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 Fahr- Cohesion. 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 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 M. 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 M. 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 this gradation 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 $\alpha$, the fluxion of the force will be as $dx (\alpha - x)$, since a number of particles proportional to $dx$ will be drawn downwards by a number proportional to $\alpha$, and upwards by a number proportional to $x$, and Cohesion.

the whole cohesion, at the given point, will be expressed by \( ax - \frac{1}{2}x^2 \); and this at last becomes \( \frac{1}{2}a^2 \), which must be equal to the undiminished cohesion in the direction of the surface; consequently the difference of the forces acting on the sides of the elementary cube will everywhere be as \( \frac{1}{2}a^2 - ax + \frac{1}{2}x^2 \), and the fluxion of the whole contractile force will be \( dx (\frac{1}{2}a^2 - ax + \frac{1}{2}x^2) \), the fluent of which, when \( x = a \), becomes \( \frac{1}{3}a^3 \), which is one third of \( a \times \frac{1}{2}a^2 \), the whole undiminished cohesion of the stratum.

We may therefore conclude in general, that the contractile force is one third of the whole cohesive force of a stratum of particles, equal in thickness to the interval to which the primitive equable cohesion extends; and if the cohesive force be not equable, we may take the interval which represents its mean extent as affording a result almost equally accurate. In the case of water, the tension of each inch of the surface is somewhat less than three grains, consequently we may consider the whole cohesive and repulsive force of the superficial stratum as equal to about nine grains. Now since there is reason to suppose the corpuscular forces of a section of a square inch of water to be equivalent to the weight of a column about 750,000 feet high, at least if we allow the cohesion to be independent of the density, their magnitude will be expressed by \( 252.5 \times 750,000 \times 12 \) grains, which is to 9 as \( 252.5 \times 1,000,000 \) to 1; consequently the extent of the cohesive force must be limited to about the 250 millionth of an inch; nor is it very probable that any error in the suppositions adopted can possibly have so far invalidated this result as to have made it very many times greater or less than the truth.

Within similar limits of uncertainty we may obtain something like a conjectural estimate of the mutual distance of the particles of vapours, and even of the actual magnitude of the elementary atoms of liquids, as supposed to be nearly in contact with each other; for if the distance at which the force of cohesion begins is constant at the same temperature, and if the particles of steam are condensed when they approach within this distance, it follows that at 60° of Fahrenheit the distance of the particles of pure aqueous vapour is about the 250 millionth of an inch; and since the density of this vapour is about one sixty thousandth of that of water, the distance of the particles must be about forty times as great; consequently the mutual distance of the particles of water must be about the ten thousand millionth of an inch. It is true that the result of this calculation will differ considerably, according to the temperature of the substances compared; for the phenomena of capillary action, which depend on the superficial tension, vary much less with the temperature than the density of vapour at the point of precipitation; thus an elevation of temperature amounting to a degree of Fahrenheit lessens the force of elasticity about one ten thousandth, the superficial tension about one thousandth, and the distance of the particles at the point of deposition about a hundredth. This discordance does not, however, wholly invalidate the general tenor of the conclusion; nor will the diversity resulting from it be greater than that of the actual measurements of many minute objects, as reported by different observers; for example, those of the red particles of blood, the diameter of which may be considered as about two million times as great as that of the elementary particles of water, so that each would contain eight or ten trillions of particles of water at the utmost. If we supposed the excess of the repulsive force of liquids above that of elastic fluids to depend rather on a variation of the law of the force than of the number of particles co-operating with each other, the extent of the force of cohesion would only be reduced to about two thirds; and, on the whole, it appears tolerably safe to conclude, that whatever errors may have affected the determination, the diameter or distance of the particles of water is between the two thousand and the ten thousand millionth of an inch.

Sect. II.—Relations of Heterogeneous Substances.

We must now return from this conjectural digression to the regions of strict mathematical argument, and inquire into the effect of the contact of substances of different kinds on the tension of their common surfaces, and on the conditions required for their equilibrium. Whatever doubts there may be respecting the variation of the number of particles co-operating when the actual density of the substance is changed, there can be none respecting the consequence of the contact of two similar substances of different densities; for the less dense must necessarily neutralise the effects of an equivalent portion of the particles of the more dense, so as to prevent their being concerned in producing any contractility in the common surface; and the remainder, acting at the same interval as when the substance remained single, must obviously produce an effect proportional to the square of the number of particles concerned, that is, of the difference of the densities of the substances. This effect may be experimentally illustrated by introducing a minute quantity of oil on the surface of the water contained in a capillary tube; the joint elevation, instead of being increased, as it ought to be according to M. Laplace, is very conspicuously diminished; and it is obvious, that since the capillary powers are represented by the squares of the density of oil and of its difference from that of water, their sum must be less than the capillary power of water, which is proportional to the square of the sum of the separate quantities.

Upon these principles we may determine the conditions of equilibrium of several different substances meeting in the same point, neglecting for a moment the consideration of solidity or fluidity, as well as that of gravitation, in estimating the contractile powers of the surfaces, and their angular situations. We suppose then three liquids, of which the densities are A, B, and C, to meet in a line situated in the plane termination of the first; the contractile forces of the surfaces will then be expressed by \((A-B)^2\), \((A-C)^2\), and \((B-C)^2\); and if these liquids be so arranged as to hold each other in equilibrium, whether with or without the assistance of any external force, the equilibrium will not be destroyed by the congelation of the first of the liquids, so that it may constitute a solid. Now, unless the joint surface of the second and third coincides in direction with that of the first, it cannot be held in equilibrium by the contractility of this surface alone; but supposing these two forces to be so combined as to produce a result perpendicular to the surface of the first substance, this force may be resisted by its direct attraction; the forces which tend to cause the oblique surface to move either way on it, balancing each other, and the perpendicular attraction being counteracted by some external force holding the solid in its situation. Consequently the force expressed by \((B-C)^2\), reduced in the proportion of the radius to the cosine of the angle, must become equal to the difference of the forces \((A-B)^2\) and \((A-C)^2\); and if the radius be called unity, this cosine must be

\[ \frac{(A-C)^2-(A-B)^2}{(B-C)^2} = \frac{2AB-2AC-(B^2-C^2)}{(B-C)^2} \]

\[ = \frac{2A-(B+C)}{B-C}, \]

which is the excess of twice the density of the solid above the sum of the densities of the liquids, divided by the difference of these densities; and when there is only one liquid, and \(C = 0\), this cosine becomes Cohesion.

\[ \frac{2A}{B} - 1, \] vanishing when \( 2A = B \), and the density of the solid is half of that of the liquid, the angle then becoming a right one, as Clairaut long ago inferred from other considerations. Supposing the attractive density of the solid to be very small, the cosine will approach to \( -1 \), and the angle of the liquid to two right angles; and on the other hand, when \( A \) becomes equal to \( B \), the cosine will be 1, and the angle will be evanescent, the surface of the liquid coinciding in direction with that of the solid. If the density \( A \) be still further increased, the angle cannot undergo any further alteration, and the excess of force will only tend to spread the liquid more rapidly on the solid, so that a thin film would always be found on its surface, unless it were removed by evaporation, or unless its formation were prevented by some unknown circumstance which seems to lessen the intimate nature of the contact of liquids with dry solids. For the case of glass and mercury we find

\[ \frac{A}{B} \approx \frac{3}{5}, \]

and the cosine \( \frac{6}{8} \), which corresponds to an angle of \( 139^\circ \); and if we add a second liquid, the expression will become

\[ \frac{-6}{-8} \cdot C, \]

which will always indicate an angle less than \( 180^\circ \), as long as \( C \) remains less than 1, or as long as the liquid added is less dense than glass. There must, therefore, have been a slight inaccuracy in the observation mentioned by M. Laplace, that the surface of mercury contained in a glass tube becomes hemispherical under water; and if we could obtain an exact measurement of the angle assumed by the mercury under these circumstances, we should at once be able to infer from it the comparative attractive density of water and glass, which has not yet been ascertained, although it might be deduced with equal ease from the comparative height of a portion of mercury contained in two unequal branches of the same tube, observed in the air and under water. The cosine is more exactly \( -735 \), in the case of the contact of glass and mercury, and

\[ \frac{2A}{B} = 265, \]

whence

\[ \frac{A}{B} = \frac{1}{755}, \]

which is a disproportion somewhat greater than that of the specific gravities; but it must probably vary with the various kinds of glass employed.

There is also another mode of determining the angle of contact of a solid with a single liquid, which has been ingeniously suggested by M. Laplace: it is derived from the principle of the invariability of the curvature of the surface at a given elevation; and its results agree with those which we have already obtained, except that it does not appear to be applicable to the case of more than one liquid in contact with the given solid. Supposing a capillary tube to be partially inserted into a liquid, if we imagine it to be continued into a similar tube of the liquid, leaving a cylinder or column of indefinite length in the common cavity, then the action of either tube upon the liquid immediately within it will have no tendency either to elevate or to depress the column; but the attraction of the portion of the tube above the column will tend to raise it with a certain force, and the lower end of the tube will exert an equal force upon the portion of the column immediately below it; and this double force will only be opposed by the single attraction of the liquid continuation of the tube drawing down the column above it, so that the weight of the column suspended will be as the excess of twice the attractive force of the solid above that of the liquid. Now supposing two plates of the solid in question to approach very near each other, so that the elevation may be very great in comparison with the radius of curvature of the surface, which in this case may be considered as uniform, the weight suspended will then be simply as Cohesion.

The elevation, which will be the measure of the efficient attractive force, and will vary with it, if we suppose the nature of the solid to vary, the radius of curvature varying in the inverse ratio of the elevation; but the radius of curvature is to half the distance of the plates, as unity to the numerical sine of half the angular extent of the surface, or the cosine of the angle of the liquid, so that this cosine will be inversely as the radius, or directly as the elevation, that is, as the efficient attractive force, which is expressed by \( 2A - B \) becoming \( -1 \) when \( A \) vanishes, and consequently being always equal to \( \frac{2A - B}{B} \), as we have already found from other considerations. If we wished to extend this mode of reasoning to the effect of a repulsive force counteracting the cohesion, we should only have to suppose the diameter of the tube diminished on each side by the interval which is the limit of the repulsion, since beyond this the repulsion could not interfere with the truth of the conclusions, for want of any particles situated in the given directions near enough to each other to exhibit it; and within the stratum more immediately in contact with the solid, the forces may be supposed to balance each other by continuing their action along its surface until they are opposed by similar forces on the outside of the tube or elsewhere; and indeed such a repulsive stratum seems in many cases to be required for affording a support to the extended surface of the liquid when the solid does not project beyond it. It may also be shown, in a manner nearly similar, by supposing the column to be divided into concentric cylinders, that the superficial curvature of the liquid will not affect the truth of the conclusion.

Sect. III.—Forms of Surfaces of Simple Curvature.

We may now proceed upon the principle admitted by all parties, of a hydrostatic pressure proportional to the curvature of the surface of the liquid, which is equivalent to a uniform tension of that surface, and which either supports the weight or pressure of the fluid within its concavity, or suspends an equal column from its convexity, whether with the assistance of the pressure of the atmosphere, or more simply by the immediate effect of the same cohesion that is capable of retaining the mercury of the barometer in contact with the summit of the tube; and on this foundation we may investigate the properties of the forms assumed by the surface, first considering the cases of simple curvature which are analogous to some of the varieties of the elastic curve, and next those of the surfaces having an axis of revolution, which will necessarily involve us in still more complicated calculations.

A. Let the height of the curve at its origin be \( a \), the horizontal abscissa \( z \), the vertical ordinate \( y \), the sine of the angular elevation of the surface \( s \), the versed sine \( v \), and the rectangle contained by the ordinate and the radius of simple curvature \( r \); then the area of the curve will be \( rs \), and \( y = \sqrt{(a^2 + 2rs)} \).

The fluxion of the curve \( z \) is jointly as the radius of curvature \( \frac{y}{r} \), and as the fluxion of the angle of elevation, which we may call \( w \), or \( dz = \frac{r}{y} dw \), and \( dx = \sqrt{(1 - s^2)} \) \( dz \)

\[ = \sqrt{(1 - s^2)} \frac{r}{y} dw; \]

but \( \sqrt{(1 - s^2)} dw = ds \), consequently \( dx = \frac{r}{y} ds \), and \( ydx \), the fluxion of the area, becomes equal to \( rds \), and the area itself to \( rs \). In order to find \( y \), we have \( dy = s dz = s r y \frac{dy}{y} = r \frac{dy}{y} \); whence \( y \frac{dy}{y} = r \frac{dy}{y} \), and \( y^2 = 2rv + av \), \( y \) becoming equal to \( a \) when \( v \) vanishes.

It may also be immediately inferred, that the area of the curve must vary as the sine of the inclination of the surface, from considering that, according to the principles of the resolution of forces, the tension being uniform, the weight which it supports must be proportional to that sine.

Scholium. The value of \( r \) for water at common temperatures is about one hundredth of a square inch, according to the results of a variety of experiments compared by Dr Young; or, more correctly, if we adopt the more recent measurement of M. Gay Lussac, \( 0.115 \); for alcohol M. Gay Lussac's experiments give \( r = 0.047 \), and for mercury \( r = 0.051 \). Dr Young had employed \( 0.05 \) for mercury, a number which appears to be so near the truth that it may still be retained, for the greater convenience of calculation. Hence, in a very wide vessel, the smallest ordinate \( a \) being supposed evanescent, and \( y = \sqrt{(2rv)} = 1.516 \sqrt{v} \), the height of the water rising against the side of the vessel, when \( v = 1 \), will be \( 1.516 \); and the utmost height at which the water will adhere to a horizontal surface, raised above its general level, will be \( 2 \sqrt{r} = 2.145 \). For mercury, \( y \) becomes in these circumstances \( \sqrt{(0.0102v)} = 1.01 \sqrt{v} \), and if \( s = 735 \), \( v = 322 \), and the depression of the surface in contact with a vertical surface of glass becomes \( 0.0573 \); and again, when \( v = 1735 \), as in the case of a large portion of mercury lying on a plate of glass, the height \( y = 133 \); and if the glass had no attraction at all for mercury, \( v \) would become 2, and the height \( 1.328 \). The actual tension of the surface of mercury is to that of water as \( 0.051 \times 13.6 \), or \( 0.0936 \) to \( 0.115 \); that is, a little more than six times as great; while the angle of contact of mercury with glass, which is more attractive than water, would have led us to expect a disproportion somewhat greater. If we take a mean of these results, and estimate it at seven times, the value of \( \sqrt{r} \) will be reduced by immersing mercury standing on glass into water in the ratio of \( \frac{6}{7} \times \sqrt{13.6} \), since the buoyant effect of the water increases the value of \( r \), so that \( \sqrt{(2r)} \) will be \( 0.09 \), and the angle approaching to \( 180^\circ \), the height will be about \( 1.27 \).

B. When the curve is infinite, the absciss \( x \) becomes \( \frac{1}{2} \sqrt{r} - \sqrt{(4r - yy)} + \sqrt{(4r - y^2)} \), reckoning from the greatest ordinate \( y = 2 \sqrt{r} \); and the excess of the length of the curve above the absciss is \( 2 \sqrt{r} - \sqrt{(4r - y^2)} \).

In this case, \( a \) being \( 0 \), \( y^2 = 2rv \); but \( \frac{dx}{dy} = \frac{1-v}{s} = \frac{1-v}{\sqrt{(2v - ve)}} = \frac{2r - 2rv}{\sqrt{(4r - 2rv)} \sqrt{(2rv)}} = \frac{2r - 2rv}{\sqrt{(4r - 2rv)} \sqrt{(2rv)}} \), and, by the common rules for finding fluents, \( x = \frac{2r}{4 \sqrt{r}} \).

HL \( \frac{2 \sqrt{r} - \sqrt{(4r - yy)} + \sqrt{(4r - y^2)}}{\sqrt{(4r - yy)} + \sqrt{(4r - y^2)}} \), which vanishes when \( y = 2 \sqrt{r} \); and for the length of the curve, since \( \frac{dz}{dy} = \frac{1}{s} = \frac{1}{\sqrt{(2v - ve)}} = \frac{2r}{\sqrt{(4r - yy)} \sqrt{(2rv)}} \); subtracting the former fluxional co-efficient from this, we have \( \frac{ydy}{\sqrt{(4r - yy)}} \) for the fluxion of the difference; and the fluent of this is \( -\sqrt{(4r - y^2)} \).

Corollary I. Hence, where the curve is vertical, we find \( x = 5.528 \sqrt{r} \); and where the inclination amounts to a second, \( x = 11.28 \sqrt{r} \); for example, in the case of water, \( \sqrt{r} \) being \( 1.072 \), the latter value of \( x \) will become \( 1.21 \), and the former \( 0.56 \); so that the surface must be considered as sensibly inclined to the horizon at the distance of more than an inch from the vessel, but scarcely at an inch and a half; and for mercury these distances will be two thirds as great. This circumstance must not be forgotten when mercury is employed for an artificial horizon, although, where the vessel is circular, the surface becomes horizontal at its centre; and in other parts the inclination is materially affected by the double curvature.

Corollary 2. The form of the surface coincides in this case with that of an elastic bar or a slender spring of infinite length, supposed to be bent by a weight fixed to its extremity; since the curvature of such a spring must always be proportional to its distance from the vertical line passing through the weight. We may therefore deduce from this proposition the correction required for the length of a pendulum like Mr Whitehurst's, consisting of a heavy ball suspended by a very fine wire. Now the radius of curvature of the spring is \( \frac{Moa}{12fy} \) (Art. Bridge, Prop. G); the modulus of elasticity, of which \( M \) is the weight, being for iron or steel about \( 10,000,000 \) feet in height; and since eighty inches of the wire weighed three grains, the thickness \( a \), supposing it to have been one third or two fifths of the breadth, as is usual in wire flattened for hair springs, must have been about \( \frac{1}{12} \) of an inch; the weight \( f \) was 12,251 grams; and the weight of \( M \) of ten million feet must have been \( \frac{3}{50} \times 12 \times 10000000 \) grains; consequently,

\[ \frac{Moa}{12fy} = \frac{3 \times 10000000}{80 \times 12251 \times 375 \times 375y} = \frac{1000}{12251 \times 375y} = \frac{1}{4594y} \]

which is analogous to \( \frac{r}{y} \) in these propositions; consequently \( \sqrt{r} = \frac{1}{12} \); and the whole value of \( \sqrt{(4r - y^2)} \) from \( y = 2 \sqrt{r} \) to \( y = 0 \), is \( \frac{1}{12} \) of an inch. Now, supposing the spring to have been firmly fixed at the axis of vibration, the excess of its length above the ordinate will always be measured by \( 2 \sqrt{r} - \sqrt{(4r - y^2)} \); but \( \sqrt{(4r - y^2)} = \sqrt{(4r - 2rv)} = \sqrt{r} \sqrt{(4 - 2v)} \), which is the chord of the supplement of the arc of vibration in the circle of which the radius is \( \sqrt{r} = \frac{1}{12} \); and the ball will be drawn above its path to a height equal to the distance between this circle and another of twice the diameter, touching it at its lowest point; but a perpendicular falling from this point on the wire would always be found in a circle twice as much curved as the first circle; and if it were made the centre of vibration, the ball would always be raised twice as far above its original path as the distance between the first circle and the second, which is the measure of the effect of the curvature; so that the pendulum must be supposed to be shortened half as much as this; that is, in the present instance, \( \frac{1}{24} \) of an inch. If, however, the spring remained, in Mr Whitehurst's experiments, at liberty to turn within the clip, and was firmly fixed at a considerable distance above, the variation of the length must have been only that which belongs to half of the arc of vibration; that is, one fourth as great as in the former case, since the versed sine is initially as the square of the arc; but since it would affect the spring both above and below the clip, it would be doubled from this cause, and would amount to \( \frac{1}{12} \) of an inch; so that the true correction would be liable to vary from \( 0.0735 \) to \( 0.0367 \), according to the mode of fixing the wire. But since this error must have affected both Mr Whitehurst's pendulums in an equal degree, and the result was deduced from the dif- Cohesion, and not the proportion of the lengths, it is free from any inaccuracy on this account. The calculation, however, sufficiently proves the necessity of attending to the effect of different modes of fixing the spring, in order that no variation may be made in the different experiments compared without a proper correction. The elasticity of such a wire as Mr Whitehurst employed, could not have produced any sensible error, by co-operating with the force of gravitation, since it did not amount to one two-millionth part of the weight of the ball.

C. The relation of the ordinate and absciss may be generally expressed by means of an infinite series.

When the curve is concave towards the absciss throughout its extent, the ordinate may be compared with the lengths of hyperbolic and elliptic arcs, as Maclaurin has shown with respect to the elastic curve (Fluxions, § 928): but his solution fails in the more ordinary cases of the problem; and even where it is applicable, the calculation is very little facilitated by it. Segner has made use of two different forms of infinite series, each having its peculiar advantages with respect to convergence in particular cases; and other forms may be found, which will sometimes be more convenient than either of these. The value of the cotangent \( \frac{dx}{dy} \) being in general \( \frac{1-v}{\sqrt{(2v-ve)}} \)

\[ = \frac{2r-2rv}{\sqrt{(4r-2re)\sqrt{(2re)}}} = \frac{2r-yy+aa}{\sqrt{(yy-aa)}} \cdot \frac{1}{\sqrt{(yy-aa)}} \]

we may retain either of these fractions, and expand the other by means of the binomial theorem.

1. In the first place, making \( 4r+aa=c^2 \), we have

\[ (c^2-y^2)^{-\frac{1}{2}} = \frac{1}{c} + \frac{1}{2} \cdot \frac{y^2}{c^3} + \frac{3}{4} \cdot \frac{y^4}{c^5} + \frac{5}{8} \cdot \frac{y^6}{c^7} + \ldots \]

and \( \frac{dx}{dy} = \frac{2r+aa}{\sqrt{(yy-aa)}} \left( \frac{1}{c} + \frac{1}{2} \cdot \frac{y^2}{c^3} + \frac{3}{4} \cdot \frac{y^4}{c^5} + \frac{5}{8} \cdot \frac{y^6}{c^7} + \ldots \right) \]

Now, in order to find the fluents of the separate terms, we have first \( \int \frac{dy}{\sqrt{(yy-aa)}} = HL(y+\sqrt{(yy-aa)}) \);

and calling this logarithm L,

\[ \int y^2 \cdot \frac{dy}{\sqrt{(yy-aa)}} = \frac{y^2}{2} \sqrt{(y^2-a^2)} + \frac{a^2}{2} L; \]

\[ \int y^4 \cdot \frac{dy}{\sqrt{(yy-aa)}} = \frac{y^4}{4} - \frac{3a^2y^2}{8} \sqrt{(y^2-a^2)} + \frac{3a^4}{8} L; \]

\[ \int y^6 \cdot \frac{dy}{\sqrt{(yy-aa)}} = \frac{y^6}{6} - \frac{5a^2y^4}{24} + \frac{5a^4y^2}{16} \sqrt{(y^2-a^2)} + \frac{5a^6}{16} L; \]

whence by substitution we have

\[ x = \frac{2r+aa}{4r+aa} L + \left( \frac{2r+aa}{2(4r+aa)^2} + \frac{1}{4r+aa} \right) \left( \frac{y^2}{2} \sqrt{(y^2-a^2)} + \frac{a^2}{2} L \right) + \ldots \]

2. If we reduce \( \frac{1}{\sqrt{(yy-aa)}} \) into a series, we have

\[ \left( \frac{1}{yy} \right)^{-\frac{1}{2}} = 1 + \frac{1}{2} \cdot \frac{a^2}{y^2} + \frac{3}{4} \cdot \frac{a^4}{y^4} + \frac{5}{8} \cdot \frac{a^6}{y^6} + \ldots \]

and \( \frac{dx}{dy} = \frac{2r+aa-yy}{\sqrt{(yy-aa)}} \cdot \left( \frac{1}{y} + \frac{1}{2} \cdot \frac{a^2}{y^3} + \frac{3}{4} \cdot \frac{a^4}{y^5} + \ldots \right) \).

Then, for the fluents,

\[ \int \frac{ydy}{\sqrt{(yy-aa)}} = \sqrt{(x^2-y^2)}; \int \frac{dy}{\sqrt{(yy-aa)}} = HL(y+\sqrt{(yy-aa)}) = L; \]

\[ \int \frac{dy}{y^2 \sqrt{(yy-aa)}} = \frac{\sqrt{(yy-aa)}}{2c yy} - \frac{1}{2cc} L; \]

\[ \int \frac{dy}{y^3 \sqrt{(yy-aa)}} = \left( \frac{1}{4c^2y^2} + \frac{3}{8c^2y^4} \right) \sqrt{(c^2-y^2)} + \frac{3}{8c^2L}; \]

\[ \int \frac{dy}{y^4 \sqrt{(yy-aa)}} = \left( \frac{1}{6c^2y^2} + \frac{5}{24c^2y^4} - \frac{5}{16c^2y^6} \right) \sqrt{(c^2-y^2)} - \frac{5}{16c^2L}; \]

and by combining these fluents we obtain a second series for \( x \).

3. These series may be employed with advantage where the initial ordinate is very small, the one being more convenient for the upper, and the other for the lower part of the curve; but where the elevation \( a \) is more considerable, the form of the curve will be more readily determined by means of fluents derived from circular arcs.

Beginning with the expressions \( \frac{dx}{dy} = \frac{1-v}{\sqrt{(2v-ve)}} \) and \( y^2 = a^2 + 2rv \), we may seek for a value of \( x \) in terms of \( v \); and since \( 2ydy = 2rdv \), \( dy = \frac{r}{y} dv = \frac{rdv}{\sqrt{(aa+2rv)}} \)

and \( dx = \frac{1-v}{\sqrt{(2v-ve)}} \cdot \sqrt{(aa+2rv)} \). The binomial \( (aa+2rv)^{-\frac{1}{2}} \) may then be expanded into a series of integral powers of \( v \), and the fluents may be found by means of the equations \( \int \frac{dv}{\sqrt{(2v-ve)}} = \int \frac{dv}{s} = w \), the arc of which \( v \) is the versed sine: \( \int \frac{rdv}{s} = s-w; \)

\[ \int \frac{v^2dv}{s} = \left( \frac{v^2}{2} - \frac{3}{4} \cdot \frac{v^4}{8} + \frac{3}{8} \cdot \frac{v^6}{16} \right) + \frac{3}{8} \cdot \frac{v^8}{32} + \ldots \]

whence \( \frac{dx}{dy} = \left( 1 - \frac{yy-aa}{2r} \right) \sqrt{(yy-aa)} \times \left( 1 + \frac{1}{4} \cdot \frac{yy-aa}{2r} + \frac{3}{8} \cdot \frac{(yy-aa)^2}{2r} + \ldots \right) \); the fluxions belonging to the series \( (y^2-a^2)^{-\frac{1}{2}} dy \), \( (y^2-a^2)^{\frac{1}{2}} dy \); and the fluents of these are

\[ HL(y+\sqrt{(y^2-a^2)}) = L; \frac{1}{2} y \sqrt{(y^2-a^2)} - \frac{1}{2} a^2 L; \]

\[ \left( \frac{1}{2} \sqrt{(y^2-a^2)} + \frac{a^2}{2} y \sqrt{(y^2-a^2)} + \frac{3a^4}{8} L; \right. \]

which afford a result somewhat resembling that which is deduced from the first method. 5. We may also express \( x \) in a series of integral powers of \( y \) only, if we suppose it to begin at some point in which the curve is inclined to the horizon, where the height is \( p \), calling it at other points \( p + y \); and making \( \frac{dx}{dy} = r \)

\[ = a + by + cy^2 + \ldots; \text{we have then } x = \beta + ay + \frac{1}{2}by^2 + \frac{1}{3}cy^3 + \ldots, \text{and the area } \int (p + y) \, dx = \gamma + pay + \frac{1}{2}py^2 + \ldots + \frac{1}{3}cy^3 + \frac{1}{4}cy^4 + \ldots, \text{which} \]

must be equal to \( rs \) (Prop. A.): but \( s = \sqrt{(dx^2 + dy^2)} \)

\[ = \sqrt{(1 + rr)}, \text{which may be developed by means of the} \]

Taylorian theorem \( \varphi(A + H) = \varphi A + \frac{d(\varphi A)}{dA} H \)

\[ + \frac{d^2(\varphi A)}{dA^2} \cdot \frac{H^2}{2} + \ldots, \text{taking } A = a, \text{and } H = by \]

\[ + cy^2 + \ldots, \text{whence } H^2 = b^2y^2 + 2bey^3 + (2bd + c^2)y^4 + \ldots \]

\( H^3 = b^3y^3 + 3b^2ey^4 + \ldots; \text{consequently } rs = rpa \ldots \]

\[ = \sqrt{(1 + aa)} + \frac{r}{da} \cdot d \sqrt{(1 + aa)} (by + cy^2 + dy^3 + \ldots) \]

\[ + \frac{r}{2da^2} \cdot d^2 \sqrt{(1 + aa)} \cdot (b^2y^2 + 2bey^3 + (2bd + c^2)y^4 + \ldots) \]

\[ + \frac{r}{2da^3} \cdot d^3 \sqrt{(1 + aa)} (b^3y^3 + 3b^2ey^4 + \ldots) \]

\[ = \gamma + pay + (\frac{1}{2}pb + \frac{1}{3}a)y^2 + \ldots; \text{and hence by comparing} \]

the homologous terms, we find \( \gamma = \sqrt{(1 + aa)} \cdot \frac{r}{da} \cdot d \)

\[ \frac{1}{\sqrt{(1 + aa)}} \cdot b = pa = \frac{-ra}{(1 + aa)^{\frac{3}{2}}} \cdot b, \text{and } b \]

\[ = -\frac{p}{r}(1 + aa)^{\frac{3}{2}}; \text{and in a similar manner we may de-} \]

termine the subsequent coefficients; but the calculation is somewhat laborious, and has no particular advantages.

6. We may still more readily obtain a similar series for \( y \) in terms of the powers of \( x \) with constant coefficients;

\[ \text{calling } \frac{dy}{dx} t, \text{and making } t = bx + cx^2 + dx^3 + \ldots \text{whence} \]

\( y = a + bx^2 + \frac{1}{2}cx^4 + \frac{1}{3}dx^5 + \ldots, \text{and the area } \int y \, dx \)

\[ = ax + \frac{1}{2}bx^2 + \frac{1}{3}cx^3 + \ldots = rs = \sqrt{(1 + u)} = rt \]

\[ (1 - \frac{1}{2}t^2 + \frac{3}{4}t^4 - \frac{5}{8}t^6 + \ldots). \text{But } t = bx^2 + \frac{1}{2}cx^3 + \frac{1}{3}dx^4 + \ldots; \text{hence we have the equation} \]

\[ \frac{ax}{r} + \frac{1}{2}bx^2 + \frac{1}{3}cx^3 + \ldots = \]

\[ bx + cx^2 + dx^3 + \ldots \]

\[ - \frac{1}{2}bx^2 - \frac{1}{3}cx^3 - \ldots \]

\[ + \frac{1}{3}bx^3 + \ldots; \text{consequently } \]

\( b = \frac{a}{r}, c = \frac{1}{23r} b + \frac{1}{2} b^2, \text{and } d = \frac{1}{45r} c + \frac{1}{3} b^2 c - \frac{1}{3} b^3. \)

It is the less necessary to enter into any further detail of these results, as we have a table calculated by Segner, with his son's assistance, which is sufficient to afford us a general idea of the forms of the curve in different circumstances. The unit of this table is the quantity \( \sqrt{r} \), which Segner calls the modulus of capillary attraction, and which for water is \( 1072 \) inch. The table begins with the extreme ordinate, where the curve is vertical: we have then the least ordinate, \( a \); the greatest ordinate, where the curve again becomes horizontal; and the absciss corresponding to the extreme ordinate and to the greatest ordinate.

| Extreme Ordinate | Least Ordinate | Greatest Ordinate | Greatest Abscissa | Terminal Abscissa | |------------------|----------------|-------------------|------------------|------------------| | 100\(\sqrt{r}\) | 99-99 | 100-01 | 0-01 | 0-000001 | | 90 | 89-99 | 90-01 | 0-01 | 0-000002 | | 80 | 79-99 | 80-01 | 0-01 | 0-000003 | | 70 | 69-99 | 70-01 | 0-01 | 0-000004 | | 60 | 59-99 | 60-02 | 0-02 | 0-000007 | | 50 | 49-98 | 50-02 | 0-02 | 0-00001 | | 45 | 44-98 | 45-02 | 0-02 | 0-00002 | | 40 | 39-97 | 40-02 | 0-02 | 0-00003 | | 35 | 34-97 | 35-03 | 0-03 | 0-00004 | | 30 | 29-96 | 30-03 | 0-03 | 0-00006 | | 25 | 24-96 | 25-04 | 0-04 | 0-0001 | | 20 | 19-95 | 20-05 | 0-05 | 0-0002 | | 15 | 14-93 | 15-07 | 0-07 | 0-0004 | | 10 | 9-90 | 10-10 | 0-10 | 0-001 | | 9 | 8-89 | 9-11 | 0-11 | 0-002 | | 8 | 7-87 | 8-12 | 0-12 | 0-003 | | 7 | 6-85 | 7-14 | 0-14 | 0-004 | | 6 | 5-83 | 6-16 | 0-16 | 0-007 | | 5 | 4-79 | 5-19 | 0-21 | 0-01 | | 4 | 3-74 | 4-24 | 0-24 | 0-02 | | 3 | 2-64 | 3-32 | 0-32 | 0-06 | | 2 | 1-41 | 2-45 | 0-45 | 0-06 | | 1-9 | 1-27 | 2-37 | 0-37 | 0-07 | | 1-8 | 1-11 | 2-29 | 0-29 | 0-08 | | 1-7 | 0-94 | 2-21 | 0-21 | 0-09 | | 1-6 | 0-75 | 2-13 | 0-13 | 0-10 | | 1-5 | 0-50 | 2-06 | 0-06 | 0-11 | | 1-47 | 0-40 | 2-04 | 0-04 | 0-12 | | 1-445 | 0-30 | 2-02 | 0-02 | 0-13 | | 1-428 | 0-20 | 2-01 | 0-01 | 0-14 | | 1-418 | 0-10 | 2-003 | 0-003 | 0-15 | | 1-4142 | 0-01 | 2-000 | 0-000 | 0-16 | | 1-4142 | 0-001 | 2-000 | 0-000 | 0-17 | | 1-4142 | 0-00001 | 2-000 | 0-00001 | 0-18 | | 1-4142 | 0-000001 | 2-000 | 0-000001 | 0-19 |

It may be observed that the last six values of the least ordinate are in geometrical progression, while the absciss increases in arithmetical progression; the difference of the abscissas 2, 3, being the hyperbolical logarithm of 10, which is the common multiplier of the ordinates. Although the table appears to be generally accurate, yet we cannot always depend on the last figures. Thus the ultimate difference of the two last columns is made 43, while it ought to be 53 (Prop. B. Cor. 1). It is scarcely necessary to remark, that if we look in the fourth column for half the distance between two parallel planes of glass, in a vertical position, the first and second columns will give us the height to which water will rise between them, where it touches the glass, and in the middle of the interval.

Sect. IV.—Surfaces of Double Curvature.

When the liquid is contained in a tube, or when it forms itself spontaneously into a drop having an axis of revolution, it becomes necessary to consider the effect of the Cohesion.

tension in a direction transverse to that of the principal section; since the curvature will cause it to exhibit an equal pressure, whatever the direction of the section to which it belongs may be; and the curvatures of the sections perpendicular to each other will either co-operate with or counteract each other, according as the convexities of both are on the same side, or on the opposite sides, of the surface. But the simple consideration of the tension supporting the weight of the parts below, or the equivalent pressure in a contrary direction, will at once afford us the equations necessary for the solution of the problem, without any immediate reference to the curvature in question.

D. The form of a surface of revolution may be determined by means of an infinite series.

The fluxion of the weight or mass of the parts contained within the cylindrical surface, of which \( x \) is the radius or absciss, and \( y \) the ordinate, being always proportional to \( ydx \), and the fluent to \( fyxzdx \); and the extent of the circumference supporting it varying also as \( x \), and the contractile force being diminished when reduced to the direction of gravitation, in the ratio of the radius unity to the sine of the elevation \( z \), it will always be proportional to \( xz \); so that we have the general equation \( fyxzdx = mxz \). Now if we suppose \( y \) incomparably greater than \( x \), and the surface extremely minute, the variation of \( y \) may be neglected, and we have in this case \( fyxzdx = \frac{1}{2}yx^2 \); and supposing also \( s = 1 \), and the curve vertical, \( \frac{1}{2}yx^2 = mxz \), and \( \frac{1}{2}yx = m \), becoming also equal to the radius of curvature. But it is easy to perceive that the height \( y \) must be twice as great, for any value of \( x \), as in the case of a simple curvature, since each portion of the circumference has here only to support a wedge, which is only half as heavy as a parallelopiped of the same height; so that \( \frac{1}{2}yx \) will be equal to \( yx \) in Proposition A, and \( m = r \).

In order to obtain a series for finding \( y \) from the equation \( fyxzdx = mxz \), we may put the tangent \( t = \frac{dy}{dz} = bx + cx^3 + dx^5 + ... \), whence \( y = a + \frac{1}{2}bx^2 + \frac{1}{4}cx^4 + \frac{1}{6}dx^6 + ... \); and the value of \( s = \sqrt{1 + t^2} \) being expanded into a series, as in Proposition C, n 6, calling \( \frac{1}{m} \) or \( \frac{1}{r} \), \( q \), we find \( s = \frac{2}{x}fyxzdx = bx + cx^3 + dx^5 + cx^7 + ... - \frac{1}{2}b^2x^2 - \frac{1}{3}b^3x^3 - \frac{1}{4}b^4x^4 + \frac{1}{5}b^5x^5 + ... + \frac{1}{6}b^6x^6 + ... \)

\( = \frac{1}{2}qax + \frac{1}{24}qbx^2 + \frac{1}{46}qcx^4 + \frac{1}{68}qdx^6 + ... \);

consequently \( b = \frac{1}{2}qa = \frac{a}{2r} \), and \( a = \frac{2b}{q} = 2rb \); and by continuing the calculation, and reducing the values, we find

\[ \begin{align*} c &= \frac{1}{24}qb + \frac{1}{2}b^2 \\ d &= \frac{1}{246}q^2b + \frac{10}{246}qb^3 + \frac{3}{24}b^5 \\ e &= \frac{1}{2468}q^3b + \frac{82}{2468}q^2b^3 + \frac{105}{2468}qb^5 + \frac{15}{246}b^7 \\ f &= \frac{1}{246810}q^4b + \frac{652}{246810}q^3b^3 + \frac{2645}{246810}q^2b^5 + ... \end{align*} \]

We may here observe, that the numerical co-efficients of the highest powers of \( b \) form the series

\[ \begin{align*} &\frac{1}{24}, \frac{3}{246}, \frac{35}{2468}, \frac{357}{246810}, \frac{6}{246810}, \frac{3579}{246810}, \frac{8}{246810}; \end{align*} \]

the ratio of the successive terms of both continually approaching to equality, and those of the next in order the series

\[ \begin{align*} &\frac{3}{24}, \frac{2}{246}, \frac{35}{2468}, \frac{4}{2468}, \frac{357}{2468}; \end{align*} \]

but the laws of the numerical co-efficients in general appear to be wholly incapable of being reduced to any simple form. It will be convenient for calculation to form tables of the logarithmic values of these co-efficients, which may be continued by means of successive differences, for as many terms as are requisite for any practical purpose. The indices, with lines drawn over them, are to be considered as negative numbers.

Logarithmic Co-efficients of the Value of the Sine.

\[ \begin{align*} s &= (0.000000 + \frac{22}{24}.10 \ldots \ldots q^{1420}) \\ &+ \frac{1}{24}.0969100 q^2 + \frac{24}{246}.0 \ldots \ldots q^{1320} \\ &+ \frac{3}{246}.7166987 q^2 + \ldots \ldots b^3x^3 \\ &+ \frac{4}{246}.0354574 q^2 + (3.8927900 q^2) \\ &+ \frac{6}{246}.1323674 q^2 + \frac{3}{246}.5337080 q^2 \\ &+ \frac{8}{246}.0531861 q^2 + \frac{4}{246}.8558231 q^2 \\ &+ \frac{11}{246}.8278768 q^2 + \frac{5}{246}.9581885 q^2 \\ &+ \frac{13}{246}.4776288 q^2 + \frac{6}{246}.9727959 q^2 \\ &+ \frac{15}{246}.0182362 q^2 + \frac{7}{246}.82 \ldots \ldots q^2 \\ &+ \frac{18}{246}.4619336 q^2 + \frac{8}{246}.57 \ldots \ldots q^2 \\ &+ \frac{21}{246}.8184809 q^2 + \frac{9}{246}.27 \ldots \ldots q^2 \\ &+ \ldots \ldots bx + \frac{11}{246}.97 \ldots \ldots q^2 \\ &+ \frac{2}{246}.3187587 q^2 + \frac{12}{246}.77 \ldots \ldots q^2 \\ &+ \frac{3}{246}.6375174 q^2 + \ldots \ldots b^2x^2 \\ &+ \frac{4}{246}.6482413 q^2 + (3.5917600 q^2) \\ &+ \frac{5}{246}.4094937 q^2 + \frac{3}{246}.4368580 q^2 \\ &+ \frac{6}{246}.1456895 q^2 + \frac{4}{246}.9595058 q^2 \\ &+ \frac{8}{246}.7008651 q^2 + \ldots \ldots b^2x^2 \\ &+ \frac{9}{246}.1510234 q^2 + \frac{3}{246}.3576707 q^2 \\ &+ \frac{11}{246}.002009 q^2 + \frac{3}{246}.3498514 q^2 \\ &+ \frac{13}{246}.7811595 q^2 + \ldots \ldots b^2x^2 \\ &+ \frac{15}{246}.9774 \ldots \ldots q^2 + (3.1657913 q^2) \\ &+ \frac{16}{246}.1026 \ldots \ldots q^2 + \ldots \ldots b^2x^2 \\ &+ \frac{18}{246}.160 \ldots \ldots q^2 + \ldots \ldots b^2x^2 \\ &+ \frac{20}{246}.16 \ldots \ldots q^2 + \ldots \ldots b^2x^2 \end{align*} \] Logarithmic Co-efficients of the Value of the Ordinate \( y \).

\[ y = \left[ \frac{2b}{q} + 0.6989700 + \frac{2}{q} \cdot 7958800 + \frac{2}{q} \cdot 5337080 q^2 + \frac{2}{q} \cdot 948500 q^3 + \frac{2}{q} \cdot 9350043 q^4 + \frac{2}{q} \cdot 1313165 q^5 + \frac{2}{q} \cdot 1769159 q^6 + \frac{2}{q} \cdot 1323674 q^7 + \frac{2}{q} \cdot 9385474 q^8 + \frac{2}{q} \cdot 5.1323674 q^9 + \frac{2}{q} \cdot 4.1323674 q^{10} + \frac{2}{q} \cdot 10.9740048 q^{11} + \frac{2}{q} \cdot 12.6817488 q^{12} + \frac{2}{q} \cdot 14.2735087 q^{13} + \frac{2}{q} \cdot 17.7629636 q^{14} + \frac{2}{q} \cdot 19.1609036 q^{15} + \frac{2}{q} \cdot 21.0969100 q^{16} + \frac{2}{q} \cdot 2.5406074 q^{17} + \frac{2}{q} \cdot 3.6482413 q^{18} + \frac{2}{q} \cdot 4.5486749 q^{19} + \frac{2}{q} \cdot 5.2918175 q^{20} + \frac{2}{q} \cdot 7.9049851 q^{21} + \frac{2}{q} \cdot 8.4062959 q^{22} + \frac{2}{q} \cdot 10.8090500 q^{23} + \frac{2}{q} \cdot 11.235822 q^{24} + \frac{2}{q} \cdot 13.3576 q^{25} + \frac{2}{q} \cdot 15.5176 q^{26} + \frac{2}{q} \cdot 17.607 q^{27} + \frac{2}{q} \cdot 19.637 q^{28} + \frac{2}{q} \cdot 21.60 q^{29} + \frac{2}{q} \cdot 23.5 q^{30} + \ldots \right] b^x \]

Hence if

\[ b = s = 0.3073 \cdot 7248 + 0.255 \]

\[ = 1.147 \cdot 7237 + 0.265 \]

\[ = 4.160 \cdot 7160 + 0.254 + 0.060 + [0.025] \]

\[ = 1.503 \cdot 7211 + 0.240 + 0.041 + [0.010] \]

\[ = 5.776 \cdot 7345 + 0.122 + 0.024 + 0.007 + [0.003] \]

\[ = 14.004 \cdot 7449 + 0.040 + 0.008 + 0.002 + [0.001] \]

It appears, upon inspection of this table, that the co-efficients of \( bx \) alone always determine \( \frac{3}{4} \) of the value of the quantity required, and these are easily calculated with perfect accuracy, so that the error must always be far less than \( \frac{1}{20} \), and in fact the actual uncertainty never exceeds \( \frac{1}{100} \) of the whole; at least in the last four examples. The differences of M. Laplace's approximatory calculations from these results are incomparably greater, so that we cannot hesitate to consider these differences as errors.

Indeed, when we recollect that in the method employed by M. Bouvard, under M. Laplace's directions, the radius of curvature of each of the small portions into which the curve has been cut up, has been determined from the ordinate at the beginning of the portion, it is obvious that the curvature thus found must be less than the truth, and that in order to obtain any required curvature of the whole surface, the depression must be increased in the same proportion; and there is no ready way of appreciating the amount of this error. Dr Young had before attempted to avoid it, in making an estimate of the same nature, by calculating for the middle of each portion; but, from some accident, the numbers of his table, published in 1807, are generally a little too small, although the method which he then employed is nearly the same as that which M. Laplace afterwards adopted, except that for the lowest portion of the curve M. Laplace had recourse to an infinite series, applicable only to that part. The elements deduced in Nicholson's Journal for 1809, from M. Gay Lussac's experiments, which are \( r = 0.0051 \) and \( s = 0.733 \), agree better with the numbers found in M. Laplace's table, than those from which it was constructed, which were \( r = 0.005038 \) and \( s = 0.729 \); the depressions being always a little larger than the true results from the elements assumed.

The value of the ordinate \( y \) depends also principally on the first variable member of the series, although the subsequent co-efficients are not so inconsiderable as in the determination of the sine. Thus, taking \( x = 2 \), and \( b = 1.503 \), we have \( y = a + 3.1323674 b^x + 2.9385474 b^{2x} + 2.1769159 b^{3x} + 1.323674 b^{4x} + 0.9385474 b^{5x} + 0.5337080 b^{6x} + 0.2958800 b^{7x} + 0.15774 b^{8x} + 0.08556 b^{9x} + 0.0489 b^{10x} + 0.0240 b^{11x} + 0.0122 b^{12x} + 0.0060 b^{13x} + 0.0030 b^{14x} + 0.0015 b^{15x} + 0.0007 b^{16x} + 0.0003 b^{17x} + 0.0001 b^{18x} \)

which is the marginal depression, leaving \( 0.964 \) for the height of the convex portion \( y - a \). We may determine the effect of any small variations in this height, in the same manner as that of the sine of the inclination; supposing them to depend on a change of the angle of contact only, the quantity \( r \) remaining unaltered, it is obvious that \( q \) and \( x \) must retain their value, while \( y \) and \( b \) only vary, and making \( Y = Ab + 3Bb^x + \ldots = b \frac{d(y-a)}{db} \),

we have \( Y : b = d(y-a) : db \). In the present instance, we find \( Y = 0.0489 + 3 \times 0.054 + 5 \times 0.015 + \ldots = 0.79 \); and supposing, as in the example suggested by M. Laplace, the variation of the height \( y - a \) to be \( 0.00394 \), which is \( \frac{1}{100} \)th of \( Y \), that of \( b \) will be \( \frac{1}{100} \)th of \( b \), or \( 0.075 \), and the variation of the central depression \( a \), \( 0.0075 \), which is somewhat less than one fifth of the alteration in the height of the convex portion; but in smaller tubes it is obvious that the variations of the depression \(a\) might much exceed that of the height of the convex portion. Nothing can be easier or more direct than this part of the calculation; and it is remarkable that M. Laplace should have considered the awkward contrivance of building up a curve, like the arch of a bridge, with fourteen blocks on each side, as possessing any thing like an "advantage" over the series in the determination of this variation.

If we wish to find the effect of a small variation of the diameter of a tube, from \(D\) to \(D' = D'\), on the depression \(a\) of the mercury contained in it, we may use for the interpolation the formula

\[ \frac{d}{a} = \frac{10}{C} - 1, \quad C \text{ being about} \]

2-9 for tubes between 1 inch and \(4/5\)th of an inch in diameter, and being elsewhere easily deduced from the depressions already known. For variations of the cohesive power, and of its measure \(r\), we may suppose the whole of the numbers of the table to be altered in the proportion of the supposed alteration of \(\sqrt{r}\), and the change produced by restoring the diameter to its former dimensions Cohesion may then be calculated like any other interpolation. There is also a more comprehensive formula, which seems to express the depression in tubes of all sizes with great accuracy; it is this, \(a = \frac{D + 48D^{3/4}}{320}\); and it might even be possible to shorten the original calculation by a comparison of the series with the expansion of this empirical formula, if it were of any farther importance to facilitate the mode of computation. But for all practical purposes, it will be sufficient to collect the results already obtained into a comparative table, arranged in chronological order; and it is remarkable, that they are all comprehended, without any material exception, between the two values assigned to each as near the truth in Dr Young's first table, the mean of those values never differing a thousandth of an inch from the result of the more correct calculation; while the error of Lord Charles Cavendish's experiments, notwithstanding their general accuracy, sometimes amounts to nearly one hundredth. (T.Y.)

**Table of the Depression of Mercury in Glass Tubes.**

| Diameter | Observed by Dr Cavendish | Dr Young, Phil. Trans. 1803 | Laplace, 1806 | Dr Young, 1807 | Nicholson's Journal, 1809 | Laplace, 1810 | Correct Calculation | Empirical Formula | Marginal Depression, Nich. 1820 | Difference | Diameter | |----------|--------------------------|-----------------------------|--------------|---------------|-------------------------|--------------|---------------------|------------------|-----------------------------|------------|----------| | Inches | Diagr. Form. | | | | | | | | | | | | 1-00 | | | | | | | | | | | | | 0-90 | | | | | | | | | | | | | 0-80 | | | | | | | | | | | | | 0-70 | | | | | | | | | | | | | 0-60 | | | | | | | | | | | | | 0-50 | | | | | | | | | | | | | 0-45 | | | | | | | | | | | | | 0-40 | | | | | | | | | | | | | 0-35 | | | | | | | | | | | | | 0-30 | | | | | | | | | | | | | 0-25 | | | | | | | | | | | | | 0-20 | | | | | | | | | | | | | 0-15 | | | | | | | | | | | | | 0-10 | | | | | | | | | | | | | 0-05 | | | | | | | | | | | |

**COHOBATION,** a term invented by Paracelsus to denote the repeated distillation of the same liquor from the same materials.