Chapter II: Part 2

_Santo Antao_ (pop. 25,000), at the extreme north-west of the
archipelago, has an area of 265 sq. m. Its surface is very rugged and
mountainous, abounding in volcanic craters, of which the chief is the
Topoda Coroa (7300 ft.), also known as the Sugar-loaf. Mineral springs
exist in many places. The island is the most picturesque, the
healthiest, and, on its north-western slope, the best watered and most
fertile of the archipelago. The south-eastern slope, shut out by lofty
mountains from the fertilizing moisture of the trade-winds, has an
entirely different appearance, black rocks, white pumice and red clay
being its most characteristic features. Santo Antao produces large
quantities of excellent coffee, besides sugar and fruit. It has
several small ports, of which the chief are the sheltered and spacious
Tarrafal Bay, on the south-west coast, and the more frequented Ponta
do Sol, on the north-east, 8 m. from the capital, Ribeira Grande, a
town of 4500 inhabitants. Cinchona is cultivated in the neighbourhood.
In 1780 the slaves on Santo Antao were declared free, but this decree
was not carried out. About the same time many white settlers, chiefly
from the Canaries, entered the island, and introduced the cultivation
of wheat.

_Sao Vicente_, or _St Vincent_ (8000), lies near Santo Antao, on the
south-east, and has an area of 75 sq. m. Its highest point is Monte
Verde (2400 ft.). The whole island is as arid and sterile as the
south-eastern half of Santo Antao, and for the same reason. It was
practically uninhabited until 1795; in 1829 its population numbered
about 100. Its harbour, an extinct crater on the north coast, with an
entrance eroded by the sea, affords complete shelter from every wind.
An English speculator founded a coaling station here in 1851, and the
town of Mindello, also known as Porto Grande or St Vincent, grew up
rapidly, and became the commercial centre of the archipelago. Most of
the business is in English hands, and nine-tenths of the inhabitants
understand English. Foodstuffs, wood and water are imported from Santo
Antao, and the water is stored in a large reservoir at Mindello. Sao
Vicente has a station for the submarine cable from Lisbon to
Pernambuco in Brazil.

_Santa Luzia_, about 5 m. south-east, has an area of 18 sq. m., and
forms a single estate, occupied only by the servants or the family of
the proprietor. Its highest point is 885 ft. above sea-level. On the
south-west it has a good harbour, visited by whaling and fishing
boats. Much orchil was formerly gathered, and there is good pasturage
for the numerous herds of cattle. A little to the south are the
uninhabited islets of Branco and Razo.

_Sao Nicolao_, or _Nicolau_ (12,000), a long, narrow, crescent-shaped
island with an area of 126 sq. m., lies farther east, near the middle
of the archipelago. Its climate is not very healthy. Maize,
kidney-beans, manioc, sugar-cane and vines are cultivated; and in
ordinary years grain is exported to the other islands. The interior is
mountainous, and culminates in two peaks which can be seen for many
leagues; one has the shape of a sugar-loaf, and is near the middle of
the island; the other, Monte Gordo, is near the west end, and has a
height of 4280 ft. All the other islands of the group can be seen from
Sao Nicolao in clear weather. Vessels frequently enter Preguica, or
Freshwater Bay, near the south-east extremity of the island, for water
and fresh provisions; and the custom-house is here. The island was one
of the first colonized; in 1774 its inhabitants numbered 13,500, but
famine subsequently caused a great decrease. The first capital, Lapa,
at the end of a promontory on the south, was abandoned during the
period of Spanish ascendancy over Portugal (1580-1640) in favour of
Ribeira Brava (4000), on the north coast, a town which now has a
considerable trade.

_Sal_ (750), in the north-east of the archipelago, has an area of 75
sq. m. It was originally named _Lana_, or _Lhana_ ("plain"), from the
flatness of the greater part of its surface. It derives its modern
name from a natural salt-spring, but most of the salt produced here is
now obtained from artificial salt-pans. Towards the close of the 17th
century it was inhabited only by a few shepherds, and by slaves
employed in the salt-works. In 1705 it was entirely abandoned, owing
to drought and consequent famine; and only in 1808 was the manufacture
of salt resumed. A railway, the first built in Portuguese territory,
was opened in 1835. The hostile Brazilian tariffs of 1889 for a time
nearly destroyed the salt trade. Whales, turtles and fish are
abundant, and dairy-farming is a prosperous industry. There are many
small harbours, which render every part of the island easily
accessible.

_Boa Vista_ (2600), the most easterly island of the archipelago, has
an area of 235 sq. m. It was named Sao Christovao by its discoverers
in the 15th century. Its modern name, meaning "fair view," is
singularly inappropriate, for with the exception of a few coco-nut
trees there is no wood, and in the dry season the island seems nothing
but an arid waste. The little vegetation that then exists is in the
bottom of ravines, where corn, beans and cotton are cultivated. The
springs of good water are few. The coast is indented by numerous
shallow bays, the largest of which is the harbour of the capital,
Porto Sal-Rei, on the western side (pop. about 1000). A chain of
heights, flanked by inferior ranges, traverses the middle of Boa
Vista, culminating in Monte Gallego (1250 ft.), towards the east. In
the north-western angle of the island there is a low tract of loose
sand, which is inundated with water during the rainy season; and here
are some extensive salt-pans, where the sea-water is evaporated by the
heat of the sun. Salt and orchil are exported. A good deal of fish is
taken on the coast and supplies the impoverished islanders with much
of their food.

_Maio_ (1000) has an area of 70 sq. m., and resembles Sal and Boa
Vista in climate and configuration, although it belongs to the
Sotavento group. Its best harbour is that of Nossa Senhora da Luz, on
the south-west coast, and is commonly known as Porto Inglez or English
Road, from the fact that it was occupied until the end of the 18th
century by the British, who based their claim on the marriage-treaty
between Charles II. and Catherine of Braganza (1662). The island is a
barren, treeless waste, surrounded by rocks. Its inhabitants, who live
chiefly by the manufacture of salt, by cattle-farming and by fishing,
are compelled to import most of their provisions from Sao Thiago, with
which, for purposes of local administration, Maio is included.

_Sao Thiago_ (63,000) is the most populous and the largest of the Cape
Verde Islands, having an area of 350 sq. m. It is also one of the most
unhealthy, except among the mountains over 2000 ft. high. The interior
is a mass of volcanic heights, formed of basalt covered with chalk and
clay, and culminating in the central Pico da Antonia (4500 ft.), a
sharply pointed cone. There are numerous ravines, furrowed by
perennial streams, and in these ravines are grown large quantities of
coffee, oranges, sugar-cane and physic-nuts, besides a variety of
tropical fruits and cereals. Spirits are distilled from sugar-cane,
and coarse sugar is manufactured. The first capital of the islands was
Ribeira Grande, to-day called Cidade Velha or the Old City, a
picturesque town with a cathedral and ruined fort. It was built in the
15th century on the south coast, was made an episcopal see in 1532,
and became capital of the archipelago in 1592. In 1712 it was sacked
by a French force, but despite its poverty and unhealthy situation it
continued to be the capital until 1770, when its place was taken by
Praia on the south-east. Praia (often written Praya) has a fine
harbour, a population of 21,000 and a considerable trade. It contains
the palace of the governor-general, a small natural history museum, a
meteorological observatory and an important station for the cables
between South America, Europe and West Africa. It occupies a basalt
plateau, overlooking the bay (Porto da Praia), and has an attractive
appearance, with its numerous coco-nut trees and the peak of Antonia
rising in the background above successive steps of tableland. Its
unhealthiness has been mitigated by the partial drainage of a marsh
lying to the east.

_Fogo_ (17,600) is a mass of volcanic rock, almost circular in shape
and measuring about 190 sq. m. In the centre a still active volcano,
the Pico do Cano, rises to a height of about 10,000 ft. Its crater,
which stands within an older crater, measures 3 m. in circumference
and is visible at sea for nearly 100 m. It emits smoke and ashes at
intervals; and in 1680, 1785, 1799, 1816, 1846, 1852 and 1857 it was
in eruption. After the first and most serious of these outbreaks, the
island, which had previously been called Sao Felippe, was renamed
Fogo, _i.e._ "Fire." The ascent of the mountain was first made in 1819
by two British naval officers, named Vidal and Mudge. The island is
divided, like Santo Antao, into a fertile and a sterile zone. Its
northern half produces fine coffee, beans, maize and sugar-cane; the
southern half is little better than a desert, with oases of cultivated
land near its few springs. Sao Felippe or Nossa Senhora da Luz (3000),
on the west coast, is the capital. The islanders claim to be the
aristocracy of the archipelago, and trace their descent from the
original Portuguese settlers. The majority, however, are negroes or
mulattoes. Drought and famine, followed by severe epidemics, have
been especially frequent here, notably in the years 1887-1889.

_Brava_ (9013), the most southerly of the islands, has an area of 23
sq. m. Though mountainous, and in some parts sterile, it is very
closely cultivated, and, unlike the other islands, is divided into a
multitude of small holdings. The desire to own land is almost
universal, and as the population numbers upwards of 380 per sq. m.,
and the system of tenure gives rise to many disputes, the peasantry
are almost incessantly engaged in litigation. The women, who are
locally celebrated for their beauty, far outnumber the men, who
emigrate at an early age to America. These emigrants usually return
richer and better educated than the peasantry of the neighbouring
islands. To the north of Brava lie a group of reefs among which two
islets (Ilheus Seccos or Ilheus do Rombo) are conspicuous. These are
usually known as the Ilheu de Dentro (Inner Islet) and the Ilheu de
Fora (Outer Islet). The first is used as a shelter for whaling and
fishing vessels, and as pasturage for cattle; the second has supplied
much guano for export.

_History_.--The earliest known discovery of the islands was made in 1456 by the Venetian captain Alvise Cadamosto (q.v.), who had entered the service of Prince Henry the Navigator. The archipelago was granted by King Alphonso V. of Portugal to his brother, Prince Ferdinand, whose agents completed the work of discovery. Ferdinand was an absolute monarch, exercising a commercial monopoly. In 1461 he sent an expedition to recruit slaves on the coast of Guinea and thus to people the islands, which were almost certainly uninhabited at the time. On his death in 1470 his privileges reverted to the crown, and were bestowed by John II. on Prince Emanuel, by whose accession to the throne in 1495 the archipelago finally became part of the royal dominions. Its population and importance rapidly increased; its first bishop was consecrated in 1532, its first governor-general appointed about the end of the century. It was enriched by the frequent visits of Portuguese fleets, on their return to Europe laden with treasure from the East, and by the presence of immigrants from Madeira, who introduced better agricultural methods and several new industries, such as dyeing and distillation of spirits. The failure to maintain an equal rate of progress in the 18th and 19th centuries was due partly to drought, famine and disease--in particular, to the famines of 1730-1733 and 1831-1833--and partly to gross misgovernment by the Portuguese officials.

The best general account of the islands is given in vols. xxiii. and
xxvii. of the _Boletim_ of the Lisbon Geographical Society (1905 and
1908), and in _Madeira, Cabo Verde, e Guine_, by J.A. Martins (Lisbon,
1891). Official statistics are published in Lisbon at irregular
intervals. See also _Uber die Capverden_ (Leipzig, 1884) and _Die
Vulcane der Capverden_ (Graz, 1882), both by C. Dolter. A useful map,
entitled _Ocean Atlantico Norte, Archipelago do Cabo Verde_, was
issued in 1900 by the _Commissao de Cartographia_, Lisbon.

CAPGRAVE, JOHN (1393-1464), English chronicler and hagiologist, was born at Lynn in Norfolk on the 21st of April 1393. He became a priest, took the degree of D.D. at Oxford, where he lectured on theology, and subsequently joined the order of Augustinian hermits. Most of his life he spent in the house of the order at Lynn, of which he probably became prior; he was certainly provincial of his order in England, which involved visits to other friaries, and he made at least one journey to Rome. He died on the 12th of August 1464.

Capgrave was an indefatigable student, and was reputed one of the most learned men of his age. The bulk of his works are theological: sermons, commentaries and lives of saints. His reputation as a hagiologist rests on his _Nova legenda Angliae_, or _Catalogus_ of the English saints, but this was no more than a recension of the _Sanctilogium_ which the chronicler John of Tinmouth, a monk of St Albans, had completed in 1366, which in its turn was largely borrowed from the _Sanctilogium_ of Guido, abbot of St Denis. The _Nova legenda_ was printed by Wynkyn de Worde in 1516 and again in 1527. Capgrave's historical works are _The Chronicle of England_ (from the Creation to 1417), written in English and unfinished at his death, and the _Liber de illustribus Henricis_, completed between 1446 and 1453. The latter is a collection of lives of German emperors (918-1198), English kings (1100-1446) and other famous Henries in various parts of the world (1031-1406). The portion devoted to Henry VI. of England is a contemporary record, but consists mainly of ejaculations in praise of the pious king. The accounts of the other English Henries are transferred from various well-known chroniclers. The _Chronicle_ was edited for the "Rolls" Series by Francis Charles Hingeston (London, 1858); the _Liber de illustrious Henricis_ was edited (London, 1858) for the same series by F.C. Hingeston, who published an English translation the same year. The editing of both the works is very uncritical and bad.

See Potthast, _Bibliotheka Med. Aev_.; and U. Chevalier, _Repertoire
des sources hist. Bio-bibliographie, s.v._

CAP HAITIEN, CAPE HAITIEN or HAYTIEN, a seaport of Haiti West Indies. Pop. about 15,000. It is situated on the north coast, 90 m. N. of Port au Prince, in 19 deg. 46' N. and 72 deg. 14' W. Its original Indian name was Guarico, and it has been known, at various times, as Cabo Santo, Cap Francais and Cape Henri, while throughout Haiti it is always called Le Cap. It is the most picturesque town in the republic, and the second in importance. On three sides it is hemmed in by lofty mountains, while on the fourth it overlooks a safe and commodious harbour. Under the French rule it was the capital of the colony, and its splendour, wealth and luxury earned for it the title of the "Paris of Haiti." It was then the see of an archbishop and possessed a large and flourishing university. The last remains of its former glory were destroyed by the earthquake of 1842 and the British bombardment of 1865. Although now but a collection of squalid wooden huts, with here and there a well-built warehouse, it is the centre of a thriving district and does a large export trade. It was founded by the Spaniards about the middle of the 17th century, and in 1687 received a large French colony. In 1695 it was taken and burned by the British, and in 1791 it suffered the same fate at the hands of Toussaint L'Ouverture. It then became the capital of King Henri Christophe's dominions, but since his fall has suffered severely in numerous revolutions.

CAPILLARY ACTION.[1] A tube, the bore of which is so small that it will only admit a hair (Lat. _capilla_), is called a capillary tube. When such a tube of glass, open at both ends, is placed vertically with its lower end immersed in water, the water is observed to rise in the tube, and to stand within the tube at a higher level than the water outside. The action between the capillary tube and the water has been called capillary action, and the name has been extended to many other phenomena which have been found to depend on properties of liquids and solids similar to those which cause water to rise in capillary tubes.

The forces which are concerned in these phenomena are those which act between neighbouring parts of the same substance, and which are called forces of cohesion, and those which act between portions of matter of different kinds, which are called forces of adhesion. These forces are quite insensible between two portions of matter separated by any distance which we can directly measure. It is only when the distance becomes exceedingly small that these forces become perceptible. G.H. Quincke (_Pogg. Ann._ cxxxvii. p. 402) made experiments to determine the greatest distance at which the effect of these forces is sensible, and he found for various substances distances about the twenty-thousandth part of a millimetre.

_Historical_.--According to J.C. Poggendorff (_Pogg. Ann._ ci. p. 551), Leonardo da Vinci must be considered as the discoverer of capillary phenomena, but the first accurate observations of the capillary action of tubes and glass plates were made by Francis Hawksbee (_Physico-Mechanical Experiments_, London, 1709, pp. 139-169; and _Phil. Trans._, 1711 and 1712), who ascribed the action to an attraction between the glass and the liquid. He observed that the effect was the same in thick tubes as in thin, and concluded that only those particles of the glass which are very near the surface have any influence on the phenomenon. Dr James Jurin (_Phil. Trans._, 1718, p. 739, and 1719, p. 1083) showed that the height at which the liquid is suspended depends on the section of the tube at the surface of the liquid, and is independent of the form of the lower part of the tube. He considered that the suspension of the liquid is due to "the attraction of the periphery or section of the surface of the tube to which the upper surface of the water is contiguous and coheres." From this he showed that the rise of the liquid in tubes of the same substance is inversely proportional to their radii. Sir Isaac Newton devoted the 31st query in the last edition of his _Opticks_ to molecular forces, and instanced several examples of the cohesion of liquids, such as the suspension of mercury in a barometer tube at more than double the height at which it usually stands. This arises from its adhesion to the tube, and the upper part of the mercury sustains a considerable tension, or negative pressure, without the separation of its parts. He considered the capillary phenomena to be of the same kind, but his explanation is not sufficiently explicit with respect to the nature and the limits of the action of the attractive force.

It is to be observed that, while these early speculators ascribe the phenomena to attraction, they do not distinctly assert that this attraction is sensible only at insensible distances, and that for all distances which we can directly measure the force is altogether insensible. The idea of such forces, however, had been distinctly formed by Newton, who gave the first example of the calculation of the effect of such forces in his theorem on the alteration of the path of a light-corpuscle when it enters or leaves a dense body.

Alexis Claude Clairault (_Theorie de la figure de la terre_, Paris, 1808, pp. 105, 128) appears to have been the first to show the necessity of taking account of the attraction between the parts of the fluid itself in order to explain the phenomena. He did not, however, recognize the fact that the distance at which the attraction is sensible is not only small but altogether insensible. J.A. von Segner (_Comment. Soc. Reg. Gotting_, i. (1751) p. 301) introduced the very important idea of the surface-tension of liquids, which he ascribed to attractive forces, the sphere of whose action is so small "ut nullo adhuc sensu percipi potuerit." In attempting to calculate the effect of this surface-tension in determining the form of a drop of the liquid, Segner took account of the curvature of a meridian section of the drop, but neglected the effect of the curvature in a plane at right angles to this section.

The idea of surface-tension introduced by Segner had a most important effect on the subsequent development of the theory. We may regard it as a physical fact established by experiment in the same way as the laws of the elasticity of solid bodies. We may investigate the forces which act between finite portions of a liquid in the same way as we investigate the forces which act between finite portions of a solid. The experiments on solids lead to certain laws of elasticity expressed in terms of coefficients, the values of which can be determined only by experiments on each particular substance. Various attempts have also been made to deduce these laws from particular hypotheses as to the action between the molecules of the elastic substance. We may therefore regard the theory of elasticity as consisting of two parts. The first part establishes the laws of the elasticity of a finite portion of the solid subjected to a homogeneous strain, and deduces from these laws the equations of the equilibrium and motion of a body subjected to any forces and displacements. The second part endeavours to deduce the facts of the elasticity of a finite portion of the substance from hypotheses as to the motion of its constituent molecules and the forces acting between them. In like manner we may by experiment ascertain the general fact that the surface of a liquid is in a state of tension similar to that of a membrane stretched equally in all directions, and prove that this tension depends only on the nature and temperature of the liquid and not on its form, and from this as a secondary physical principle we may deduce all the phenomena of capillary action. This is one step of the investigation. The next step is to deduce this surface-tension from a hypothesis as to the molecular constitution of the liquid and of the bodies that surround it. The scientific importance of this step is to be measured by the degree of insight which it affords or promises into the molecular constitution of real bodies by the suggestion of experiments by which we may discriminate between rival molecular theories.

In 1756 J.G. Leidenfrost (_De aquae communis nonnullis qualitatibus tractatus_, Duisburg) showed that a soap-bubble tends to contract, so that if the tube with which it was blown is left open the bubble will diminish in size and will expel through the tube the air which it contains. He attributed this force, however, not to any general property of the surfaces of liquids, but to the fatty part of the soap which he supposed to separate itself from the other constituents of the solution, and to form a thin skin on the outer face of the bubble.

In 1787 Gaspard Monge (_Memoires de l'Acad. des Sciences_, 1787, p. 506) asserted that "by supposing the adherence of the particles of a fluid to have a sensible effect only at the surface itself and in the direction of the surface it would be easy to determine the curvature of the surfaces of fluids in the neighbourhood of the solid boundaries which contain them; that these surfaces would be _linteariae_ of which the tension, constant in all directions, would be everywhere equal to the adherence of two particles, and the phenomena of capillary tubes would then present nothing which could not be determined by analysis." He applied this principle of surface-tension to the explanation of the apparent attractions and repulsions between bodies floating on a liquid.

In 1802 John Leslie (_Phil. Mag._, 1802, vol. xiv. p. 193) gave the first correct explanation of the rise of a liquid in a tube by considering the effect of the attraction of the solid on the very thin stratum of the liquid in contact with it. He did not, like the earlier speculators, suppose this attraction to act in an upward direction so as to support the fluid directly. He showed that the attraction is everywhere normal to the surface of the solid. The direct effect of the attraction is to increase the pressure of the stratum of the fluid in contact with the solid, so as to make it greater than the pressure in the interior of the fluid. The result of this pressure if unopposed is to cause this stratum to spread itself over the surface of the solid as a drop of water is observed to do when placed on a clean horizontal glass plate, and this even when gravity opposes the action, as when the drop is placed on the under surface of the plate. Hence a glass tube plunged into water would become wet all over were it not that the ascending liquid film carries up a quantity of other liquid which coheres to it, so that when it has ascended to a certain height the weight of the column balances the force by which the film spreads itself over the glass. This explanation of the action of the solid is equivalent to that by which Gauss afterwards supplied the defect of the theory of Laplace, except that, not being expressed in terms of mathematical symbols, it does not indicate the mathematical relation between the attraction of individual particles and the final result. Leslie's theory was afterwards treated according to Laplace's mathematical methods by James Ivory in the article on capillary action, under "Fluids, Elevation of," in the supplement to the fourth edition of the _Encyclopaedia Britannica_, published in 1819.

In 1804 Thomas Young (Essay on the "Cohesion of Fluids," _Phil. Trans._, 1805, p. 65) founded the theory of capillary phenomena on the principle of surface-tension. He also observed the constancy of the angle of contact of a liquid surface with a solid, and showed how from these two principles to deduce the phenomena of capillary action. His essay contains the solution of a great number of cases, including most of those afterwards solved by Laplace, but his methods of demonstration, though always correct, and often extremely elegant, are sometimes rendered obscure by his scrupulous avoidance of mathematical symbols. Having applied the secondary principle of surface-tension to the various particular cases of capillary action, Young proceeded to deduce this surface-tension from ulterior principles. He supposed the particles to act on one another with two different kinds of forces, one of which, the attractive force of cohesion, extends to particles at a greater distance than those to which the repulsive force is confined. He further supposed that the attractive force is constant throughout the minute distance to which it extends, but that the repulsive force increases rapidly as the distance diminishes. He thus showed that at a curved part of the surface, a superficial particle would be urged towards the centre of curvature of the surface, and he gave reasons for concluding that this force is proportional to the sum of the curvatures of the surface in two normal planes at right angles to each other.

The subject was next taken up by Pierre Simon Laplace (_Mecanique celeste_, supplement to the tenth book, pub. in 1806). His results are in many respects identical with those of Young, but his methods of arriving at them are very different, being conducted entirely by mathematical calculations. The form into which he threw his investigation seems to have deterred many able physicists from the inquiry into the ulterior cause of capillary phenomena, and induced them to rest content with deriving them from the fact of surface-tension. But for those who wish to study the molecular constitution of bodies it is necessary to study the effect of forces which are sensible only at insensible distances; and Laplace has furnished us with an example of the method of this study which has never been surpassed. Laplace investigated the force acting on the fluid contained in an infinitely slender canal normal to the surface of the fluid arising from the attraction of the parts of the fluid outside the canal. He thus found for the pressure at a point in the interior of the fluid an expression of the form

p = K + 1/2H(1/R + 1/R')

where K is a constant pressure, probably very large, which, however, does not influence capillary phenomena, and therefore cannot be determined from observation of such phenomena; H is another constant on which all capillary phenomena depend; and R and R' are the radii of curvature of any two normal sections of the surface at right angles to each other.

In the first part of our own investigation we shall adhere to the symbols used by Laplace, as we shall find that an accurate knowledge of the physical interpretation of these symbols is necessary for the further investigation of the subject. In the _Supplement to the Theory of Capillary Action_, Laplace deduced the equation of the surface of the fluid from the condition that the resultant force on a particle at the surface must be normal to the surface. His explanation, however, of the rise of a liquid in a tube is based on the _assumption_ of the constancy of the angle of contact for the same solid and fluid, and of this he has nowhere given a satisfactory proof. In this supplement Laplace gave many important applications of the theory, and compared the results with the experiments of Louis Joseph Gay Lussac.

The next great step in the treatment of the subject was made by C.F. Gauss (_Principia generalia Theoriae Figurae Fluidorum in statu Aequilibrii_, Gottingen, 1830, or _Werke_, v. 29, Gottingen, 1867). The principle which he adopted is that of virtual velocities, a principle which under his hands was gradually transforming itself into what is now known as the principle of the conservation of energy. Instead of calculating the direction and magnitude of the resultant force on each particle arising from the action of neighbouring particles, he formed a single expression which is the aggregate of all the potentials arising from the mutual action between pairs of particles. This expression has been called the force-function. With its sign reversed it is now called the potential energy of the system. It consists of three parts, the first depending on the action of gravity, the second on the mutual action between the particles of the fluid, and the third on the action between the particles of the fluid and the particles of a solid or fluid in contact with it.

The condition of equilibrium is that this expression (which we may for the sake of distinctness call the potential energy) shall be a minimum. This condition when worked out gives not only the equation of the free surface in the form already established by Laplace, but the conditions of the angle of contact of this surface with the surface of a solid.

Gauss thus supplied the principal defect in the great work of Laplace. He also pointed out more distinctly the nature of the assumptions which we must make with respect to the law of action of the particles in order to be consistent with observed phenomena. He did not, however, enter into the explanation of particular phenomena, as this had been done already by Laplace, but he pointed out to physicists the advantages of the method of Segner and Gay Lussac, afterwards carried out by Quincke, of measuring the dimensions of large drops of mercury on a horizontal or slightly concave surface, and those of large bubbles of air in transparent liquids resting against the under side of a horizontal plate of a substance wetted by the liquid.

In 1831 Simeon Denis Poisson published his _Nouvelle Theorie de l'action capillaire_. He maintained that there is a rapid variation of density near the surface of a liquid, and he gave very strong reasons, which have been only strengthened by subsequent discoveries, for believing that this is the case. He proceeded to an investigation of the equilibrium of a fluid on the hypothesis of uniform density, and arrived at the conclusion that on this hypothesis none of the observed capillary phenomena would take place, and that, therefore, Laplace's theory, in which the density is supposed uniform, is not only insufficient but erroneous. In particular he maintained that the constant pressure K, which occurs in Laplace's theory, and which on that theory is very large, must be in point of fact very small, but the equation of equilibrium from which he concluded this is itself defective. Laplace assumed that the liquid has uniform density, and that the attraction of its molecules extends to a finite though insensible distance. On these assumptions his results are certainly right, and are confirmed by the independent method of Gauss, so that the objections raised against them by Poisson fall to the ground. But whether the assumption of uniform density be physically correct is a very different question, and Poisson rendered good service to science in showing how to carry on the investigation on the hypothesis that the density very near the surface is different from that in the interior of the fluid.

The result, however, of Poisson's investigation is practically equivalent to that already obtained by Laplace. In both theories the equation of the liquid surface is the same, involving a constant H, which can be determined only by experiment. The only difference is in the manner in which this quantity H depends on the law of the molecular forces and the law of density near the surface of the fluid, and as these laws are unknown to us we cannot obtain any test to discriminate between the two theories.

We have now described the principal forms of the theory of capillary action during its earlier development. In more recent times the method of Gauss has been modified so as to take account of the variation of density near the surface, and its language has been translated in terms of the modern doctrine of the conservation of energy.[2]

J.A.F. Plateau (_Statique experimentale et theorique des liquides_), who made elaborate study of the phenomena of surface-tension, adopted the following method of getting rid of the effects of gravity. He formed a mixture of alcohol and water of the same density as olive oil, and then introduced a quantity of oil into the mixture. It assumes the form of a sphere under the action of surface-tension alone. He then, by means of rings of iron-wire, disks and other contrivances, altered the form of certain parts of the surface of the oil. The free portions of the surface then assume new forms depending on the equilibrium of surface-tension. In this way he produced a great many of the forms of equilibrium of a liquid under the action of surface-tension alone, and compared them with the results of mathematical investigation. He also greatly facilitated the study of liquid films by showing how to form a liquid, the films of which will last for twelve or even for twenty-four hours. The debt which science owes to Plateau is not diminished by the fact that, while investigating these beautiful phenomena, he never himself saw them, having lost his sight in about 1840.

G.L. van der Mensbrugghe (_Mem. de l'Acad. Roy. de Belgique_, xxxvii., 1873) devised a great number of beautiful illustrations of the phenomena of surface-tension, and showed their connexion with the experiments of Charles Tomlinson on the figures formed by oils dropped on the clean surface of water.

Athanase Dupre in his 5th, 6th and 7th Memoirs on the Mechanical Theory of Heat (_Ann. de Chimie et de Physique_, 1866-1868) applied the principles of thermodynamics to capillary phenomena, and the experiments of his son Paul were exceedingly ingenious and well devised, tracing the influence of surface-tension in a great number of very different circumstances, and deducing from independent methods the numerical value of the surface-tension. The experimental evidence which Dupre obtained bearing on the molecular structure of liquids must be very valuable, even if our present opinions on this subject should turn out to be erroneous.

F.H.R. Ludtge (_Pogg. Ann._ cxxxix. p. 620) experimented on liquid films, and showed how a film of a liquid of high surface-tension is replaced by a film of lower surface-tension. He also experimented on the effects of the thickness of the film, and came to the conclusion that the thinner a film is, the greater is its tension. This result, however, was tested by Van der Mensbrugghe, who found that the tension is the same for the same liquid whatever be the thickness, as long as the film does not burst. [The continued coexistence of various thicknesses, as evidenced by the colours in the same film, affords an instantaneous proof of this conclusion.] The phenomena of very thin liquid films deserve the most careful study, for it is in this way that we are most likely to obtain evidence by which we may test the theories of the molecular structure of liquids.

Sir W. Thomson (afterwards Lord Kelvin) investigated the effect of the curvature of the surface of a liquid on the thermal equilibrium between the liquid and the vapour in contact with it. He also calculated the effect of surface-tension on the propagation of waves on the surface of a liquid, and determined the minimum velocity of a wave, and the velocity of the wind when it is just sufficient to disturb the surface of still water.

THEORY OF CAPILLARY ACTION

When two different fluids are placed in contact, they may either diffuse into each other or remain separate. In some cases diffusion takes place to a limited extent, after which the resulting mixtures do not mix with each other. The same substance may be able to exist in two different states at the same temperature and pressure, as when water and its saturated vapour are contained in the same vessel. The conditions under which the thermal and mechanical equilibrium of two fluids, two mixtures, or the same substance in two physical states in contact with each other, is possible belong to thermodynamics. All that we have to observe at present is that, in the cases in which the fluids do not mix of themselves, the potential energy of the system must be greater when the fluids are mixed than when they are separate.

It is found by experiment that it is only very close to the bounding surface of a liquid that the forces arising from the mutual action of its parts have any resultant effect on one of its particles. The experiments of Quincke and others seem to show that the extreme range of the forces which produce capillary action lies between a thousandth and a twenty-thousandth part of a millimetre.

We shall use the symbol [epsilon] to denote this extreme range, beyond which the action of these forces may be regarded as insensible. If [chi] denotes the potential energy of unit of mass of the substance, we may treat [chi] as sensibly constant except within a distance [epsilon] of the bounding surface of the fluid. In the interior of the fluid it has the uniform value [chi]0. In like manner the density, [rho], is sensibly equal to the constant quantity [rho]0, which is its value in the interior of the liquid, except within a distance [epsilon] of the bounding surface. Hence if V is the volume of a mass M of liquid bounded by a surface whose area is S, the integral _ _ _ / / / M = | | | [rho] dx dy dz, (1) _/_/_/

where the integration is to be extended throughout the volume V, may be divided into two parts by considering separately the thin shell or skin extending from the outer surface to a depth [epsilon], within which the density and other properties of the liquid vary with the depth, and the interior portion of the liquid within which its properties are constant.

Since [epsilon] is a line of insensible magnitude compared with the dimensions of the mass of liquid and the principal radii of curvature of its surface, the volume of the shell whose surface is S and thickness [epsilon] will be S[epsilon], and that of the interior space will be V - S[epsilon].

If we suppose a normal [nu] less than [epsilon] to be drawn from the
surface S into the liquid, we may divide the shell into elementary
shells whose thickness is d[nu], in each of which the density and
other properties of the liquid will be constant.

The volume of one of these shells will be Sd[nu]. Its mass will be
S[rho]d[nu]. The mass of the whole shell will therefore be
_
/ [epsilon]
S | [ro]d[nu],
_/0

and that of the interior part of the liquid (V - S[epsilon])[rho]0. We
thus find for the whole mass of the liquid
_
/ [epsilon]
M = V [rho]0 - S | ([rho]0 - [rho]) d[nu]. (2)
_/0

To find the potential energy we have to integrate
_ _ _
/ / /
E = | | | [chi][rho] dx dy dz (3)
_/_/_/

Substituting [chi][rho] for [rho] in the process we have just gone
through, we find
_
/ [epsilon]
E = V[chi]0[rho]0 - S | ([chi]0[rho]0 - [chi][rho]) d[nu]. (4)
_/0

Multiplying equation (2) by [chi]0, and subtracting it from (4),
_
/ [epsilon]
E - M[chi]0 = S | ([chi] - [chi]0) d[nu]. (5)
_/0

In this expression M and [chi]0 are both constant, so that the
variation of the right-hand side of the equation is the same as that
of the energy E, and expresses that part of the energy which depends
on the area of the bounding surface of the liquid. We may call this
the surface energy.

The symbol [chi] expresses the energy of unit of mass of the liquid at
a depth [nu] within the bounding surface. When the liquid is in
contact with a rare medium, such as its own vapour or any other gas,
[chi] is greater than [chi]0, and the surface energy is positive. By
the principle of the conservation of energy, any displacement of the
liquid by which its energy is diminished will tend to take place of
itself. Hence if the energy is the greater, the greater the area of
the exposed surface, the liquid will tend to move in such a way as to
diminish the area of the exposed surface, or, in other words, the
exposed surface will tend to diminish if it can do so consistently
with the other conditions. This tendency of the surface to contract
itself is called the surface-tension of liquids.

+-----------------------+
|///////////////////////|
|///////////////////////|
|//A+---------------+A//|
|///| |///|
|///| |///|
B =========================== B
|///|C C|///|
+---+ +---+

Dupre has described an arrangement by which the surface-tension of a
liquid film may be illustrated. A piece of sheet metal is cut out in
the form AA (fig. 1). A very fine slip of metal is laid on it in the
position BB, and the whole is dipped into a solution of soap, or M.
Plateau's glycerine mixture. When it is taken out the rectangle AACC
if filled up by a liquid film. This film, however, tends to contract
on itself, and the loose strip of metal BB will, if it is let go, be
drawn up towards AA, provided it is sufficiently light and smooth.

Let T be the surface energy per unit of area; then the energy of a
surface of area S will be ST. If, in the rectangle AACC, AA = a, and
AC = b, its area is S = ab, and its energy Tab. Hence if F is the
force by which the slip BB is pulled towards AA,

d
F = --- Tab = Ta, (6)
db

or the force arising from the surface-tension acting on a length a of
the strip is Ta, so that T represents the surface-tension acting
transversely on every unit of length of the periphery of the liquid
surface. Hence if we write
_
/ [epsilon]
T = | ([chi] - [chi]0) [rho] d[nu], (7)
_/0

we may define T either as the surface-energy per unit of area, or as
the surface-tension per unit of contour, for the numerical values of
these two quantities are equal.

If the liquid is bounded by a dense substance, whether liquid or
solid, the value of [chi] may be different from its value when the
liquid has a free surface. If the liquid is in contact with another
liquid, let us distinguish quantities belonging to the two liquids by
suffixes. We shall then have
_
/ [epsilon]1
E1 - M1[chi]01 = S | ([chi]1 - [chi]01) [rho]1 d[nu]1, (8)
_/ 0
_
/ [epsilon]2
E2 - M2[chi]02 = S | ([chi]2 - [chi]02) [rho]2 d[nu]2. (9)
_/ 0

Adding these expressions, and dividing the second member by S, we
obtain for the tension of the surface of contact of the two liquids
_ _
/ [epsilon]1 / [epsilon]2
T1.2 = | ([chi]1 - [chi]01) [rho]1 d[nu]1 + | ([chi]2 - [chi]02) [rho]2 d[nu]2. (10)
_/0 _/0

If this quantity is positive, the surface of contact will tend to
contract, and the liquids will remain distinct. If, however, it were
negative, the displacement of the liquids which tends to enlarge the
surface of contact would be aided by the molecular forces, so that the
liquids, if not kept separate by gravity, would at length become
thoroughly mixed. No instance, however, of a phenomenon of this kind
has been discovered, for those liquids which mix of themselves do so
by the process of diffusion, which is a molecular motion, and not by
the spontaneous puckering and replication of the bounding surface as
would be the case if T were negative.

It is probable, however, that there are many cases in which the
integral belonging to the less dense fluid is negative. If the denser
body be solid we can often demonstrate this; for the liquid tends to
spread itself over the surface of the solid, so as to increase the
area of the surface of contact, even although in so doing it is
obliged to increase the free surface in opposition to the
surface-tension. Thus water spreads itself out on a clean surface of
glass. This shows that
_
/ [epsilon]
| ([chi] - [chi]0) [rho] d[nu]
_/0

must be negative for water in contact with glass.

_On the Tension of Liquid Films._--The method already given for the investigation of the surface-tension of a liquid, all whose dimensions are sensible, fails in the case of a liquid film such as a soap-bubble. In such a film it is possible that no part of the liquid may be so far from the surface as to have the potential and density corresponding to what we have called the interior of a liquid mass, and measurements of the tension of the film when drawn out to different degrees of thinness may possibly lead to an estimate of the range of the molecular forces, or at least of the depth within a liquid mass, at which its properties become sensibly uniform. We shall therefore indicate a method of investigating the tension of such films.

Let S be the area of the film, M its mass, and E its energy; [sigma]
the mass, and e the energy of unit of area; then

M = S[sigma], (11)

E = Se. (12)

Let us now suppose that by some change in the form of the boundary of
the film its area is changed from S to S + dS. If its tension is T the
work required to effect this increase of surface will be TdS, and the
energy of the film will be increased by this amount. Hence

TdS = dE = Sde + edS. (13)

But since M is constant,

dM = Sd[sigma] + [sigma]dS = 0. (14)

Eliminating dS from equations (13) and (14), and dividing by S, we
find

de
T = e - [sigma]--------, (15)
d[sigma]

In this expression [sigma] denotes the mass of unit of area of the
film, and e the energy of unit of area.

If we take the axis of z normal to either surface of the film, the
radius of curvature of which we suppose to be very great compared with
its thickness c, and if [rho] is the density, and [chi] the energy of
unit of mass at depth z, then
_
/ c
[sigma] = | [rho] dz, (16)
_/0

and
_
/ c
e = | [chi] [rho] dz. (17)
_/0

Both [rho] and [chi] are functions of z, the value of which remains
the same when z - c is substituted for z. If the thickness of the film
is greater than 2 [epsilon], there will be a stratum of thickness c -
2 [epsilon] in the middle of the film, within which the values of
[rho] and [chi] will be [rho]0 and [chi]0. In the two strata on either
side of this the law, according to which [rho] and [chi] depend on the
depth, will be the same as in a liquid mass of large dimensions. Hence
in this case
_
/ [epsilon]
[sigma] = (c - 2[epsilon]) [rho]0 + 2 | [rho]d[nu], (18)
_/0
_
/ [epsilon]
e = (c - 2[epsilon]) [chi]0[rho]0 + 2 | [chi][rho]d[nu], (19)
_/ 0

d[sigma] de de
-------- = [rho]0, -- = [chi]0[rho]0, .: -------- = [chi]0,
dc dc d[sigma]
_ _
/ [epsilon] / [epsilon]
T = 2 | [chi][rho]d[nu] - 2[chi]0 | [rho]d[nu] =
_/0 _/ 0
_
/ [epsilon]
2 | ([chi] - [chi]0)[rho]d[nu]. (20)
_/0

Hence the tension of a thick film is equal to the sum of the tensions
of its two surfaces as already calculated (equation 7). On the
hypothesis of uniform density we shall find that this is true for
films whose thickness exceeds [epsilon].

The symbol [chi] is defined as the energy of unit of mass of the
substance. A knowledge of the absolute value of this energy is not
required, since in every expression in which it occurs it is under the
form [chi] - [chi]0, that is to say, the difference between the
energy in two different states. The only cases, however, in which we
have experimental values of this quantity are when the substance is
either liquid and surrounded by similar liquid, or gaseous and
surrounded by similar gas. It is impossible to make direct
measurements of the properties of particles of the substance within
the insensible distance [epsilon] of the bounding surface.

When a liquid is in thermal and dynamical equilibrium with its vapour,
then if [rho]' and [chi]' are the values of [rho] and [chi] for the
vapour, and [rho]0 and [chi]0 those for the liquid,

[chi]' - [chi]0 = JL - p(1/[rho]' - 1/[rho]0), (21)

where J is the dynamical equivalent of heat, L is the latent heat of
unit of mass of the vapour, and p is the pressure. At points in the
liquid very near its surface it is probable that [chi] is greater than
[chi]0, and at points in the gas very near the surface of the liquid
it is probable that [chi] is less than [chi]', but this has not as yet
been ascertained experimentally. We shall therefore endeavour to apply
to this subject the methods used in Thermodynamics, and where these
fail us we shall have recourse to the hypotheses of molecular physics.

We have next to determine the value of [chi] in terms of the action
between one particle and another. Let us suppose that the force
between two particles m and m' at the distance f is

F = mm' ([phi](f) + Cf^-2), (22)

being reckoned positive when the force is attractive. The actual force
between the particles arises in part from their mutual gravitation,
which is inversely as the square of the distance. This force is
expressed by mm' Cf^-2. It is easy to show that a force subject to
this law would not account for capillary action. We shall, therefore,
in what follows, consider only that part of the force which depends on
[phi](f), where [phi](f) is a function of f which is insensible for
all sensible values of f, but which becomes sensible and even
enormously great when f is exceedingly small.

If we next introduce a new function of f and write
_
/ [oo]
| [phi](f)df = [Pi](f), (23)
_/f