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ular intervals, the next after that of 1860 having been in 1877. The International Statistical Congress, which consists of eminent statisticians from all countries, has done much to improve the taking of censuses, and now several countries, such as Austria, Belgium, Italy, Prussia, Russia, and Switzerland, have statistical bureaus for the purpose, amongst other things, of controlling the taking of the periodical census. In a few countries information as to the religion of the population, and in some cases additional particulars, are obtained, such as the census of useful domestic animals' in Norway.

or

In the United Kingdom the practice is for parliament to pass special acts directing the taking of each census. These acts provide that the registrars of births and deaths shall be the officers through whom the census is to be taken by enumerators, of whom at the census of 1881 there were upwards of 30,000 employed in England alone. All the registrars' districts are so subdivided that no enumerator has more houses than he can conveniently visit in one day. The enumerators have to deliver schedules at all houses, requiring particulars concerning every person who is alive at midnight preceding the census day, and on the census day to collect them. Account has also to be taken of all persons not dwelling in houses wherever found, and of persons travelling, and persons in ships, barges, &c. The enumerators are authorised to require the information necessary for the census, and persons refusing to answer wilfully giving false answers to the questions are rendered liable to penalties. The particulars to be required in each census are specified in the act directing it to be taken. The Census Act, 1880, required that the census of the following year should show 'the name, sex, age, rank, profession or occupation, condition as to marriage, relation to head of family, and birthplace of every living person who abode in every house on the night of Sunday the 3d of April 1881, and also whether any were blind or deaf and dumb, or imbecile or lunatic.' When the schedules have been collected they are transmitted to the census office, where the work of tabulation, which takes about two years, is carried out. The census when finished is presented to parliament in the form of several bulky volumes. Hardly any two countries agree as to the subjects on which information is demanded; thus some census schedules contain inquiries as to whether there are in the household infirm persons, blind, deaf and dumb, idiots, insane persons, persons who have been convicted of crime; how many languages are spoken by the persons entered; how many are at school; how many exercise the franchise; how many rooms and windows there are in the house, and so on. In 1851 an attempt was made to obtain religious statistics of the United Kingdom; since that year the census shows the religious statistics of Ireland only.

The census of the United States aims at giving a specially full conspectus of the condition of the people, and is illustrated by a large number of maps bearing on almost every branch of the census inquiries. Thus there are maps showing the prevalence of certain diseases; others the area occupied by various crops. The United States census of 1880 extended to 22 volumes, embracing statistics of population, agriculture, manufactures, mining, taxation, public indebtedness, with special reports on cotton-growing, petroleum, coal, coke, building. stones, iron and steel products, &c. The census of 1890 is comparatively limited in the scope of its inquiries.

Cent and Centime (Lat. centum, 'a hundred'), names of coins. The Dutch cent is a copper coin, the 100th part of the guilder (1s. 8d.);

CENTENARIAN

the United States cent is a bronze coin, the 100th part of the dollar, or nearly one halfpenny English, and the Canadian cent has the same value. The centime, the 100th part of the French franc, and of the value of th of an English penny, has been adopted in Belgium, and, under other names, in Greece, Italy, and Switzerland; and the Spanish real (24d.) also is divided into 100 centimes. The cental in the United States, legalised in 1878, is 100 lb. avoirdupois (cf. CENTNER). See DECIMAL SYSTEM.

Centaurea, a palæarctic genus of Compositæ, containing about two hundred species, all herbaceous annual and perennial, of which five or six are natives of Britain. The species most familiar, on account of its beauty, is the blue C. cyanus (see CORN-FLOWER), which is sometimes sown as an annual; while its larger perennial ally, C. montana, with white or purple ray florets, is a familiar denizen of old-fashioned gardens; C. americana is a showy lilac-purple annual (3 or 4 feet); while the oriental Sweet Sultan (C. moschata) and Yellow Sultan (C. amberboa) are also not uncommon; the latter two being often sold under the name of Amberboa. Among perennials, the large, downy C. babylonica, with yellow flowers, is often cultivated; also C. ragusina and C. candidissima, of which the silver-white pinnate leaves furnish an admired contrast to bright-coloured bedding-plants. Several species (C. calcitrapa, &c.) bear the naine of Star-thistle, from their spiny involucre. Some are common wayside weeds, often troublesome in Knapweed, also called Horse Knot in Scotland; pastures, notably C. nigra, the Common or Black and the closely allied C. Scabiosa. The flowers or roots of several species were formerly used in dyeing, and the astringent roots employed by herbalists.

Cen'taurs ('bull-killers '), a wild race of men who inhabited, in early times, the forests and mountains of Thessaly, and whose chief occupation was bull-hunting. Homer, the first who mentions them, describes them merely as savage, gigantic, and covered with hair. They do not appear as monsters, half-man and half-horse, until the of age Pindar. The most ancient account of the Hippocentaurs, sometimes considered as distinct, but more often confounded with the Centaurs, is that they were the offspring of Magnesian mares and Centaurus, himself the offspring of Ixion and a cloud. The Centaurs are celebrated in Greek

The most

mythology on account of their struggles with the Lapithæ (q.v.), and with Hercules. famous was Chiron, the teacher of Achilles and other heroes. In works of art the Centaurs were represented as men from the head to the loins, with the rest of the body that of a horse. It is worth mentioning that the Mexicans, who had no native horses, when they first saw the Spaniards on horseback, believed that the horse and man together made but one animal.

Centaury (Erythræa), a pretty little annual, genus of Gentianaceae, with pink or rose coloured flowers. They possess the tonic and other medicinal virtues of gentian, and the Common Centaury (E. Centaurium) has especially been esteemed in medicine since the days of Dioscorides and Galen ; and although no longer in the pharmacopoeia, its flower-tops are still sometimes gathered and dried by country-people in England and the Continent; while the allied Sabbatia angularis enjoys similar repute in the United States and Canada. The Yellow Centaury is Chlora perfoliata; but plants belonging to the wholly distinct composite genus Centaurea (q.v.) are also sometimes called Centaury.

Centenarian. See LONGEVITY.

CENTENARY

Centenary, consisting of a hundred (Lat. centum), a period of a hundred years, is now usually employed to signify a commemoration of an event, as the birth (sometimes the death) of a great man. The centenary of Burns's birth was celebrated in 1859; the bi-centenary of Pope in 1888; the tercentenary of Shakespeare in 1864. The centenary of American Independence was celebrated by a Čentennial Exhibition in 1876; the octo-centenary of the Bologna University in 1888.

Centering, the framework upon which an arch or vault of stone, brick, or iron is supported during its construction. The simplest form of centering is that used by masons and bricklayers for the arches of common windows and doors. This is merely a deal-board of the required shape, upon the curved edge of which the bricks or stones of the arch are supported until they are keyed in. In building bridges or other structures where arches of great span are to be constructed, the centering is usually made of framed timbers, or timbers and iron combined. The arrangement of the timbers should be such that the strain upon each shall be mainly a thrust in the direction of its length, for if the strain were transverse, a comparatively slight force would snap it, and if a longitudinal pull, the whole structure would be no stronger than the joints holding the pieces of timber together. In arches of great span, a longi. tudinal pulling strain is almost inevitable in some parts, as a beam of great length would bend to some extent under a thrusting strain. In such cases great skill and care are demanded in the designing and construction of the joints. As an arch is built from the piers towards the keystone, the weight upon the haunches during construction tends to push the crown upwards, and therefore the problem of designing a framed centering involves the resistance of this tendency, as well as the supporting of the weight of the materials. Occasionally, when a very great span is required, and the navigation will permit, piers are built on the bed of the river, or piles are driven into it, to support the centering directly, simplifying it, and at the same time facilitating a more rigid disposition than in centering supported only from the sides. See article BRIDGE for descriptions and illustrations of three types of centering: (1) that for the bridge over the Dee at Chester, Vol. II., page 437, supported directly from the bed of the river; also the centering for the Ballochmyle Bridge; (2) inclined struts in pairs supported from the sides, as Rennie's centering for Waterloo Bridge, page 438; (3) trussed wooden girders supported from the sides, as Rennie's centering for London Bridge, page 438.

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to more than twelve times as many. In counting the rings the ventral surface should be looked to, for the dorsal shields often overlap. The rings are flattened from above downwards, and each bears a pair of appendages. Dorsally and ventrally the skin is hard and horny. Glands occur in various positions. (b) Appendages.-The head, which is covered by a flat shield above, bears (1) a pair of antennæ, usually of considerable length, and consisting of from twelve to over one hundred joints; (2) a pair of small, strong, toothed, and bristly mandibles; (3) a pair of under-jaws, usually with palps. The next appendages are limb-like, and are followed by a modified pair of legs, the basal pieces of which generally meet in the middle line, while the strong joints terminate in a sharp claw, at which a poison-gland opens. These appendages are obviously of use for seizing and killing the prey. The legs of the other segments are usually seven-jointed, sometimes bear spurs and glands, and are generally clawed. The last pair differ in size and form from the rest, and are turned backwards. (c) Internal Anatomy.-The large brain is connected as usual with a ventral chain of ganglia. Compound eyes occur in one family, simple eyes in many, while the feelers, certain bristles, and portions of the skin are also sensory. In some cases there is a special well-defined sense-organ of undecided func tion in front of the head, or on one of the jaws. The alimentary canal is straight, and has associated with it salivary and digestive glands, and The heart is excretory (Malpighian) tubules. represented by a chambered dorsal vessel. Trachea or air-tubes open on the sides of the body, sometimes on each ring, often on alternate segments, ramify throughout the tissues, and are connected together on each side by a longitudinal stem. The reproductive organs are usually tubular, and open on the last ring of the body. A distinct penis is sometimes present.

Life and Habit.-Centipedes are darkness-loving animals, nocturnal in their food-hunting, lurking under stones or among rotten wood and the like during the day. Their powers of vision are very poorly developed, and most of the sensory work is tactile. Only in one family (Scutigeridae) are there compound eyes, in most forms only simple eyes, in many none at all. The recent researches of Plateau distinguish light from darkness, but do not need eyes to do this; species with eyes do not apparently get on much better than those without them; those with eyes seem to perceive bright objects reflecting much white light, and in some cases conspicuous movements, but probably in no case the forms of objects. Moving actively about at nights, feeling their way by means of their antenna, which function as a blind man's staff, they light upon in

and others have shown that these creatures can

Cupolas, like those of the Pantheon and St Peter's at Rome, St Paul's in Londen, or the flat domes of the Turkish mosques, require very effec-sects, worms, and other small animals, which they tive centerings.

Centigrade. See CELSIUS and THERMOMETER. Centipede, a general name for the members of one of the orders of the class Myriopoda. Popularly they are sometimes called Galley-worms, technically Chilopoda. Like the Millipedes, which form the most important neighbour order, the Centipedes are segmented animals bearing jointed appendages, having a well-defined head furnished with feelers and jaws, and breathing by means of air-tubes or tracheæ.

Structure.-The Centipede is like a primitive insect in its general structure.-(a) Externals.— The body is divided into well-marked rings, but the region behind the distinct head is practically uniform, and not divisible into thorax and abdomen. Just behind the head a few rings appear to be fused. The number of rings varies from twelve

seize and kill with their poison-bearing appendages. They are all voracious carnivorous forms, not vegetarian like the Millipedes. Some forms can run with some rapidity, and wriggle about in curious serpent-like fashion.

The

Development.-In some cases the males are said to deposit their reproductive elements in packets (spermatophores) fixed by a web to the ground. In most cases copulation probably occurs. Scolopendra is viviparous, the others lay eggs. eggs develop into larvae, which are either miniature adults (Scolopendrida and Geophilidae), or differ from the full-grown forms in having only seven pairs of legs (Scutigeridæ and Lithobiida).

Classification and Forms of Interest.-The order of Centipedes is one of the three or four divisions of Myriopoda (q. v.), and, like the class, generally represents a somewhat low grade of development among animals breathing by air-tubes. The most

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evolved centipede is a very uniform and oldfashioned animal when compared with any normal insect or spider. In the order itself we distinguish four families-(1) Scutigeridæ, (2) Lithobiidæ, (3) Scolopendridæ, (4) Geophilidæ. The first of these includes curious forms with compound eyes, very long feelers, eight shields along the back, and fifteen pairs of very long legs. The feelers and the last pair of legs are longer than the body; there are external generative appendages. In Scutigera, and apparently in some other centipedes, there are peculiar lung-like' dorsal aggregations of air-tubes opening on the back, and perhaps the beginning of the 'pulmonary chambers of some arachnids. Scutigera is represented by about a score of species, widely distributed in warm countries, and common in houses. In Lithobiidæ, as in the two other families, simple eyes alone are present; there are fifteen pairs of legs, antennæ measuring a third or more of the body length, and fifteen dorsal shields. The genus Lithobius includes over one hundred species-L. forficatus (of a reddish-brown colour, and about an inch long) is very common throughout Europe and America; our most familiar British species, L. mutabilis, also very common, has the habit of feigning death. The bite occasions considerable irritation, like that due to nettle-stings.

Giant Centipede (Scolopendra gigas).

The Scolopendrida have over a score of legs, short many-jointed antennæ, not more than one-fifth of the total length of the body; and simple eyes, not over four pairs in number, or altogether absent. About one hundred species are known, distributed over sixteen genera. They are especially at home in warm countries, where they often attain large size, the Scolopendra gigas, for instance, being sometimes a foot long. The poisonous bite of some of the larger forms is really dangerous to man. Scolopendra is the most important genus. Lastly, the Geophilidae are very long, worm-like centipedes, of somewhat sluggish habit, with 31 to 173 pairs of legs, short feelers, and no eyes. Some 22 species and 9 genera have been recorded, especially abundant in warm climates. Geophilus electricus and another species, G. longicornis, both found in Britain, shine in the dark. This is probably due to a viscid fluid ex reted all over the ventral surface. Himantarium, found round the Mediterranean, is the largest form of Geophilidæ. Well-developed spinning glands are seen in this family, and their secretion cements together ova and spermatozoa.

Distribution. The centipedes are world-wide, but abound especially in warm regions. Some what unsatisfactory fossil remains have been obtained from the American Carboniferous strata; petter preserved possible centipedes have been got from the Solenhofen strata, but it cannot yet be said with certainty that centipedes are known before Tertiary times.

CENTRAL AMERICA

Practical Import.-The centipedes have some direct practical importance as voracious devourers of injurious insects, larvæ, snails, and the like, while some of the large tropical forms are known in a somewhat different connection as animals able to give a painful and poisonous bite. In his Personal Narrative, Humboldt says he saw Indian children draw large centipedes out of the ground and eat them.

Literature.-Newport, Monograph of the class Myriapoda, order Chilopoda (Trans. Linnæan Society, vol xix. 1845); Haase, Schlesiens Chilopoden (1880-81); Latzel, Die Myriapoden Oesterreichs (1880-84).

Centlivre, SUSANNAH, an English dramatic authoress, was the daughter of a Lincolnshire gentleman named Freeman, of Holbeach, and born (say some authorities) in Ireland about 1667. Her early history is obscure; but such were her wit and beauty that on her arrival in London, though a destitute orphan, and only sixteen years of age, she won the heart of a nephew of Sir Stephen Fox, who died shortly after their marriage. Her second husband, an officer named Carroll, lost his life in a duel. Left in extreme poverty, his widow endeavoured to support herself by writing for the stage, and after producing a tragedy called The Perjured Husband (performed first in 1700), made her appearance on the stage at Bath. She afterwards married (1706) Joseph Centlivre, headcook to Queen Anne, with whom she lived happily until the time of her death, December 1, 1723. Her plays-The Busybody (with Marplot' for leading character, 1709), and A Bold Stroke for a Wife (1717)-are lively in their plots, and have kept their place on the stage. Nineteen in all, they were collected in 3 vols. 1761, with a biography, and reprinted 1872.

Centner is, with metallurgists, a weight of 100 lb., and it often has this value in commerce. The German centner is however 50 kilogrammes or 110 lb. avoirdupois; the metric or doppel centner is 100 kilogrammes. The cental of the United States is 100 lb.

Cento, a town of Central Italy, 16 miles N. by W. of Bologna, on a fertile plain near the Reno, the birthplace of Guercino (q.v.). Pop. 4975.

Cento, a name applied to literary trivialities in the form of poems manufactured by putting together distinct verses or passages of one author, or of several authors, so as to make a new meaning. After the decay of genuine poetry among the Greeks, this worthless verse-manufacture came into vogue, as is proved by the Homero-centones (ed. by Teucher, Leip. 1793), a patchwork of lines taken from Homer and forming a consecutive history of the fate and redemption of man. It was much more common, however, among the Romans in the later times of the Empire, when Virgil was frequently abused in this fashion, as in the Cento Nuptialis of Ausonius, and especially in the Cente Vergilianus, constructed in the 4th century by Proba Falconia, wife of the Proconsul Adelfius, and giving, in Virgil's misplaced words, an epitome of sacred history. The cento was a favourite recreation in the middle ages. In the 12th century a monk at Tegernsee, named Metellus, contrived to make a cento of spiritual hymns out of Horace and Virgil. See Delepierre, Tableau de la Littérature du Centon (1875).

Central America, a name applied to that part of the American continent which lies between the isthmuses of Tehuantepec, Mexico, and Panama, Colombia. Specifically the new Greater Republic of Central America, formed by treaty at Amapala in 1895 and formally recognised by President Cleveland December 23, 1896, embraces the republics of Salvador, Nicaragua, and Honduras, with provisions

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CENTRAL CITY

for the admission of Costa Rica and Guatemala. See under separate articles; also AMERICA.

Central City, the name of several villages and hamlets in the United States, and of the capital of Gilpin county, Colorado, 40 miles W. of Denver by rail, with quartz-mills and rich gold

mines, and (1890) 2480 inhabitants; also of a mining town of Lawrence county, South Dakota, in the Black Hills, 280 miles SW. of Bismarck, with formerly some twenty quartz-mills for gold. Pop. (1880) 1008; (1890) 519.

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Central India is the official term for a group of feudatory states in India, which fall into nine political agencies, but are all under the supervision of the governor-general's agent. The region in which these states lie is to the north of the British Central Provinces' of India, and touches the North-west Provinces, Rajputana, Khandesh in the Bombay Province, and Chutia-Nagpur in Bengal. The total area is about 75,000 sq. m.; pop. in 1891, 10,314,787. The nine subordinate agencies comprised in the Central India Agency are the Indore, Bhil or Bhopawar, Deputy Bhil, Western Malwa, Bhopal, Gwalior, Guna, Bundelkhand, and Baghelkhand agencies. The intrusion of two British districts, those of Jhansi and Lalitpur, belonging to the North-west Provinces, separates these nine agencies into two divisionsnative Bundelkhand and Baghelkhand on the east, and Central India proper or Malwa on the west; but the whole country lies between the Nerbudda, the Ganges, and the Chambal rivers, and is mostly fertile and well tilled. The Malwa western division is mainly a tableland 2000 feet above the sea; but its rich black soil produces fine wheat and much opium. The climate of Malwa is on the whole mild and equable; but the northern part of Central India is torrid, and unhealthy during the rainy season. The mineral wealth of Central India is great: iron, coal, copper, and lime are plentiful, and diamonds are found in some parts of Bundelkhand. The inhabitants are very diverse in origin, comprising Mahrattas (the ruling race), Rajputs, Bundelas, Baghelas, Jats, Kols, and hill-tribes such as the Gonds (414,000) and Bhils (217,000). The population is mainly Hindu in religion, only 510,718 being Mohammedans. The agent to the governorgeneral of India, whose headquarters are at Indore, has very high and very various duties and powers. He is the adviser of all the native chiefs, and their guardian during minority; exercises the functions of a court of appeal; has at his command large bodies of troops; as 'opiumagent' supervises the opium-tax throughout the agency; and he is of course the medium of communication between the imperial government and the native authorities. The principal states and agencies have separate articles. See INDORE, BAGHELKHAND, &c. The Central Provinces (q.v.) are a British commissionership.

Centralisation, a term which has come into general use for expressing a tendency to aiminister by the sovereign or the central government matters which would otherwise be under local management. The centralising tendency has been a feature in most of the great states recorded in history, though not in all of them. The oriental empires admitted of a large degree of local independence among the subject peoples. Roman empire was one of the most remarkable instances of centralisation the world has ever seen. That empire grew out of the subjugation of all the states round the Mediterranean by the city of Rome, and the control of it passed by the inevitable tendency of events into the hands of a single chief, whose power rested on the army, and

The

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who centred in himself all the great functions of
government. In the later days of the empire the
Amid the chaos
tendency increased, until the system broke down
with the power that wielded it.
that followed the downfall of Rome various systems
arose for the restoration of order, political or
religious, or both. Of these the greatest is still
the Papacy; the greatest in bygone history was
the empire of Charlemagne. In those times of
struggle, the natural method was centralisation
based on military supremacy,

Modern attempts to found a great monarchy in
Europe on the model of the Roman empire have
failed. There have grown up instead a group of
powerful states, in the history of which the central-
Centralisation
ising tendency is strongly marked.
was necessary, for in the great struggles which
have incessantly been going on, success or even self-
preservation could be secured only through a strong
organisation repressing internal division, and
through large and efficient armies. As an adequate
revenue was required for these objects, there was
further involved a strong control by the central
power of the economic and industrial functions of
the state. Thus it will be seen that centralisation
is more or less inevitable in the struggle for exist-
ence on the European continent. The most notable
examples of the opposite tendency at present are
apparent in the colonial empire of Great Britain,
and in the United States, where we find extensive
groups of self-governing communities with only a
limited measure of control by the central govern-
ment. Such control is most limited of all in the
British colonies.

On the other hand, in the French commune and in the Russian mir we see, under governments otherwise strongly centralised, a form of local activity which had been long extinct in Britain. The municipal reform of 1835 has done much to revive local action in the town life of England. The aim of the reform of local government begun in 1888 is to revive, extend, and systematise local responsibility and freedom of action, particularly in rural districts. It is now recognised that efficiency in the central government can be best secured by transferring local interests to local management by decentralisation. A wise decentralisation may be subservient to an effective centralisation, a principle which holds good also on the European continent. No absolute rules can, however, be laid down for marking off the respective provinces of the central and local powers. Each country must solve the problem in its own way, as its interests and circumstances require.

Central Provinces, a chief-commissionership of India, lying between 17° 50′ and 24° 27′ N. lat., and between 76° and 85° 15′ E. long., and embracing 18 British districts and 15 native states. Area, 115,936 sq. m.; pop. (1891) 12,932,330. The surface is very broken, straggling ranges of hills cropping up even in the level portions. In the north extend the Vindhyan and Satpura (2000 feet) tablelands, with the Nerbudda between; south of these stretches the great Nagpur plain, with the Chatisgarh plain to the east, and a wild forestregion beyond, reaching almost to the Godavari. Besides the two mentioned, the chief rivers of the province are the Wardha and Wainganga; all four are rapid streams, with their crystal waters leaping from point to point, and rushing headlong through the narrow mountain-gorges of their upper course. The climate is hot and dry, except during the south-west monsoon, from June to September, when 41 of the mean annual 45 inches of rain fall. Wheat is grown chiefly in the Nerbudda valley, rice in the Nagpur plain; these are the principal crops, but oil-seeds, cotton, and tobacco are also raised. The only manufactures of note are

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weaving and the smelting and working of iron ores. Iron is abundant, especially in the south, and there are also large coalfields, but the coal is of a very inferior quality. There is considerable trade, but its progress is retarded by the want of means of communication; this drawback, however, is being removed, roads are being made, and the railway system steadily pushed forward. Of the population, three-fourths are Hindus, and one-seventh belong to the so-called aboriginal or non- -Aryan tribes, who have found a refuge in the Satpura plateau, and still adhere to their primitive faiths (see GONDS). From these hill-tribes the Hindus throughout the province have contracted beliefs and habits which they have grafted upon the usual worship of their sect; adoration of the dead, worship of the goddess of smallpox, and belief in witchcraft are universal. The population is almost entirely rural, only 6 per cent. residing in the 52 towns of above 5000 inhabitants, of which three -Nagpur, Jubbulpore, and Kampti-have over 50,000 inhabitants. Central India (q.v.) is a term of quite distinct meaning.

Centre and Central Forces.-CENTRE OF INERTIA (MASS).If m, and m, be the masses of two particles placed at the points A, and A,, and if the right line AA, be divided in B1, so that

B1

Ba

A1

Fig. 1.

-Ag

m1A,B1 = m,A,B1, the point B1 is called the centre of inertia, or centre of mass, of the two particles. If my be a third mass at A, and if ВÀ, be divided in B2, so that

(m2 + m ̧)B1B2 = m ̧A,B,

B2 is called the centre of inertia of the three particles. In general, if there be any number of particles, a continuation of the above process will enable us to find their centre of inertia. Every body may be supposed to be made up of a multitude of particles connected by cohesion." From this it is obvious that the centre of inertia is a definite point for every piece of matter.

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tre of inertia must therefore be O, the point of intersection of CP and AQ. It is easily proved by elementary geo

=

P

Fig. 2.

Q

B

metry that OP one-third of CP. Hence, the centre of inertia of a triangular plate is obtained by joining a vertex to the middle point of the opposite side and taking the point two-thirds of this line measured from the vertex. By a similar method the centre of inertia of other plane figures may be obtained.

CENTRE OF GRAVITY.-If a body be sufficiently small, relatively to the earth, the weights of its par ticles may be considered as constituting a system of parallel forces acting on the body. Now, the magnitude of the weight of a particle is proportional to its mass. Hence, the line of action of the resultant of the parallel forces will approximately pass through the centre of inertia. For this reason such bodies are said to have a centre of gravity. Strictly speaking, there is no such point of necessity for every body, since the directions of the forces acting on the body are not accurately parallel. Hence, it is only approximately that we can say of a body that it has a centre of gravity. On the other hand, every piece of matter has, as is shown above, a centre of inertia. For all heavy bodies of moderate dimensions it is, however, sufficiently accurate to assume that the centre of inertia and gravity coincide. For example, the centre of gravity of a uniform homogeneous cylinder with parallel ends is the middle point of its axis, that of a uniformly thin circular lamina its centre, and so on.

The centre of gravity of a body of moderate dimensions may be approximately determined by suspending it by a single cord in two different positions, and finding the single point in the body which, in both positions, is intersected by the axis of the cord.

The term centre of gravity is also used in a stricter sense than the one just explained. Thus, if a body attracts and is attracted by all other gravitating matter as if its whole mass were concentrated in one point, it is said to have a true centre of gravity at that point, and the body itself is called a centrobaric body. A spherical shell of uniform gravitating matter attracts an external particle as if its whole mass were condensed at its centre. Such a body has a true centre of gravity. When such a point exists, it necessarily coincides with the centre of inertia.

In general, the determination of the centre of inertia requires the use of the integral calculus. In the case of some bodies, such as those which have a simple geometrical form and are of uniform density, elementary mathematical methods will generally be sufficient. Any straight line or plane that divides a homogeneous body symmetrically must contain its centre of inertia. For the particles of the body may be arranged in pairs of equal mass and at equal distances from the straight line or plane; and, since the centre of inertia of each pair lies in the line or plane, the centre of inertia of the whole must also lie in the same line or plane. For example, the centre of inertia of a uniform thin straight rod is its middle point; that of a uniform thin rod bent in CENTRE OF OSCILLATION.-A heavy particle the form of a parallelogram, the point of intersec- suspended from a point by a light inextensible tion of its diagonals; that of a lamina, uniform in string constitutes what is called a simple or mathethickness and density and in form a circle, ellipse, matical pendulum. For such a pendulum it is or parallelogram, its centre of figure; that of a uni- easily proved that the time of an oscillation from form spherical shell, its centre; that of a homogene-side to side of the vertical is proportional to the ous sphere, its centre; that of a parallelopiped, the square root of its length for any small arc of vibra intersection of its diagonals; that of a circular tion. A simple pendulum is, however, a thing of cylinder with parallel ends, the middle point of its theory, as in all physical problems we have to deal axis. with a rigid mass, and not a particle, oscillating about a horizontal axis. In a pendulum of this kind the time of oscillation will not vary as the square root of the length of the string, for it is obvious that those particles of the body which are nearest the point of suspension will have a tendency to vibrate more rapidly than those more

An important case is that of a uniformly thin triangular plate. Let ABC be the plate. Bisect AB in P and join CP. Let the triangle be divided by right lines parallel to AB into an indefinitely great number of indefinitely narrow strips. The centre of inertia of each strip is its middle point.

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