases without limit as \( y \) decreases; whence it follows that the curve has an asymptote in the level surface of the fluid.

In like manner, we may investigate the curve formed by the intersection of the fluid between two parallel plates and a vertical plane perpendicular to the plates. Let \( y \) denote the height above the natural level of a point in the curve, and \( x \) the distance of \( y \) from the middle of the plates, or from the point where \( y \) is least. Suppose two vertical planes, PO and QR (fig. 5), parallel to the plates and at equal distances from them; then, as before, the fluid on the outside of the planes PO and QR will attract the ends of the superficial canal between them with a force equal to \( K + K \sin. \theta \); and, as the part \( K \) is alone sufficient for the equilibrium of the canal, it follows that the other part \( K \sin. \theta \) will compress the fluid in contact with the two planes, producing thereby a lateral pressure that tends upward and sustains the weight of the suspended fluid. Hence the weight of the fluid suspended between the planes PO and QR, in every unit of the horizontal length, is equal to \( 2 \times \frac{1}{2} H \sin. \theta \); and its bulk is equal to \( 2\beta^2 \sin. \theta \).

But the same bulk is also equal to \( 2 \times \int ydx \), the fluent vanishing with \( x \). Therefore, by putting \( z = \sin. \theta \), and equating the two expressions of the same bulk, we get the equations

\[ \int ydx = \beta z \\ \frac{dy}{dz} = \frac{z}{\sqrt{1-z^2}} \]

which determine the nature of the curve.

By combining the two equations, we get \( ydy = \frac{\beta^2 dz}{\sqrt{1-z^2}} \); now let \( u = \sin. \frac{1}{2}\theta \); then \( z = \sin. \theta = 2u \sqrt{1-u^2} \), and

\[ \frac{dz}{\sqrt{1-z^2}} = \frac{2du}{\sqrt{1-u^2}} ; \text{ hence } ydy = 4\beta^2 du \] \[ y = 2\beta \sqrt{q^2 + w^2} \]

\( y = 2\beta \) being the height of the lowest point of the curve above the level.

Again, \( dx = \frac{\beta dz}{y} = \beta \cdot \frac{du(1 - 2u^2)}{\sqrt{(1 - u^2)(q^2 + w^2)}} \); therefore,

\[ x = \beta \int \frac{du(1 - 2u^2)}{\sqrt{(1 - u^2)(q^2 + w^2)}}; \]

and \( x \) will be obtained by the rectification of the conic sections.

In the case of a capillary tube, conceive an imaginary tube, of which the sides are PO and QR (fig. 5), within the real one, and let \( \theta \) denote the inclination of the curve surface to the horizon at the points P and Q. The elevated column within the imaginary tube is supported by the attraction of the fluid between the two tubes, in the same manner that the whole capillary column is supported by the attraction of the solid matter of the real tube. The fluid in contact with the imaginary tube on the outside having the shape of a wedge contained in the obtuse angle 90° + \( \theta \), will attract the fluid in the inside in a horizontal direction with a force equal to \( \frac{1}{2}K + \frac{1}{2}K \sin \theta \); and of this force, the part \( \frac{1}{2}K \sin \theta \) will compress the fluid ring on which it acts, producing, by this means, a lateral tendency upward, proportional to \( \frac{1}{2}H \sin \theta \), which supports the weight of the suspended column. If \( r \) denote the radius of the imaginary tube, then \( \frac{1}{2}H \sin \theta \times r \sigma \) will be equal to the weight, and \( \beta \sin \theta \times r \sigma \) to the bulk of the elevated column within that tube; and if \( y \) be the vertical ordinate of a point in the curve surface, or the height above the natural level, and \( r \) the horizontal distance of \( y \) from the axis of the tube, the bulk of the same column will be equal to \( \int y r dr \), the fluent vanishing with \( r \).

Therefore, by equating the equivalent expressions, we shall get the following equations, which determine the nature of the curve surface, viz.

\[ \beta \sigma z = \int y r dr, \]

\[ \frac{dy}{dr} = \frac{z}{\sqrt{1 - z^2}}. \]

If these equations be combined so as to exterminate \( y \), a differential equation between \( r \) and \( z \) will be obtained, viz.

\[ \frac{d}{dr} \left( \frac{r z}{\sigma} \right) = \frac{1}{\beta} \times \frac{z}{\sqrt{1 - z^2}}. \]

Having explained the most remarkable instances of elevation by capillary action, we must now turn our attention to the cases where a fluid is depressed below the level by the same cause. It has been shown that an elevation will always take place when \( K' \) and \( H' \) are greater than \( \frac{1}{2}K \) and \( \frac{1}{2}H \), and that the fluid will remain level when the same quantities are equal. It follows, therefore, that the fluid will sink below the level when the former quantities are less than the latter; or otherwise there could not be an equilibrium. The shortest and most perspicuous manner of explaining the cases when a fluid is depressed is to compare them with the similar cases of an elevation. Suppose that AB and ab (fig. 7 and 8) are two plates of different kinds of matter immersed in the same fluid, which they attract with intensities equally above and below the mean quantity \( \frac{1}{2}K \), we shall prove that the same curve which is formed above the level on the surface of the one, will be in equilibrium by the action of the other when placed upon its surface in a reversed position below the level.

Conceive the two curves to be intersected at intervals Capillary equal to the range of the corpuscular force by an indefinite number of planes parallel to the plates; and let the curves at L and l have the same inclination, \( \theta \), to the horizon. In the curve above the level (fig. 7), it has been shown that the force which tends upward, and supports the part of the ring below L, is equal to \( \frac{1}{2}H \sin \theta \); and, in like manner, the force which supports the part of the ring below O, indefinitely near L, is equal to \( \frac{1}{2}H \cdot (\sin \theta + d \sin \theta) \). Therefore the difference of these forces, or \( \frac{1}{2}H \cdot d \sin \theta \), which may be considered as a force urging the curvilinear element OL upward, is equal to the weight of the fluid elevated between the planes passing through O and L. In effect, if we put \( y \) and \( x \) to denote the vertical and horizontal ordinates of the point L, and equate the two expressions of the bulk of the small portion of fluid above mentioned, we shall get

\[ \beta \sigma d \sin \theta = \beta \sigma dz = y dx, \]

which is no other than the differential of the equation formerly obtained (sect. 7). We may therefore conceive that every element of the curve is urged upward with a force equal to the weight of the elevated fluid below it, the attraction of the plate supplying the force necessary to sustain the accumulated weight of all the suspended fluid.

In the curve below the level (fig. 8), the fluid on the same side of the plane \( \theta \) with the plate ab attracts the particles on the other side of that plane; and as the attracting fluid forms an acute-angled wedge contained in the angle \( \theta = 90^\circ - \theta \), the horizontal attraction will be equal to \( \frac{1}{2}K - \frac{1}{2}K \sin \theta \); and the lateral force thence arising, and acting vertically, to \( \frac{1}{2}H - \frac{1}{2}H \sin \theta \). The point l of the curve is therefore urged upward by the attraction of the fluid between it and the plate ab, with a force equal to \( \frac{1}{2}H - \frac{1}{2}H \sin \theta \); and in like manner, the point o indefinitely near l, tends upwards with the force \( \frac{1}{2}H - \frac{1}{2}H \cdot (\sin \theta + d \sin \theta) \). The difference of these forces, which may be considered as a force applied to the curvilinear element ol, is equal to \( -\frac{1}{2}Hd \sin \theta \); and it is the same in quantity as in the other curve, but has an opposite direction. The difference in the directions of the two forces acting upon the elements of the two curves arises from this, that, in the curve above the level, the force acting upward continually increases from the level surface to the plate, whereas, in the curve below the level, it continually decreases. Again, because OL and ol are placed at equal distances above and below the general level of the fluid, the weight drawing the element OL downward will be just equal to the vertical pressure caused by the superincumbent fluid, and urging the element ol upward. It thus appears that the forces which act upon the like elements of the two curves are the same in quantity, but that they have their directions reversed; which proves that, because the one curve is in equilibrium, the other will be so too; at least this will be the case if the attraction of the plate ab be sufficient to maintain the lowest point of the convex curve in its place.

The parts of the curves between the level surface of the fluid and the planes TL and \( \theta \) are kept in their places by the horizontal attraction of the fluid on the other side of the same planes. These attractions are respectively equal to \( \frac{1}{2}K + \frac{1}{2}K \sin \theta \), and \( \frac{1}{2}K - \frac{1}{2}K \sin \theta \); and the one as much exceeds the mean quantity \( \frac{1}{2}K \) as the other falls short of it. Now, in place of the attractions of the fluid particles contained in the wedges KLT and klt, we may substitute the attractions of two solid plates that act upon the fluid with equal forces; and these plates will come under the condition we have supposed with respect to the attractions of the plates AB and ab. It follows, therefore, that, because the attraction of the plate AB main- Capillary tains the concave curve in its place, the attraction of the plate \( ab \) will be sufficient to maintain the convex curve in its place.

It is evident that the same reasoning which has been applied to two solid plates will apply equally in all other cases, and we may lay down this general proposition, viz. If two solid bodies, perfectly equal and similar, but composed of different kinds of matter, be immersed in a fluid which they attract with intensities equally different from the mean quantity \( \frac{1}{2} K \), the fluid will be raised above the level by the action of the one, and depressed below the level by the action of the other, and the convex curve below the level will differ from the concave curve above the level in no respect, except that it will have a reversed position.

As no bounds can be set to the attractive force which a solid body exerts upon the particles of a fluid, it may be asked, will the weight displaced by capillary action increase in proportion to the attraction of the solid? or, are there any conditions that confine the effect within a certain limit, however great may be the attraction of the solid? In answer to this, it must be observed, that the action of the solid matter is confined to a thin film of the fluid in contact with it; and that it is this film alone which acts on the particles beyond it, and keeps them suspended by means of the force of cohesion. Hence the weight maintained above the level can never exceed what this last force is able to support. The elevation of the fluid will there be regulated by the attractive force of the solid matter, only so long as that force is less than the mutual attraction of the fluid particles; and the fluid will always rise to the same height when the attraction of the solid matter is either equal to or greater than the fluid's cohesion. In all these cases, the solid is wetted by the fluid, and we may conceive that it becomes covered with a coating of sufficient thickness to shield the particles on the outside from the attraction of the solid matter, a new body being thus formed, which attracts the fluid with a force equal to its own cohesive power.

From the relation that has been shown to take place between the cases of equal elevation and depression, it follows that the greatest depression will take place when a solid has no attraction for the particles of a fluid. If we go beyond this limit, and suppose that the solid matter repels the fluid, the capillary effect will not be heightened; for the repulsive force will be confined to the particles within the range of its action; beyond this insensible distance the repelling power will produce no effect, and the fluid will be left to assume the same figure it would do if no such power existed.

In the several cases that have been considered, the weight of the fluid suspended below that point of the curve surface which is inclined to the horizon in the angle \( \delta \), has been found to be equal to \( \frac{1}{2} H \sin \delta \); therefore, if \( \epsilon \) denote the angle of contact, or the angle in which the surface of the fluid intersects the solid, then \( 90^\circ - \epsilon \) will be the limit of \( \delta \), or what \( \delta \) becomes at the surface of the solid; and consequently the weight of the whole fluid suspended by capillary action will be equal to \( \frac{1}{2} H \cos \epsilon \); but, as has likewise been proved (sect. 7), the same weight is also equal to \( H' - \frac{1}{2} H \); and hence, by equating the equivalent quantities, we get

\[ H' = H \cos \frac{\epsilon}{2}. \]

This expression is possible only when \( H' \) is not greater than \( H \); but we must not infer that the theory leads to any contradiction in the case where a solid body attracts the particles of a fluid with an intensity greater than their own mutual action upon one another. The equation is a consequence of the equality that takes place between the vertical force \( H' - \frac{1}{2} H \) derived from the attraction of the solid, and the weight of the displaced fluid deduced from the figure which the attraction of its own particles causes it to assume. It is, therefore, only the effective part of the force \( H' - \frac{1}{2} H \), or that which is really employed in displacing the fluid, that can enter into the equation; and when a part of the same force has no effect in elevating or depressing the fluid, that part must be neglected. Now it has been proved that, however great the force \( H' - \frac{1}{2} H \) may be, the capillary effect can never exceed that produced by the force \( H' - \frac{1}{2} H \) (sect. 9); and hence it appears, from the principle on which the investigation proceeds, that, in the equation, \( H' \) must be limited not to exceed \( H \), which must be taken for its value in all cases when the solid matter acts upon the fluid particles with an intensity either equal to or greater than their own mutual attraction.

From this equation it follows that the angle of contact is always the same when different solids of the same attractive powers are immersed in the same fluid, a property that was first noticed by Dr Young.

The weight of the displaced fluid being equal to \( \frac{1}{2} H \cos \epsilon \), is in every case proportional to the cosine of the angle of contact.

In order still farther to illustrate and confirm the principles of the theory we have been explaining, we shall conclude this article with applying them to demonstrate Laplace's formula for the attraction of a fluid mass bounded by a curve surface.

Conceive a fluid mass bounded by a curve surface concave outward, and let the plane MAN (fig. 9) be a tangent, and the straight line AO normal, to the curve surface at any point A; through AO draw any two planes perpendicular to one another, which cut the surface of the fluid in the curve lines BA and CA, and let DC and DB be two other sections of the fluid's surface made by planes parallel to the first planes, and indefinitely near them. Put \( ds \) and \( ds' \) for the small curve lines AB and AC; and \( db \) and \( db' \) for the measures of the small angles which the tangents drawn to the curve lines from the points B and C make with the tangent plane MAN. The four planes intersecting the fluid contain within them a rectangular prism, standing upon the base ABDC, and extending into the interior of the fluid mass at right angles to the curve surface: it is required to find the force which urges the prism outward above the tangent plane.

Conceive a surface intersecting the prism at a depth below its base equal to the range of the corpuscular force; then all the fluid of the prism below this imaginary surface being an equilibrium with regard to the attractions to which it is subjected, we have only to examine the forces that act upon the superficial stratum. It is attracted by the particles below the imaginary surface, and by the fluid on the outside of the force bounding planes. The attraction of the particles below the imaginary surface is at every point perpendicular to that surface; and therefore the stratum would be in equilibrium, if the attractions upon its four sides were equal. The fluid on the outside of each of the two planes AB and AC is a rectangular wedge; and consequently the attractions upon the particles within the stratum causing them to press perpendicularly upon these planes, are each proportional to \( \frac{1}{2} K \). On the outside of the plane DC, the fluid is a wedge contained in the obtuse angle \( 90^\circ + d\theta \); and on the outside of the plane BD, it is a wedge contained in the obtuse angle \( 90^\circ + d\theta \); the attraction is therefore proportional to \( \frac{1}{2} K + \frac{1}{2} Kd\theta \) in the first case, and to \( \frac{1}{2} K + \frac{1}{2} Kd\theta \) in the other case. Hence, after the attractions upon the sides of the stratum are equalized, there is an excess of force perpendicular to each of the planes CD and DB, In a fluid mass, which is subjected to no forces but the capillary attractions of its own particles, and which is in equilibrium, if we conceive a slender canal passing through the interior and forming a communication between any two points of the surface, the canal will be in equilibrium taken separately (fig. 1). Of the forces in action at the ends, those which arise from the direct attraction of the whole mass, being equal and opposite, counteract one another in all positions of the canal; but the other forces, which depend on the curvature, and which in reality are nothing more than the lateral tendencies outward, produced by the direct attraction of the particles surrounding the two orifices, cannot be equal to one another in all positions of the canal, unless the function

$$\frac{1}{R} + \frac{1}{R'}$$

have the same value at all points of the curve surface, which is the case in no solid figure except a sphere. Such a body of fluid, therefore, cannot be in equilibrium, unless its form be perfectly spherical.

The formula of Laplace must be considered as a great step made in this branch of natural philosophy, not only because it ascertains the connection between the pressure and the curvature, in which it agrees with the hypothesis of Segner and Dr Young; but also because it brings into view the forces $K$ and $H$, and draws the attention to the relation they have to one another, and to the primitive attraction of the particles. The labours of philosophers have discovered the facts of capillary action, which have been verified by innumerable experiments; but if the truth is to be told, it may be affirmed, that, reckoning back from the present time to the speculations of the Florentine academicians, the formula of Laplace, and the remark of Professor Leslie relating to the lateral force, are the only approaches that have been made to a sound physical account of the phenomena.

**Method of computing the Depression of the Mercury in the Tubes of Barometers.**

It is a problem of no small difficulty to determine the vertical ordinates of the curve surface in a capillary tube from the differential equations that have been investigated. The research possesses considerable interest, as it applies to the correction of the observed heights of the mercury in a barometer, by enabling us to compute the depression arising from capillary action. It is more particularly with a view to this application that the problem is here very briefly considered.

Resuming the equations of the curve surface in a tube, found in sect. 8, we get

$$y = \beta z \left( \frac{dz}{dr} + \frac{z}{r} \right)$$

$$d^2 \left( \frac{dz}{dr} \right)^2 = \frac{1}{\beta^2} \cdot \frac{z}{\sqrt{1 - z^2}};$$

and if we put $x = \frac{r}{\beta}$, these equations will become

$$y = \beta z \left( \frac{dz}{dx} + \frac{z}{x} \right)$$

$$d^2 \left( \frac{dz}{dx} \right)^2 = \frac{z}{\sqrt{1 - z^2}} \cdots \cdots (1).$$

In these equations, when $x = 0$, we have $\frac{dz}{dx} = \frac{z}{x}$; and Capillary hence, if \( q \) denote the elevation or depression, or the least value of \( y \), then \( \frac{q}{2} = \frac{z}{x} \) when \( x = 0 \).

When \( z \) is small, the equation (1) will coincide very nearly with the more simple equation

\[ \frac{d^2w}{dx^2} = w \quad \ldots \quad (2). \]

And if, in this last equation, we put \( w = \lambda x, x^2 = t \), we shall get

\[ \frac{d^2\lambda}{dt^2} = \lambda; \]

hence,

\[ \lambda = 1 + \frac{t}{1273} + \frac{1}{192934} e^6 + \ldots + \infty. \]

Again, if we put \( \lambda = e^{\int dt} \), \( e \) being the base of the hyperbolic logarithms, we shall get by substitution,

\[ \frac{1}{2} \frac{dv}{dt} + \frac{v^2}{4} + v = 1; \]

and hence,

\[ v = 1 - \frac{1}{6} t + \frac{1}{24} e^6 - \frac{1}{90} e^{12} + \frac{13}{4320} e^{18} - \ldots \]

Thus we have these two expressions of \( w \), viz.

\[ w = r \lambda = 2 \sqrt{\lambda}; \]

\[ w = xe^{\int dt} = 2 \sqrt{\lambda} e^{\int dt} \]

each of which, being multiplied by a constant quantity, will exhibit the general value, on the supposition that \( w \) vanishes with \( x \); but the constant quantity is not necessary for the purpose we have in view.

Now, let \( z = \frac{ws}{2} \); then, on account of the equations (1) and (2), we shall readily get

\[ \frac{dds}{dx^2} + \frac{ds}{wdx} \cdot \frac{ds}{dx} = \frac{s}{\sqrt{1 - w^2 s^2}} - s \quad \ldots \quad (A) \]

which may be thus written:

\[ \frac{d}{dx} \left( \frac{ds}{dx} \right)^2 + 2 \frac{dw}{wdx} \cdot \frac{ds}{dx} x^2 = \frac{s}{\sqrt{1 - w^2 s^2}} - s; \]

but \( xdx = 2dt, \frac{ds}{wdx} x^2 = 2 \frac{ds}{dt}, \text{ and } 2 \frac{dw}{wdx} = \frac{1}{2} \cdot \frac{1}{t} + \frac{1}{t} v; \)

therefore, we have

\[ \frac{d}{dt} \left( \frac{ds}{dt} \right) + \frac{ds}{dt} + v \frac{ds}{dt} = \frac{s}{\sqrt{1 - w^2 s^2}} - s; \]

and if we multiply both sides by \( e^{\int dt} \), and expand the radical on the right hand side, we shall get

\[ \frac{d}{dt} \left( \frac{ds}{dt} e^{\int dt} \right) = \frac{t}{2} \cdot s \cdot e^{\int dt} \]

\[ + \frac{3t^2}{8} \cdot s^5 \cdot e^{\int dt} \]

\[ + \frac{5t^3}{16} \cdot s^7 \cdot e^{\int dt} \]

\[ + \ldots \]

In order to integrate this expression, assume

\[ s = k + k^2 N + k^3 N^2 e^{\int dt} + k^4 N^3 e^{\int dt} + \ldots \]

then by substituting these values, and equating the terms containing the like powers of \( k \), we shall get

\[ \frac{dM}{dt} + vM = \frac{e^{\int dt}}{2} \]

\[ \frac{dN}{dt} = \frac{M}{t^2} \]

\[ \frac{dM'}{dt} + 2vM' = \frac{3}{8} e^{\int dt} + \frac{3t}{2} N \]

\[ \frac{dN'}{dt} + vN' = \frac{M'}{t^2} \]

\[ \ldots \]

Now, if we expand \( e^{\int dt} = \lambda^2 \) in a series, we shall get, by means of the first two equations, first a value of \( M \), and then one of \( N \), each in a series; and by a like procedure with the other equations, it will be found that

\[ N = t \cdot Q = t \left\{ \frac{t}{12} + \frac{t^3}{36} + \frac{11t^5}{1440} + \frac{t^7}{1200} + \frac{19t^9}{120960} + \ldots \right\} \]

\[ N' = t \cdot Q' = t \left\{ \frac{t}{32} + \frac{t^3}{128} + \frac{17t^5}{5760} + \frac{t^7}{7560} + \ldots \right\} \]

\[ N'' = t \cdot Q'' = t \left\{ \frac{t}{64} + \frac{3t^3}{640} + \frac{163t^5}{161280} + \frac{823t^7}{2257920} + \ldots \right\} \]

These formulæ will enable us to compute the value of \( Q, Q', Q'' \) with sufficient exactness when \( t \) is not extremely large. By substituting in the assumed value of \( s = \frac{2z}{w} \), and observing that \( \lambda^2 e^{\int dt} \), we shall get

\[ \frac{2z}{w} = k + k^2 O^2 + \frac{Q}{\lambda^2} + k^3 \lambda^4 \frac{Q'}{\lambda^2} + \ldots \]

and hence if we put \( f = \frac{kx}{2z} \), we shall have

\[ 1 = f + f^2 \frac{Q^2}{\lambda^2} + f^3 \frac{Q'^2}{\lambda^2} + f^4 \frac{Q''^2}{\lambda^2} + \ldots \]

In this method of proceeding the co-efficients in the series for \( f \) are in every case very small, and decrease so fast, that a few of the first terms determine the value of \( f \) with sufficient exactness. In reality, as \( t \) increases, each of the co-efficients increases from 0 to a certain limit; whence it follows that \( f \) will decrease from 1 to a certain limit, which is greater than \( \frac{24}{25} \).

In order to prove what has been advanced, and to determine the limit of \( f \), assume \( w = \frac{u}{\sqrt{x}} \), and substitute in the equation (2); then,

\[ \frac{du}{dx} = \left( 1 + \frac{3}{4} \cdot \frac{1}{x^2} \right) u; \]

Again, put \( u = c \); then,

\[ \frac{du}{dx} + \frac{u}{x} = 1 + \frac{3}{4} \cdot \frac{1}{x^2}; \]

and hence,

\[ \frac{1}{x} = 1 + \frac{3}{8} \cdot \frac{1}{x^2} + \frac{3}{8} \cdot \frac{1}{x^3} + \frac{63}{128} \cdot \frac{1}{x^4} + \ldots \] In consequence of the different assumptions, we have

\[ w = x \cdot \lambda = \frac{e^{\int f dx}}{\sqrt{x}}. \]

The expression \( \sqrt{x} \) will represent every value of \( w \) that vanishes with \( x \); for it vanishes with \( x \), and we conceive that \( \int f dx \) contains an arbitrary constant not necessary to be determined here.

If now we substitute this value of \( w \) in the equation (A), and observe that \( \frac{2dw}{wdx} = -\frac{1}{x} + 2z \), we shall get

\[ \frac{dds}{dx} + 2z \frac{ds}{dx} = \frac{s}{\sqrt{1 - \frac{s^2}{4x}}} \cdot e^{2f dx} - s. \]

and, by multiplying both sides by \( e^{2f dx} \), and expanding the radical, we have

\[ \frac{d}{dx} \left( \frac{ds}{dx} \right) = \frac{1}{4} \cdot \frac{3s}{8} \cdot e^{4f dx} + \frac{1}{16} \cdot \frac{3s}{8} \cdot e^{4f dx} + \ldots \]

In order to integrate this expression, we may assume

\[ \frac{ds}{dx} = \frac{k^3}{4} \cdot M \cdot e^{2f dx} + \frac{k^5}{16} \cdot M' \cdot e^{4f dx} + \ldots \]

then, by substituting and proceeding as before, we shall get

\[ \frac{dM}{dx} + 4z \cdot M = \frac{1}{2} \cdot \frac{1}{x}, \] \[ \frac{dN}{dx} + 2z \cdot N = M, \] \[ \frac{dM'}{dx} + 6z \cdot M' = \frac{3}{8} \cdot \frac{1}{x^2} + \frac{3}{2} \cdot \frac{N}{x}, \] \[ \frac{dN'}{dx} + 4z \cdot N' = M', \] \[ \ldots \]

By means of the first two equations we get

\[ N = \frac{1}{16} \cdot \frac{1}{x} + \frac{3}{64} \cdot \frac{1}{x^2} + \frac{1}{128} \cdot \frac{1}{x^3} - \frac{9}{128} \cdot \frac{1}{x^4} - \ldots \]

This series will coincide with its first term in the extreme case when \( x \) is very great; and by applying the like method of investigation to the other quantities sought, it will be found that

\[ N = \frac{1}{16} \cdot \frac{1}{x}, \quad N' = \frac{5}{256} \cdot \frac{1}{x^2}, \quad N'' = \frac{119}{12288} \cdot \frac{1}{x^3}, \] \[ N''' = \frac{393}{65536} \cdot \frac{1}{x^4}. \]

Now, let these quantities be substituted in the assumed value of \( s \), and, because

\[ w = x \cdot \lambda = \frac{e^{\int f dx}}{\sqrt{x}}, \]

we shall get

\[ \frac{2z}{x \lambda} = k + \frac{1}{16} \cdot \frac{k^3}{4} + \frac{5}{256} \cdot \frac{k^5}{16} + \ldots \]

and hence, by putting \( f = \frac{k \lambda}{2z} \), \( 1 = f + f' = \frac{3}{16} + f' \cdot \frac{5}{256} \)

\[ + f'' \cdot \frac{119}{12288} + f''' \cdot \frac{393}{65536} + \ldots \]

from which we derive

\[ f = 1 - \frac{z^2}{16} - \frac{z^4}{128} - \frac{35z^6}{12288} - \frac{137z^8}{98304} - \ldots \]

This is the limit to which \( f \) tends as \( x \) increases, and with which it coincides when \( x \) is infinitely great.

It remains now to apply the formulae that have been investigated. If, in the equation \( \beta zr = \int ydr \) (sect. 8), we substitute \( q + y' \) for \( y \), we shall get

\[ \beta z = \frac{q}{2} + \frac{\int ydr}{r}, \]

and the smaller the diameter of the tube, the more nearly will this equation approach to \( \beta z = \frac{1}{2} qr \). Therefore, \( l \) being the diameter of the tube, the value of \( 4\beta z \) will be equal to \( ql \), that is, to the product of the elevation or depression by the diameter of the tube, when the bore is very small. When mercury is contained in tubes of glass, the value of \( 4\beta z \), assigned by the English philosophers, is -015; and Laplace, from the experiments of Gay Lussac, makes it equal to -01469. There is also some uncertainty in the value of \( z \), or the cosine of the angle of contact, which seems to be between the limits 0-75 and 0-729. We may assume \( 4\beta z = -015 \), and \( z = -735 \), whence \( \beta = \frac{1}{14} \); these numbers being recommended by their simplicity, and lying between the limits of the errors of observation.

Now, \( t = \frac{x^2}{4} = \frac{z^2}{4\beta^2} = \left(\frac{l}{\beta}\right)^2 = \left(\frac{L}{2}\right)^2 \): the series denoted by \( \lambda \), and the co-efficients of the series for \( f \), will therefore be known in numbers, and hence \( f \) may be found.

Again, when \( x = 0 \), we have \( s = \frac{2z}{x \lambda} = \frac{2z}{x} = k = \frac{q}{\beta} \); and because \( x = \frac{r}{\beta} = \frac{l}{\beta} \) we get \( f = \frac{k \lambda}{2z} = \frac{ql}{4\beta^2} \); and hence

\[ q = \frac{4\beta z}{l \cdot \lambda} \times f = \frac{-015}{l \cdot \lambda} \times f \ldots (4). \]

If we compute the value of the limit to which \( f \) approaches when \( l \) is very great, we shall find \( f = -09635 \); and hence, in the case of tubes with very large diameters, we have

\[ q = \frac{-015}{l \cdot \lambda} \times -09635 = \frac{-01445}{l \cdot \lambda} \ldots (5). \]

It remains to ascertain in what cases this last formula may be safely used.

If we make \( l \) successively equal to -3 and -4, we shall find

\[ l = -3; \quad t = 1-1025; \quad f = -09696; \quad q = -02916; \] \[ l = -4; \quad t = 1-96; \quad f = -09649; \quad q = -01534. \]

Now, this last value of \( f \) approaches very nearly to the ultimate value; and if \( q \) be computed by the formula (5), we shall find

\[ q = -01532. \]

We may therefore use the formula (5) in all cases when the diameter of the tube is greater than four tenths of an inch. In other cases, we must compute the depression by the formula (4), having first found \( f \) by means of the following expression, in which all the quantities too small to affect the exactness of the result are left out, viz. To compute $f$ from this formula, assume $f = 1 - \alpha$:

Then, $\alpha$ being always less than $\frac{1}{25}$, its square and higher powers may be neglected.

By the procedure just described, the following table has been constructed, in which all the numbers may be reckoned exact, with the uncertainty of one unit in the last place of figures.

| Diameter of the Tube. | Depression. | |-----------------------|-------------| | Inches. | Inches. | | 0·05 | 0·29494 | | 0·10 | 1·40288 | | 0·15 | 0·86328 | | 0·20 | 0·58511 | | 0·25 | 0·40755 | | 0·30 | 0·29116 | | 0·35 | 0·21110 | | 0·40 | 0·15344 | | 0·45 | 0·11117 | | 0·50 | 0·08835 | | 0·60 | 0·04433 | | 0·70 | 0·02228 | | 0·80 | 0·01119 (K.K.) |