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The Orange party so profited by these disasters that the Perpetual Edict was revoked, and Prince William assumed the office of stadtholder. De Witt's policy was thus finally defeated, and he himself became an object of general and intense hatred. All sorts of monstrous charges were brought against him, and believed; and his brother Cornelius was falsely accused of conspiring against the life of the stadtholder. Brought to the Hague, Cornelius was there, on July 24, 1672, tortured and condemned to perpetual banishment. In the same town De Witt was assaulted by a band of assassins, who left him lying on the ground under the impression that he was dead. Summoned by a pretended message from Cornelius, De Witt went to visit him in prison, when a mob assembled and murdered the brothers amid circumstances of revolting cruelty.

De Witt is one of the greatest figures of Dutch history. His action in connection with the Triple Alliance proves that he thoroughly understood the central tendencies of European politics; and, whether he is to be praised or blamed for his life-long opposition to the house of Orange, there can be no doubt as to the greatness and purity of his motives. As an administrator he displayed extraordinary energy and resource; and personally he was a man of steady, upright character, loyal and fearless. His Memoirs were published at the Hague in 1667; and in 1725, at Amsterdam, appeared Lettres et Négociations entre Jean De Witt et les Plenipotentiaires des Provinces Unies aux Cours de France, &c., depuis l'an 1652 jusqu'a 1669.” A Life of the two brothers, by Madame Zoutelande, was published at Utrecht in 1709.

(J. SI.)

DEWSBURY, a market-town in the West Riding of Yorkshire, situated at the foot of a hill, on the left bank of the Calder, eight miles S. by W. of Leeds, on the Manchester and Leeds railway. The chief industries are the making of blankets, carpets, druggets, and worsted yarn. A mile from the town is Batley, the centre of the shoddy manufacture. Coal is worked in the neighbour hood of Dewsbury. The parish church of All Saints was for the most part rebuilt in the latter half of the 18th century; the portions still preserved of the original structure are of great antiquity. Paulinus, first archbishop of York, about the year 627 preached in the district of Dewsbury, where Edwin, king of Northumbria, whom he converted to Christianity, had a royal mansion. Dewsbury is said to have been originally called Duis burgh, or the town of Dui, the tutelar god of the Brigantes. At Kirklees, in the parish of Dewsbury, is the tomb of Robin Hood. The population of the municipal borough of Dewsbury in 1851 was 14,049; in 1871 it was 24,764, while that of the parliamentary borough, which has an extended area, was 54,940. The municipal charter of the town was granted in 1862. It returns one member to Parliament.

DEXTRIN, or British gum, C6H10O5, is a substance produced from starch by the action of dilute acids, alkalies, and diastase or malt extract, and by roasting it at a temperature between 140° and 160° C. (284°-320° Fahr.) till it is of a light brown colour, .and smells like over-baked bread. Its name has reference to its powerful dextro-rotatory action on polarized light. The purest dextrin is prepared by boiling 25 parts of sulphuric acid with 125 of water, and adding by degrees a mixture of 100 parts of starch and 125 of cold water. The liquid is then lowered in temperature to 60° or 70° C. (140°-158° Fahr.), at which it is kept for some time; it is next neutralized with chalk, filtered, and evaporated. Dextrin is an uncrystallizable, insipid, odourless, yellowish-white, translucent substance, brittle and friable when thoroughly dried. It dissolves in water and dilute alcohol; by strong alcohol it is precipitated from its solutions as the hydrated compound,

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CHO.H2O. Unlike starch it is not coloured blue Ly iodine. Diastase converts it eventually into maltose, C12H22O11; and by boiling with dilute acids and alkalies it is transformed into dextrose, or ordinary glucose, CH12O. It does not ferment in contact with yeast, and in the pure state has probably no reducing action on an alkaline copper solution. If heated with strong nitric acid it gives oxalic, and not mucic acid. Limpricht has shown that dextrin is present in the flesh-juice of the hor 2. Dextrin much resembles gum arabic, instead of which it is generally substituted for a great variety of purposes. It is employed for sizing paper, for stiffening cotton goods, and for thickening colours in calico printing, also in the making of lozenges, adhesive stamps and labels, and surgical bandages. In most technical operations the crude solution only is used.

DHAR, a small native state of Malwá, in Central India, under the political superintendence of the British Government. Area, 2500 square miles; population, 150,000 souls. The state contains mueh fertile ground, the principal agricultural products being wheat, opium, gram, sugarcane, Indian corn, and cotton. The Rájá is a Puar or Pramar Rájput, who claims descent from the famous King Vikramaditya; but the family only received possession of Dhar in 1749, by gift from the Marhattá Peshwa, Báji Ráo. Towards the close of the last, and in the early part of the present century, the state was subject to a series of spoliations by Sindhia and Holkar, and was only preserved from destruction by the talents and courage of the adoptive mother of the fifth Rájá. By a treaty in 1819, Dhar passed under British protection, and bound itself to act in subordinate co-operation. The state was confiscated for rebellion in 1857, but subsequently restored to Rájá Anand Ráo Puar, then a minor, with the exception of the detached district of Bairusia, which was granted to the Begam of Bhopal. The revenue of the state is estimated at £80,000 per annum, inclusive of jagirs. The military force consists of 276 cavalry, with 800 infantry (including police), 2 guns, and 21 artillery. Road-making is being pushed on. Fifteen schools were attended in 1874 by 550 scholars. Two charitable dispensaries afford gratuitous medical relief. The town of Dhar, situated on the road from Mau (Mhow) to Baroda, extends 34 miles in length by half a mile in breadth, and is surrounded by a mud wall. The fort, built of red stone, forms a conspicuous object outside the city, and contains the Rájá's palace.

DHARWÁR, a district of British India in the presidency of Bombay, situated between 14° 6' and 15° 53′ N. lat., and 74° 50′ and 75° 58′ E. long. It contains a total area of 4536 square miles, and a population of 988,037 inhabitants. The district is about 116 miles long, with an average width of 77 miles. It is bounded on the N. by the Belgaum and Kaládgi districts, on the E. by the Nizám's territory and Bellári district, on the S. by Mysore, and on the W. by Belgaum and North Kanara districts. The western portions of the district, in the neighbourhood of the Sabyádri range, are rugged and hilly; but towards the east, the land falls away into plains of black soil in many parts very fertile and well suited to the growth of cotton, with occasionally a detached peak or group of hills. The chief rivers are the Malprabhá on the north and the Tungbhadrá on the south. The hills are principally composed of hornblende and chloritic schists, gneiss, and mica slate, large interstratified beds of silicious and ferruginous schists (as at and near Dharwár) often forming their ridges. Seams and beds of a crystalline white marble occur, which, near their junction with the hornblende slate, are often coloured green, Gold is found sparingly in the Dumbal hills, and chick mulgund, and also iron pyrites. Iron ore is worked to a small extent at Tegur, where there was once

all the territories then in his possession should be guaranteed to him, and protected from invasion by Sindhia. This protection was subsequently withdrawn, the Ráná having been guilty of treachery. In 1783, Madhoji Sindhi succeeded in recapturing the fortress of Gwalior, and crushed his Ját opponent by seizing the whole of Gohad. In 1803, however, the family were restored to their ancestral possessions of Gohad by the British Government; but, owing to the opposition of Sindhia, the Ráná agreed to relinquish possession of Gohad, in exchange for his present territory of Dholpur. By the treaty of 1804, the state was taken under the protection of the British Government, the chief becoming bound to act in subordinate co-operation with the paramount power, and to refer all disputes with neighbouring princes to the British Government. The annual revenue of Dholpur, including jagirs, amounts to about £110,000. The military force consists of 2000 men. The town of Dholpur is situated on the Agra and Gwalior road.

a considerable trade in native iron. The most influential | conclusion of peace between the English and the Marhattás, classes of the community are Brahmans and Lingayats. The Lingayats number 380,919, or 44 per cent. of the Hindu population; they worship the symbol of Siva, and males and females both carry this emblem about their person in a silver case. The manufactures of the district are not numerous; they consist of cotton and silk cloth, glass bracelets, and articles of ironware. In four towns of the district cotton and mixed silk and cotton fabrics, for male and female attire, are delicately and tastefully woven, Agriculture is the chief industry of the district, the principal products being cotton, exotic and native jawári, molasses, and oil of various kinds. Of a total of 1,662,040 acres of Government arable and assessed land, 1,530,235 acres were in 1874 under cultivation as follows :--Rice, 90,896 acres; cotton, 283,810; jawári, 497,312; bájri, 6126; wheat, 112,169; sugar-cane, 2909; tobacco, 790; til seed, 29,647; linseed, 7966; gram, 23,411; miscellaneous products, 294,491; fallow land, 182,869 acres. The cotton trade of Dharwár has great commercial importance. The land revenue realized in 1875 amounted to £196,064. The district contains six municipalities.

The territory comprised within the district appears at the earliest recorded period to have formed part of the Brahmanical realm of Vijayanagar. On the overthrow of its king at Talikot in 1565, the lands of Dhárwár became part of the Mussulman kingdom of Bijapur. In 1675 the district seems to have been overrun and partially conquered by Sivaji, becoming thereby subject to the king of Satára, and subsequently to the Peshwa. In 1776 the province was overrun by Hyder Ali, the usurping sultan of Mysore. In 1778 Dharwár was taken from the Marhattás by Hyder Ali, and in 1791 retaken by a British force. On the final overthrow of the Peshwá in 1818, Dharwár was incorporated with the territory of the East India Company.

DHOLPUR, a native state of Rajputáná, in Upper India, under the political superintendence of the British Government, is situated between 26° 30′ and 26° 57′ N. lat., and 77° 32′ and 78° 20' E. long. The state is bounded on the N. and N.E. by the British district of Agra, on the E. and S. by the Gwalior state, from which it is separated by the Chambal river, and on the W. by the state of Karauli It contains an area of about 1600 square miles, and an estimated population of upwards of 500,000 souls. It is a crop-producing country, without any special manufactures. All along the bank of the Chambal the country is deeply intersected by ravines; low ranges of hills in the western portion of the state supply inexhaust ible quarries of fine-grained and easily-worked red sandstone. The chief, who has the title of Ráná, belongs, like most of his subjects, to the tribe of Deswáli Játs, who are believed to have formed a portion of the Indo-Scythian wave of invasion which swept over Northern India about 100 A.D. The earliest recorded ancestor of the family is one Jeyt Sinh, who in 1068 held certain territories south of Alwar. His descendant in 1505, Singan Deo, having distinguished himself in an expedition against the freebooters of the Deccan, was rewarded by the sovereignty of the small territory of Gohad, with the title of Ráná. The family gradually extended their possessions until they included 56 estates, yielding an annual revenue said to amount to 66 lakhs of rupees (£660,000). Upon the defeat of the Marhattás at Panipat in 1761, Ráná Bhim Sinh, the tenth in descent from Ráná Singan Deo, seized upon the fortress of Gwalior. Political relations between the Ráná and the East India Company commenced in 1779 during the Marhattá war, when an offensive and defensive alliance was entered into. The Ráná joined the British Lorces against Sindhia, on receiving a promise that, at the

DIABETES (from Sá, through, and Baívw, to pass), a disease characterized by a habitually excessive discharge of urine. Two forms of this complaint are described, viz.— Diabetes Mellitus, or Glycosuria, where the urine is not only increased in quantity, but also contains a greater or less amount of sugar, and Diabetes Insipidus, or Polyuria, where the urine is simply increased in quantity, and contains no abnormal ingredient. The former of these is the disease to which the term diabetes is most commonly applied, and is by far the more serious and important ailment.

Although sometimes classed by medical writers among diseases of the kidneys, diabetes mellitus is rather to be regarded as a constitutional disorder. Its cause is still a matter of uncertainty, but there is sufficient evidence to connect it with a defect in the process of the assimilation of food, more especially that stage in which the function of the liver is concerned. The important researches of Claude Bernard, and subsequently those of Schiff, Harley, Pavy, M'Donell, and others, have shown that this organ, besides the secretion of bile, has the additional function of forming in large quantity a substance to which the names of glycogen, dextrin, or amyloid substance have been given. This matter is capable of being converted by the action of ferments into glucose, or grape sugar, and such a change is supposed by some to take place normally in the blood where the sugar thus formed is consumed by oxidation in the course of the circulation, while by other authorities it is held that the glycogen is not directly converted into sugar, but is transformed into other compounds.

The theories of diabetes founded on these views ascribe its production either to an excessive formation of glycogen or to some defect in its transformation, the result being that grape sugar passes out of the body by the kidneys. It has long been known, both by experiment and by observation in disease, that injuries to certain parts of the nervous system, particularly the floor of the fourth ventricle in the brain, and that portion of the sympathetic nerve which sends branches to the liver and regulates its blood supply, are followed by the appearance of sugar in the urine. Hence certain pathologists seek an explanation for the disease in a morbid state of the parts of the nervous system whereby these particular nerves are either irritated or paralyzed and the flow of blood through the liver temporarily or permanently increased. It must, however, be remarked that, although in some instances the portions of the nervous system above mentioned are found after death to be involved in disease, this is by no means constant, and that in many cases of diabetes the post mortem appearances are entirely negative. While, therefore, considerable light

has by modern research been thrown upon this disease, its pathology cannot be regarded as yet settled.

It ought to be mentioned that small quantities of sugar are frequently found in the urine in many diseases, and even in health after articles of food rich in sugar or starch have been eaten, as also in some forms of poisoning.

Little is known regarding the exciting causes of diabetes. Exposure to wet and cold, privation, depressing mental emotions, or mental overwork, the abuse of alcohol and of saccharine and starchy substances, have all been assigned as causes. It appears to be in some instances hereditary. It is most common among adults, and occurs much more frequently in males than in females.

The symptoms of diabetes are usually gradual in their onset, and the patient may suffer for a length of time before he thinks it necessary to apply for medical aid. The first symptoms which attract attention are failure of strength, and emaciation, along with great thirst and an increased amount and frequent passage of urine. From the normal quantity of from two to three pints in the twenty-four hours it may be increased to 10, 20, or 30 pints, or even more. It is usually of pale colour, and of thicker consistence than normal urine, possesses a decidedly sweet taste, and is of high specific gravity (1.03 to 1.05). It frequently gives rise to considerable irritation of the urinary passages.

By simple evaporation crystals of sugar may be obtained from diabetic urine, which also yields the characteristic chemical tests of sugar, while the amount of this substance can be accurately estimated by certain analytical processes. The quantity of sugar passed may vary from a few ounces to two or more pounds per diem, and it is found to be markedly increased after saccharine or starchy food has been taken. Sugar may also be found in the blood, saliva, tears, and in almost all the excretions of persons suffering from this disease. One of the most distressing sympt is is intense thirst, which the patient is constantly seeking to allay, the quantity of liquid consumed being in general enormous, and there is usually, but not invariably, a voracious appetite. The mouth is always parched, and a faint, sweetish odour may be evolved from the breath. The effect of the disease upon the general health is very marked, and the patient becomes more and more emaciated. He suffers from increasing muscular weakness, the temperature of his body is lowered, the skin is dry and harsh, the teeth are loosened or decay, while dyspeptic symptoms, constipation, and loss of sexual power are common accompaniments. There is in general great mental depression or irritability. Diabetes as a rule advances comparatively slowly except in the case of young persons, in whom its progress is apt to be rapid. Various complications arise in the course of the disease, among which may be mentioned cataract, various cutaneous eruptions, kidney diseases, inflammatory chest affections, and especially pulmonary consumption, which is one of the most frequent modes of fatal termination in diabetes. Occasionally death occurs suddenly from exhaustion.

Diabetes is a very fatal form of disease, recovery being exceedingly rare. Nevertheless much may be done by appropriate treatment to mitigate the severity of the symptoms and to prolong life.

Cases may thus continue for years without material change to the worse, and in some rare instances it would seem that the disease is cured. The unfavourable cases are chiefly those occurring in young persons, also where serious chest or other complications arise, and especially where the disease itself is of severe character, the quantity of sugar passed being persistently large, and the patient losing flesh and strength rapidly.

With respect to the treatment of diabetes, the regula

tion of the diet has by all authorities been regarded as a matter of the first importance, inasmuch as it has been proved beyond question that certain kinds of food have a powerful influence in aggravating the disease, more particularly those consisting largely of saccharine and starchy matter; and it may be stated generally that the various methods of treatment proposed aim at the elimination as far as possible of these constituents from the diet. Hence it is recommended that such articles as bread, potatoes, and all farinaceous foods, turnips, carrots, parsnips, and most fruits should be avoided; while animal food and soups, green vegetables, milk, cream, cheese, eggs, butter, and tea and coffee without sugar, may be taken with advantage. As a substitute for ordinary bread, which most persons find it difficult to do without for any length of time, bran bread, gluten bread, almond biscuits, and even well-browned toast or rusks are recommended. Alcoholic stimulants are of little or no use, but if prescribed should be in those forms containing the least saccharine matter, such as claret, Burgundy, brandy, or bitter ale. Thirst may be mitigated by iced water, or water slightly acidulated with phosphoric acid. The employment of a diet consisting entirely of skimmed milk has been recommended by Dr Donkin of London, who has obtained good results from this method of treatment. The milk is administered in quantities of from 8 to 12 pints in the twenty-four hours, all other articles of diet being excluded.

The plan of treatment once proposed, of administering sugar in large quantity in diabetes, proved to be highly 'injurious, and is now abandoned.

Numerous medicinal substances have been employed in diabetes, but few of them are worthy of mention as possessed of any efficacy. Opium is often found of great service, its administration being followed with marked amelioration in all the symptoms, and, according to some high authorities, with cure of the disease. It is borne in diabetes in larger doses than usual, and from 5 to 12 grains or more may be taken in the twenty-four hours. lu like manner codeia (one of the constituents of opium), in doses of half a grain increased to two or three grains three times a day, has been used with good effect.

In most cases, however, it is the dieting of the patient to which the physician has to look in dealing successfully with this formidable disease; and sufferers ought always to be impressed with the necessity of strictly abstaining from those articles of food which by general consent are allowed to exercise a hurtful influence in aggravating the symptoms.

In diabetes insipidus, there is constant thirst and an excessive flow of urine, which, however, is not found to contain any abnormal constituent. Its effects upon the system are often similar to those of diabetes mellitus, except that they are much less marked, the disease being in general very slow in its progress. In some cases the health appears to suffer very slightly. It is rarely a direct cause of death, but from its debilitating effects may predispose to serious and fatal complications. Little is known as to its pathology, but it is generally supposed to own a similar origin to diabetes mellitus. It is best treated by tonics and generous diet. Opium and valerian have been found beneficial. (J. O. A.)

DIAGORAS, born at Melos, was a writer of dithyrambic poetry. Religious in his youth, he became an atheist because a great wrong done upon him (the details of which are unknown) was left unpunished by the gods. In consequence of his blasphemous speeches, and especially from his publication of the Mysteries, he was condemned to death at Athens, and a price set upon his head. During his flight he perished by shipwreck. Aristophanes alludes to his atheism in the Clouds, 830 sqq., and to his condemnation in the Dirds, 1073. His date is not exactly known.

He could not have been, as is usually stated, a pupil of Democritus, as he was older than this philosopher, or certainly not younger. The circumstances of his death may have been confused with those of Protagoras. The writing in which he disclosed the Mysteries bore the name opúyio λόγοι, οι ἀποπυργίζοντες. These are all the facts which are known of him, and none of his actual opinions are preserved. See Zeller, Geschichte der Griechischen Philosophie. DIAGRAMS. A diagram is a figure drawn in such a manner that the geometrical relations between the parts of the figure help us to understand relations between other objects. A few have been selected for description in this article on account of their greater geometrical significance. Diagrams may be classed according to the manner in which they are intended to be used, and also according to the kind of analogy which we recognize between the diagram and the thing represented.

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Diagrams are also employed in an entirely different way -namely, for purposes of measurement. The plans and designs drawn by architects and engineers are used to determine the value of certain real magnitudes by measuring certain distances on the diagram. For such purposes it is essential that the drawing be as accurate as possible.

We therefore class diagrams as diagrams of illustration, which merely suggest certain relations to the mind of the spectator, and diagrams drawn to scale, from which measurements are intended to be made.

Methods in which diagrams are used for purposes of measurement are called Graphical methods.

Diagrams of illustration, if sufficiently accurate, may be used for purposes of measurement; and diagrams for measurement, if sufficiently clear, may be used for purposes of demonstration.

There are some diagrams or schemes, however, in which the form of the parts is of no importance, provided their connections are properly shown, Of this kind are the diagrams of electrical connections, and those belonging to that department of geometry which treats of the degrees of cyclosis, periphraxy, linkedness, and knottedness. Diagrams purely Graphic and mixed Symbolic and Graphic. Diagrams may also be classed either as purely graphical diagrams, in which no symbols are employed except letters or other marks to distinguish particular points of the diagrams, and mixed diagrams, in which certain magnitudes are represented, not by the magnitudes of parts of the diagram, but by symbols, such as numbers written on the diagram.

Thus in a map the height of places above the level of the sea is often indicated by marking the number of feet above the sea at the corresponding places on the map.

There is another method in which a line called a contour line is drawn through all the places in the map whose height above the sea is a certain number of feet, and the number of feet is written at some point or points of this line.

By the use of a series of contour lines, the height of a

great number of places can be indicated on a map by means of a small number of written symbols. Still this method is not a purely graphical method, but a partly symbolical method of expressing the third dimension of objects on a diagram in two dimensions.

Diagrams in Pairs.

In order to express completely by a purely graphical method the relations of magnitudes involving more than two variables, we must use more than one diagram. Thus in the arts of construction we use plans and elevations and sectious through different planes, to specify the form of objects having three dimensions.

In such systems of diagrams we have to indicate that a point in one diagram corresponds to a point in another diagram. This is generally done by marking the corresponding points in the different diagrams with the same letter. If the diagrams are drawn on the same piece of paper we may indicate corresponding points by drawing a line from one to the other, taking care that this line of correspondence is so drawn that it cannot be mistaken for

a real line in either diagram.

In the stereoscope the two diagrams, by the combined use of which the form of bodies in three dimensions is recognized, are projections of the bodies taken from two points so near each other that, by viewing the two diagrams simultaneously, one with each eye, we identify the corre sponding points intuitively.

The method in which we simultaneously contemplate two figures, and recognize a correspondence between certain points in the one figure and certain points in the other, is one of the most powerful and fertile methods hitherto known in science. Thus in pure geometry the theories of similar, reciprocal, and inverse figures have led to many extensions of the science. It is sometimes spoken of as the method or principle of Duality.

DIAGRAMS IN KINEMATICS.

The study of the motion of a material system is much assisted by the use of a series of diagrams representing the configuration, displacement, and acceleration of the parts of the system.

Diagram of Configuration.

In considering a material system it is often convenient to suppose that we have a record of its position at any given instant in the form of a diagram of configuration.

The position of any particle of the system is defined by drawing a straight line or vector from the origin, or point of reference, to the given particle. The position of the particle with respect to the origin is determined by the magnitude and direction of this vector.

If in the diagram we draw from the origin (which need not be the same point of space as the origin for the material system) a vector equal and parallel to the vector which determines the position of the particle, the end of this vector will indicate the position of the particle in the diagram of configuration.

If this is done for all the particles, we shall have a system of points in the diagram of configuration, each of which corresponds to a particle of the material system, and the relative positions of any pair of these points will be the same as the relative positions of the material particles which correspond to them.

We have hitherto spoken of two origins or points from which the vectors are supposed to be drawn-one for the material system, the other for the diagram. These points, however, and the vectors drawn from them, may now be omitted, so that we have on the one hand the material

system and on the other a set of points, each point corresponding to a particle of the system, and the whole repreBenting the configuration of the system at a given instant. This is called a diagram of configuration.

Diagram of Displacement.

Let us next consider two diagrams of configuration of the same system, corresponding to two different instants. We call the first the initial configuration and the second the final configuration, and the passage from the one configuration to the other we call the displacement of the system. We do not at present consider the length of time during which the displacement was effected, nor the intermediate stages through which it passed, but only the final result-a change of configuration. To study this change we construct a diagram of displacement.

Let A, B, C be the points in the initial diagram of configuration, and A', B, C be the corresponding points in the final diagram of configuration.

From o, the origin of the diagram of displacement, draw a vector oa equal and parallel to AA', ob equal and parallel to BB', oc to CC, and so on.

The points, a, b, c, &c., will be such that the vector ab indicates the displacement of b relative to a, and so on. The diagram containing the points a, b, c, &c., is therefore called the diagram of displacement.

In constructing the diagram of displacement we have hitherto assumed that we know the absolute displacements of the points of the system. For we are required to draw a line equal and parallel to A,A2, which we cannot do unless we know the absolute final position of A, with respect to its initial position. In this diagram of displacement there is therefore, besides the points a, b, c, &c., an origin, o, which represents a point absolutely fixed in space. This is necessary because the two configurations do not exist at the same time; and therefore to express their relative position we require to know a point which remains the same at the beginning and end of the time.

But we may construct the diagram in another way which does not assume a knowledge of absolute displacement or of a point fixed in space.

another system in which the velocities are equal to the velocities of the given system at the given instant and continue constant for a unit of time. The diagram of displacements for this imaginary system is the required diagram of relative velocities of the actual system at the given instant.

It is easy to see that the diagram gives the velocity of any one point relative to any other. but cannot give the absolute velocity of any of them.

Diagram of Acceleration.

By the same process by which we formed the diagram of displacements from the two diagrams of initial and final configuration, we may form a diagram of changes of relative velocity from the two diagrams of initial and final velocities. This diagram may be called that of total accelerations in a finite interval of time.

By the same process by which we deduced the diagram of velocities from that of displacements we may deduce the diagram of rates of acceleration from that of total acceleration.

We have mentioned this system of diagrams in elementary kinematics because they are found to be of use especially when we have to deal with material systems containing a great number of parts, as in the kinetic theory of gases. The diagram of configuration then appears as a region of space swarming with points representing molecules, and the only way in which we can investigate it is by considering the number of such points in unit of volume in different parts of that region, and calling this the density of the gas.

In like manner the diagram of velocities appears as a region containing points equal in number but distributed in a different manner, and the number of points in any given portion of the region expresses the number of molecules whose velocities lie within given limits. We inay speak of this as the velocity-density.

Path and IIodograph.

When the number of bodies in the system is not so great, we may construct diagrams each of which represents some property of the whole course of the motion.

On the diagram of velocity the point corresponding to the moving particle will trace another continuous line called the hodograph of the particle.

Assuming any point and calling it a, draw ak parallel Thus if we are considering the motion of one particle and equal to B, A, in the initial configuration, and from k relative to another, the point on the diagram of configuradraw kb parallel and equal to A,B, in the final configura- tion which corresponds to the moving particle will trace tion. It is easy to see that the position of the point bout a continuous line called the path of the particle. relative to a will be the same by this construction as by the former construction, only we must observe that in this second construction we use only vectors such as A,B,, A,B,, which represent the relative position of points both of which exist simultaneously, instead of vectors such as A1A2, B1B2, which express the position of a point at one instant relative to its position at a former instant, and which therefore cannot be determined by observation, because the two ends of the vector do not exist simultaneously

It appears therefore that the diagram of displacements, when drawn by the first construction includes an origin o, which indicates that we have assumed a knowledge of absolute displacements. But no such point occurs in the second construction, because we use such vectors only as we can actually observe. Hence the diagram of displacements without an origin represents neither more nor less than all we can ever know about the displacement of the material system.

Diagram of Velocity.

If the relative velocities of the points of the system are constant, then the diagram of displacement corresponding to an interval of a unit of time between the initial and the final configuration is called a diagram of relative velocity.

If the relative velocities are not constant, we suppose

The hodograph was invented and used with great success by Sir W. R. Hamilton as a method of studying the motions of bodies.

DIAGRAMS OF STRESS.

Graphical methods are peculiarly applicable to statical questions, because the state of the system is constant, so that we do not need to construct a series of diagrams corresponding to the successive states of the system.

The most useful of these applications relates to the equilibrium of plane framed structures. Two diagrams are used, one called the diagram of the frame and the other called the diagram of stress.

The structure itself consists of a number of separable pieces or links jointed together at their extremities. In practice these joints have friction, or may be made purposely stiff, so that the force acting at the extremity of a piece may not pass exactly through the axis of the joint; but as it is unsafe to make the stability of the structure depend in any degree upon the stiffness of joints, we assume in our calculations that all the joints are perfectly smooth, and therefore that the force acting on the end of any link passes through the axis of the joint.

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