COHESION (1842) — Encyclopaedia Britannica Historical Editions

COHESION

Volume 7 · 12,110 words · 1842 Edition

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 $ \left( a^2 - ax + \frac{1}{2}x^2 \right) $, and the fluxion of the whole contractile force will be $ dx \left( a^2 - ax + \frac{1}{2}x^2 \right) $, the fluent of which, when $ x = a $, becomes $ \frac{1}{2}a^2 $, 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 Mr 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 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

$$\frac{A}{B} \approx \frac{1}{2},$$

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

$$\frac{6}{8}C,$$

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 Mr 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 Mr 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 absciss $z$, 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 $y = \sqrt{(a^2 + 2rv)}$.

The fluxion of the curve $z$ is jointly as the radius of curvature $\frac{r}{y}$ 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 $y dx$, the fluxion of the area, becomes equal to $r ds$, and the area itself to $rs$. In order to 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}{3}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}{3}a^2 - ax + \frac{1}{2}x^2 $, and the fluxion of the whole contractile force will be $ dx \left( \frac{1}{3}a^2 - ax + \frac{1}{2}x^2 \right) $, the fluent of which, when $ x = a $, becomes $ \frac{1}{3}a^2 $, which is one third of $ a \times \frac{1}{2}a^2 $, the whole undiminished cohesion of the stratum.

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° 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.

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

\[ = \frac{2A - (B+C)}{B-C}, \text{ which is the excess of twice the density} \]

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

\[ \frac{A}{B} \approx \frac{1}{3}, \]

and the cosine $ \frac{2}{3} $, which corresponds to an angle of $ 180^\circ $; and if we add a second liquid, the expression will become

\[ \frac{6-C}{B-C}, \]

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 Mr 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 Mr 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 $ \frac{2A-B}{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.

A. Let the height of the curve at its origin be $ a $, the horizontal absciss $ z $, the vertical ordinate $ y $, the sine of the angular elevation of the surface $ s $, the versed sine $ r $, 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 + 2re)} $.

The fluxion of the curve $ z $ is jointly as the radius of curvature $ \frac{r}{y} $, 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 $ ra $. In order to find $ y $, we have $ dy = dz = s \frac{r}{y} \frac{dv}{y} $; whence $ ydy = rde $, and $ y^2 = 2rv + aa $, $ 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 Mr Gay Lussac, $ -0115 $; for alcohol Mr Gay Lussac's experiments give $ r = -0047 $, and for mercury $ r = -0051 $. Dr Young had employed $ -005 $ 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{(2re)} = -1516 \sqrt{r} $, the height of the water rising against the side of the vessel, when $ v = 1 $, will be $ -1516 $; and the utmost height at which the water will adhere to a horizontal surface, raised above its general level, will be $ 2 \sqrt{r} = -2145 $. For mercury, $ y $ becomes in these circumstances $ \sqrt{(-0102e)} = -101 \sqrt{r} $, and if $ s = -735 $, $ v = -322 $, and the depression of the surface in contact with a vertical surface of glass becomes $ -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 $ is $ -183 $; and if the glass had no attraction at all for mercury, $ v $ would become $ 2 $, and the height $ -1428 $. The actual tension of the surface of mercury is to that of water as $ -0051 : 13-6 $, or $ -06936 : -0115 $; 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{\frac{13-6}{12-6}} $ since the buoyant effect of the water increases the value of $ r $, so that $ \sqrt{(2r)} $ will be $ -03 $, and the angle approaching to $ 180^\circ $, the height will be about $ -127 $.

B. When the curve is infinite, the absciss $ x $ becomes $ \frac{1}{2} \sqrt{r} $ HL $ \frac{2\sqrt{r} - \sqrt{(4r - yy)}}{2\sqrt{r} + \sqrt{(4r - yy)}} + \sqrt{(4r - yy)} $, 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 - yy)} $.

In this case, $ a $ being $ 0 $, $ y^2 = 2rv $; but $ dx = \frac{1-v}{s} $

$ = \frac{1-v}{\sqrt{(2v - vv)}} = \frac{2r - 2rv}{\sqrt{(4r - 2rv)} \sqrt{(2rv)}} = \frac{2r - yy}{\sqrt{(4r - yy)} y} $,

and, by the common rules for finding fluents, $ x = \frac{2r}{4 \sqrt{r}} $

HL $ \frac{2\sqrt{r} - \sqrt{(4r - yy)}}{2\sqrt{r} + \sqrt{(4r - yy)}} + \sqrt{(4r - yy)} $, which vanishes when $ y = 2 \sqrt{r} $; and for the length of the curve, since $ \frac{dz}{dy} = \frac{1}{s} = \frac{2r}{\sqrt{(2v - vv)}} = \frac{2r}{\sqrt{(4r - yy)} y} $; 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 - yy)} $.

Corollary 1. Hence, where the curve is vertical, we find $ x = -5328 \sqrt{r} $; and where the inclination amounts Cohesion to a second, $ x = -1128 \sqrt{r} $; for example, in the case of water, $ \sqrt{r} $ being $ -1072 $, the latter value of $ x $ will become $ -121 $, and the former $ -056 $; 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{Maa}{12gy} $ (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}{375} $ of an inch; the weight $ f $ was 12,251 grains; and the weight of $ M $ of ten million feet must have been $ \frac{3}{50} \times 12 \times 10000000 $ grains; consequently,

$ \frac{Maa}{12gy} = \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}{\sqrt{y}} $; and the whole value of $ \sqrt{(4r - yy)} $ from $ y = 2 \sqrt{r} $ to $ y = 0 $, is $ \frac{1}{\sqrt{y}} $ 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 - yy)} $; but $ \sqrt{(4r - yy)} = \sqrt{(4r - 2rv)} = \sqrt{r} \sqrt{(4 - 2r)} $, which is the chord of the supplement of the arc of vibration in the circle of which the radius is $ \sqrt{r} = \frac{1}{\sqrt{y}} $; 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}{4} $ 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}{375} $ of an inch; so that the true correction would be liable to vary from $ -00733 $ to $ -00367 $, 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.

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)}} $

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

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

1. In the first place, making $ 4r+a^2=c^2 $, we have

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

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

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

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

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

whence by substitution we have

\[ x = \frac{2r+aa}{4r+aa} L + \left( \frac{2r+aa}{(4r+aa)^3} + \frac{1}{4r+aa} \right) \left( \frac{y^2}{2} \sqrt{(y^2-a^2)} + \frac{a^2}{2} L \right) + \ldots \]

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

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

Then, for the fluents,

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

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

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

\[ \int \frac{dy}{y^6 \sqrt{(yy-aa)}} = \left( -\frac{1}{6c^3y^3} + \frac{5}{24c^3y^5} - \frac{5}{16c^3} \right) \sqrt{(y^2-a^2)} - \frac{5}{16c^3} L; \]

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 + 2re $, we may seek for a value of $ x $ in terms of $ v $; and since $ 2ydy = 2rdx, dy = \frac{r}{y} de = \frac{rde}{\sqrt{(aa+2re)}} $

and $ dx = \frac{1-v}{\sqrt{(2v-ve)}} \cdot \sqrt{(aa+2re)} $. The binomial $ (aa+2re)^{-\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{de}{\sqrt{(2v-ve)}} = \int \frac{de}{s} = w $, the arc of which $ v $ is the versed sine:

\[ \int \frac{r^2de}{s} = \left( \frac{e}{2} - \frac{3}{4} \cdot \frac{e^3}{8} + \frac{3}{8} \cdot \frac{e^5}{16} \right) s + \frac{3}{8} \cdot \frac{e^7}{16} w; \int \frac{r^4de}{s} = \left( \frac{e^3}{3} - \frac{5}{24} \cdot \frac{e^5}{16} \cdot \frac{8}{8} \right) s - \frac{5}{16} \cdot \frac{e^7}{8w}; \int \frac{r^6de}{s} = \left( \frac{e^5}{4} - \frac{7}{86} \cdot \frac{e^7}{864} \cdot \frac{8}{8642} \right) s + \frac{7}{8642} \cdot \frac{16w}. \]

4. Another series may be obtained by the expansion of $ \sqrt{(2v-ve)} $ into $ \sqrt{(2v)} \left( 1 + \frac{1}{2}v + \frac{3}{8}v^2 + \frac{5}{16}v^3 + \ldots \right) $,

whence $ \frac{dx}{dy} = \left( 1 - \frac{yy-aa}{2r} \right) \sqrt{\frac{r}{yy-aa}} \times \left( 1 + \frac{1}{2} \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, (y^2-a^2)^{\frac{3}{2}}dy $; and the fluents of these are

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

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

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 $

\[ = a + by + cy^2 + \ldots; \text{we have then } x = \beta + ay + \frac{1}{2}by^2 + \frac{1}{3}cy^3 + \ldots, \]

and the area $ \int (p + y) \, dx = \gamma + pay + \frac{1}{2}py^2 + \ldots + \frac{1}{3}ay^3 + \frac{1}{23}b^2y^4 + \frac{1}{34}cy^4 + \ldots, $

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

\[ = \sqrt{(1 + rr)}, \text{which may be developed by means of the Taylorian theorem } \varphi(A + H) = \varphi A + \frac{d(\varphi A)}{dA} H \]

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

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

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

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

\[ + ey^4 + \ldots) \]

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

\[ + 2becy^3 (2bd + c^2)y^4 + \ldots) \]

\[ + \frac{r}{23da^3} \cdot d^3 \sqrt{(1 + aa)} \left( \frac{b^3y^3}{\sqrt{(1 + aa)}} \right) \]

\[ + 3b^2ey^4 + \ldots) \]

\[ = \gamma + pay + \left( \frac{1}{2}pb + \frac{1}{2}a \right)y^2 \]

\[ + \left( \frac{1}{2}pc + \frac{1}{2}b \right)y^3 + \ldots; \text{and hence by comparing the homologous terms, we find } \gamma = \sqrt{(1 + aa)} \cdot \frac{r}{da} \cdot d \]

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

\[ = -\frac{p}{r} (1 + aa)^{\frac{3}{2}}; \text{and in a similar manner we may determine the subsequent co-efficients; but the calculation is somewhat laborious, and has no particular advantages.} \]

6. We may still more readily obtain a similar series for $ y $ in terms of the powers of $ x $ with constant co-efficients; calling $ \frac{dy}{dx} t $, and making $ t = bx + cx^2 + dx^3 + \ldots $ whence

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

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

\[ (1 - \frac{1}{2}t^2 + \frac{3}{4}t^3 - \frac{3}{8}t^4 + \ldots). \text{But } t = b^2x^2 + \ldots \]

\[ + 3b^2ex^3 + \ldots \text{and } e = b^2x^3 + \ldots; \text{hence we have the equation} \]

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

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

\[ - \frac{1}{2}b^2x^2 - \frac{1}{3}3b^2ex^3 + \ldots \]

\[ + \frac{1}{3}b^2x^3 + \ldots; \text{consequently} \]

\[ b = \frac{a}{r}, c = \frac{1}{23r} b + \frac{1}{2} b^3, \text{and } d = \frac{1}{43r} c + \frac{1}{3} b^3 \]

\[ - \frac{1}{3} b^3. \]

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 |------------------|----- | 100√r | 99-99 | 90 | 89-99 | 80 | 79-99 | 70 | 69-99 | 60 | 59-99 | 50 | 49-98 | 45 | 44-98 | 40 | 39-97 | 35 | 34-97 | 30 | 29-96 | 25 | 24-96 | 20 | 19-95 | 15 | 14-93 | 10 | 9-90 | 9 | 8-89 | 8 | 7-87 | 7 | 6-85 | 6 | 5-83 | 5 | 4-79 | 4 | 3-74 | 3 | 2-64 | 2 | 1-41 | 1-9 | 1-27 | 1-8 | 1-11 | 1-7 | .94 | 1-6 | .75 | 1-5 | .50 | 1-47 | .40 | 1-445 | .30 | 1-428 | .20 | 1-418 | .10 | 1-4142 | .01 | 1-4142 | .0001 | 1-4142 | .00001 | 1-4142 | .000001 | 1-4142 | .0000001 | Least Ordinate | Greatest Ordinate | Greatest Abciss | Terminal Abciss | ----------|------------------|----------------|----------------| | 100-01 | -01 | -000001 | | 90-01 | -01 | -000002 | | 80-01 | -01 | -000003 | | 70-01 | -01 | -000004 | | 60-02 | -02 | -000007 | | 50-02 | -02 | -000001 | | 45-02 | -02 | -000002 | | 40-02 | -02 | -000003 | | 35-03 | -03 | -000004 | | 30-03 | -03 | -000006 | | 25-04 | -04 | -00001 | | 20-05 | -05 | -00002 | | 15-07 | -07 | -00004 | | 10-10 | -10 | -001 | | 9-11 | -11 | -002 | | 8-12 | -12 | -003 | | 7-14 | -14 | -004 | | 6-16 | -16 | -007 | | 5-19 | -21 | -01 | | 4-24 | -26 | -02 | | 3-32 | -37 | -06 | | 2-45 | -65 | -22 | | 2-37 | -71 | -27 | | 2-29 | -79 | -33 | | 2-21 | -91 | -47 | | 2-13 | -1-10 | -65 | | 2-06 | -1-40 | -96 | | 2-04 | -1-64 | -118 | | 2-02 | -1-86 | -144 | | 2-01 | -2-24 | -182 | | 2-003 | -2-92 | -249 | | 2-000 | -5-22 | -480 | | 2-000 | -7-52 | -709 | | 2-000 | -9-82 | -939 | | 2-000 | -12-12 | -1170 | | 2-000 | -14-43 | -1400 |

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 abscisses 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. I). 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 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 flexion of the weight or mass of the parts contained within the cylindrical surface, of which $ x $ is the radius or abscissa, and $ y $ the ordinate, being always proportional to $ y^2 dx $, and the fluent to $ y^2 dx $; 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 $ s $, it will always be proportional to $ x s $; so that we have the general equation

\[ f(yx)dx = mxz. \]

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 $ f(yx)dx = \frac{1}{2} y^2 dx $; and supposing also $ s = 1 $, and the curve vertical,

\[ \frac{1}{2} y^2 dx = mxz, \]

$ y $ 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} y^2 $ will be equal to $ y^2 $ in Proposition A, and $ m = r $.

In order to obtain a series for finding $ y $ from the equation $ f(yx)dx = mxz $, we put the tangent $ t = \frac{dy}{dx} = bx + cx^3 + dx^5 + ex^7 + \ldots $,

and $ f(yx)dx = \frac{1}{2} y^2 dx + \frac{1}{2} bx^4 + \frac{1}{4} cx^6 + \frac{1}{6} dx^8 + \ldots $;

and the value of $ s = \frac{t}{\sqrt{(1+t)}} $ being expanded into a series, as in Proposition C, n 6, calling $ \frac{1}{m} $ or $ \frac{1}{r} $,

we find $ s = \frac{2}{x} f(yx)dx = bx + cx^3 + dx^5 + ex^7 + \ldots $

\[ -\frac{1}{2} bx^3 - \frac{1}{2} bx^5 - \frac{1}{2} bx^7 - \ldots \]

\[ + \frac{1}{2} bx^3 + \frac{1}{2} bx^5 + \frac{1}{2} bx^7 + \ldots \]

\[ = \frac{1}{2} qax + \frac{1}{2} qbx^3 + \frac{1}{4} qcx^5 + \frac{1}{6} qdx^7 + \ldots ; \]

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

\[ c = \frac{1}{2} q^2 + \frac{1}{2} b^2 \]

\[ d = \frac{1}{2} q^2 + \frac{10}{24} q^2 + \frac{3}{24} b^2 \]

\[ e = \frac{1}{2} q^2 + \frac{82}{24} q^2 + \frac{105}{24} q^2 + \frac{15}{24} b^2 \]

\[ f = \frac{1}{2} q^2 + \frac{652}{24} q^2 + \frac{2645}{24} q^2 + \ldots \]

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

\[ \frac{3}{24}, \frac{3}{24}, \frac{3}{24}, \frac{6}{24}, \frac{3}{24}, \frac{8}{24}, \ldots \]

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

\[ \frac{3}{24}, \frac{3}{24}, \frac{4}{24}, \frac{3}{24}, \frac{6}{24}, \ldots \]

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.

| $ s $ | $ \log_2 \sin s $ | |-------|------------------| | 0 | 0.000000 | | 1 | 1.0969100 | | 2 | 3.7166987 | | 3 | 4.0354574 | | 4 | 4.1329674 | | 5 | 4.051861 | | 6 | 4.8278768 | | 7 | 4.4776288 | | 8 | 5.0182362 | | 9 | 4.4619336 | | 10 | 4.8148499 | | 11 | 4.3187587 | | 12 | 3.6375174 | | 13 | 4.6482413 | | 14 | 5.4694937 | | 15 | 6.1456895 | | 16 | 7.008651 | | 17 | 9.1510294 | | 18 | 11.5080299 | | 19 | 13.7811595 | | 20 | 15.9774 |

\[ + \frac{1}{24} q^2 + \frac{1}{24} q^2 + \frac{1}{24} q^2 + \ldots \] Logarithmic Co-efficients of the Value of the Ordinate $ y $.

\[ y = \left[ \frac{2b}{q} + 0.6989700 \right] + \frac{2}{3} \cdot 7958800 + \frac{2}{3} \cdot 5337080 q^2 \\ + \frac{4}{3} \cdot 4948500 q^3 + \frac{4}{3} \cdot 1313165 q^4 \\ + \frac{5}{3} \cdot 9385474 q^5 + \frac{5}{3} \cdot 1769159 q^6 \\ + \frac{7}{3} \cdot 1323674 q^7 + \frac{7}{3} \cdot 87 q^8 \\ + \frac{10}{3} \cdot 9740048 q^9 + \frac{10}{3} \cdot 51 q^{10} \\ + \frac{12}{3} \cdot 6817488 q^{11} + \frac{12}{3} \cdot 35 q^{12} \\ + \frac{14}{3} \cdot 2735087 q^{13} + \frac{14}{3} \cdot 18 q^{14} \\ + \frac{17}{3} \cdot 7629636 q^{15} + \frac{17}{3} \cdot 10 q^{16} \\ + \frac{19}{3} \cdot 1609036 q^{17} + \frac{19}{3} \cdot 6 q^{18} \\ + \ldots b^x + \frac{2}{3} \cdot 5917600 \\ + (1) \cdot 0969100 + \frac{2}{3} \cdot 6709412 q^2 \\ + \frac{2}{3} \cdot 5460674 q^3 + \frac{2}{3} \cdot 1056338 q^4 \\ + \frac{3}{3} \cdot 6482413 q^5 + \frac{3}{3} \cdot 43688579 \\ + \frac{4}{3} \cdot 5486749 q^6 + \frac{4}{3} \cdot 4959794 q^7 \\ + \frac{5}{3} \cdot 2918175 q^8 + \frac{5}{3} \cdot 3119193 \\ + \frac{7}{3} \cdot 9049851 q^9 + \frac{7}{3} \cdot 3119193 \\ + \frac{8}{3} \cdot 4062959 q^{10} + \frac{8}{3} \cdot 3119193 \\ + \frac{10}{3} \cdot 8090509 q^{11} + \frac{10}{3} \cdot 3119193 \\ + \frac{11}{3} \cdot 1235822 q^{12} + \frac{11}{3} \cdot 3119193 \\ + \frac{13}{3} \cdot 3576 q^{13} + \frac{13}{3} \cdot 3119193 \\ + \frac{15}{3} \cdot 5176 q^{14} + \frac{15}{3} \cdot 3119193 \\ + \frac{17}{3} \cdot 607 q^{15} + \frac{17}{3} \cdot 3119193 \\ + \frac{19}{3} \cdot 637 q^{16} + \frac{19}{3} \cdot 3119193 \\ + \frac{21}{3} \cdot 60 q^{17} + \frac{21}{3} \cdot 3119193 \\ + \frac{23}{3} \cdot 5 q^{18} + \frac{23}{3} \cdot 3119193 \\ + \ldots b^x \]

E. The elevation or depression of a liquid contained in a given tube may be found by reversing the series.

Having a given value of $ x $, the semidiameter of the tube, and also of $ s $, the elevation or depression of the surface of the liquid at the point of contact with the solid, we obtain an equation of the form $ s = Ab + Bb^2 + Cb^3 + \ldots $; and from this we may determine the central elevation or depression $ a = 2rb $ by the well-known method of the reversion of series, which give us the value $ b = \frac{1}{A} $

\[ - \frac{B}{A^2} s^2 - \left( \frac{C}{A^3} - \frac{3B^2}{A^4} \right) s^3 - \left( \frac{D}{A^4} - \frac{3BC}{A^5} + \frac{12B^3}{A^6} \right) s^4 - \ldots \]

But it is more convenient to assume an approximate value of $ b $, a little less than $ \frac{s}{A} $, and to find the corresponding value of $ s $; then, since $ ds = Adb + Bbd^2 + Cb^3d + \ldots $, if we make $ Ab + 3Bb^2 + 5Cb^3 + \ldots = 2 $, we shall have

\[ \frac{ds}{db} = \frac{3}{b} \]

consequently the small increments of $ s $ and $ b $ will be to each other as $ s $ to $ b $, and we obtain the correction of $ b $ from the error of the calculated value of $ s $; and if the calculation be repeated with the corrected value of $ b $, the second result will always be sufficiently near to the truth.

In order to judge of the accuracy of this mode of calculation, which Mr Laplace appears to have thought liable to some undefined objection, it will be necessary to enter into the details of its different elements, which will sufficiently show the degree of convergence of the series, and the greatest possible amount of error.

Values of the Co-efficients of $ s $ for Tubes of different Diameters, $ r $ being '005, and $ s = '75 $.

\[ D = 2z \quad s = bx \times + b^2x^2 \times + b^3x^3 \times + b^4x^4 \times + b^5x^5 \times + b^6x^6 \times \]

$ 1^0 \quad 47\cdot176 \quad 7190 $

Hence if $ b = s $

\[ 0\cdot3073 \quad 7248 + 0\cdot252 \\ + 1\cdot147 \quad 7237 + 0\cdot265 \\ + 4\cdot160 \quad 7160 + 0\cdot254 + 0\cdot060 + [0\cdot026] \\ + 1\cdot503 \quad 7211 + 0\cdot240 + 0\cdot041 + [0\cdot010] \\ + 5\cdot776 \quad 7345 + 0\cdot122 + 0\cdot024 + 0\cdot007 + [0\cdot003] \\ + 14\cdot004 \quad 7449 + 0\cdot040 + 0\cdot008 + 0\cdot002 + [0\cdot001] \]

It appears, upon inspection of this table, that the coefficients of $ bx $ alone always determine $ \frac{3}{2} $ 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 Mr 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 Mr Bouvard, under Mr 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 Mr Laplace afterwards adopted, except that for the lowest portion of the curve Mr Laplace had recourse to an infinite series, applicable only to that part. The elements deduced in Nicholson's Journal for 1809, from Mr Gay Lussac's experiments, which are $ r = '0051 $ and $ s = '7533 $, agree better with the numbers found in Mr Laplace's table, than those from which it was constructed, which were $ r = '005038 $ and $ s = '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 number 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 = '1503 $, we have $ y = a + '813 bx^2 + '99 b^2x^3 + '297 b^3x^4 + \ldots = '01503 + '0489 + '0054 + '0015 + [0\cdot0006] = '0714 $, which is the marginal depression, leaving '0564 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^2 + \ldots = b \frac{d(y-a)}{db} $,

we have $ Y : b = d(y-a) : db $. In the present instance, we find $ Y = '0489 + 3 \times '0054 + 5 \times '0015 + \ldots = '079 $; and supposing, as in the example suggested by Mr Laplace, the variation of the height $ y = a $ to be '00394, which is $ \frac{3}{20} $th of $ Y $, that of $ b $ will be $ \frac{3}{20} $th of $ b $, or '075, and the variation of the central depression $ a = '00075 $, 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 Mr 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 $\frac{d}{a} = 10 \text{ cm}^2 - 1$, $C$ being about 2.9 for tubes between 1 inch and $\frac{1}{4}$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 Cohorn 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, $a = \frac{D + 48D^{3/2}}{50D^{3/2} + 30D}$; 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. (L.T.)

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

| Diameter | Observed by Lt. Col. | Dr. Young, Phil. Trans. 1806 | Laplace, 1806 | Dr. Young, Nicholson's Journal, 1807 | Laplace, 1815 | Correct Calculation | Empirical Formula | Marginal Depression, Nicholson's Journal | Difference | Diameter | |----------|----------------------|-----------------------------|--------------|-------------------------------------|--------------|-------------------|-----------------|----------------------------------------|------------|----------| | 1-inch | | | | | | | | | | | | 1.00 | Diagr. Form. | | | | | | | | | | | 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 | | | | | | | | | | |

**Cohorn, Menno, Baron de,** the Vauban of Holland, and the contemporary as well as rival of the celebrated French engineer, was born at Leeuwarden, in Friseland, in 1641. His father, an officer of distinguished merit, early inspired him with a taste for military science; and having studied mathematics, in which he made great proficiency, under his uncle Fullenius, professor at Franeker, he entered the service with the rank of captain, at the age of only sixteen. In 1673, he attracted notice at the siege of Maestricht, and afterwards distinguished himself so much in the bloody battles of Seneff, Cassel, St Denis, and Fleurus, that he was promoted to the rank of colonel. In the campaign of 1675, he performed an important service at the siege of Grave, by successfully employing the means which he had invented for crossing the fosses of fortified places, and thus carrying a bastion without a counterscarp, the access to which was defended only by the river Meuse; a service for which he received, on the spot, the warmest commendations of Vauban. At a subsequent period Cohorn ably applied his theory of fortification to the fortress of Coerden, the works of which he directed; and when war broke out in 1689 between Holland and France, he distinguished himself by new exploits.

At the siege of Namur he was opposed to Vauban, and at the head of his own regiment defended Fort William, which he had himself constructed, in the most gallant and determined manner; but he was at last compelled to yield to superior force, and surrendered the work to his great rival. He had his revenge, however, in 1695, when he participated in the capture of the same place, which had been fortified by himself, but which Boufflers was unable to hold out against King William. The alternate taking and retaking of Namur, under the direction respectively of Vauban and Cohorn, is considered as illustrative of the different styles or systems of these celebrated engineers; Vauban employing no more artillery than was absolutely necessary, using his influence to moderate the ardour of the troops, whom he never permitted to advance except under the cover of works, and placing his glory in the most rigid economy of life; whilst Cohorn, on the other hand, accumulating artillery of all kinds, and sacrificing everything to the desire of abridging the siege, by striking the besieged with surprise and terror, was equally prodigal of means and of men. "Vauban avait cerné, resserré, coupé, morcelé les assiégés; Cohorn ne s'était occupé que de les accabler. C'était la force substituée à l'industrie,"