# Equal Sentence Examples

She is quite

**equal**to Marya Antonovna.His kiss was ardent and she returned it with

**equal**passion.Only this was

**equal**to a hundred thunderstorms.She nodded with

**equal**sincerity.For simplicity, he assumes I + and I_ each

**equal**0.25 X106 electrostatic units.AdvertisementLisa waited for his reply in tense silence, but his response filled her with

**equal**anger and pain.He accepted it with

**equal**caution.Falafel, black bean burgers and tempeh share

**equal**ranking on the menu with chicken wings, locally raised beef and salmon.I pointed this out to everybody with provoking persistency, but no one seemed

**equal**to the task of providing the doll with eyes.He would perhaps have placed alder branches over the narrow holes in the ice, which were four or five rods apart and an

**equal**distance from the shore, and having fastened the end of the line to a stick to prevent its being pulled through, have passed the slack line over a twig of the alder, a foot or more above the ice, and tied a dry oak leaf to it, which, being pulled down, would show when he had a bite.AdvertisementIt struck him as a surprise that Alexander treated Bonaparte as an

**equal**and that the latter was quite at ease with the Tsar, as if such relations with an Emperor were an everyday matter to him.He was armed with a musketoon (which he carried rather as a joke), a pike and an ax, which latter he used as a wolf uses its teeth, with

**equal**ease picking fleas out of its fur or crunching thick bones.But besides this, even if, admitting the remaining minimum of freedom to

**equal**zero, we assumed in some given case--as for instance in that of a dying man, an unborn babe, or an idiot--complete absence of freedom, by so doing we should destroy the very conception of man in the case we are examining, for as soon as there is no freedom there is also no man.The quality, and presentation of the dishes are

**equal**to restaurants that charge twice the price.Boron fluoride also combines with ammonia gas,

**equal**volumes of the two gases giving a white crystalline solid of composition BF 3 NH 3 i with excess of ammonia gas, colourless liquids BF 3.2NH 3 and BF 3.3NH 3 are produced, which on heating lose ammonia and are converted into the solid form.AdvertisementThis is the highest point in the northern Apennines, and belongs to a group of summits of nearly

**equal**altitude; the range which is continued thence between Tuscany and what are now known as the Emilian provinces presents a continuous ridge from the mountains at the head of the Val di Mugello (due north of Florence) to the point where they are traversed by the celebrated Furlo Pass.Hence this part of the country has a cold winter climate, so that while the mean summer temperature of Milan is higher than that of Sassari, and

**equal**to that of Naples, and the extremes reached at Milan and Bologna are a good deal higher than those of Naples, the mean winter temperature of Turin is actually lower than that of Copenhagen.They were in January 1908

**equal**in value to the metallic currency of gold and silver.A race was formed strong enough to keep the empire itself in check, strong enough, except for its own internecine contests, to have formed a nation

**equal**to its happier neighbors.Meanwhile the podest still subsists; but he is no longer

**equal**to the task d maintaining an equilibrium of forces.AdvertisementInstead of opposing Francesco Sforza in Milan, he lent him his prestige and influence, foreseeing that the dynastic future of his own family and the pacification of Italy might be secured by a balance of power in which Florence should rank on

**equal**terms with Milan and Naples.Many regard Magna Carta as giving

**equal**rights to all Englishmen.The celebrated Gascoigne's powder, which was sold as late as the middle of the 19th century in the form of balls like sal prunella, consisted of

**equal**parts of crabs' eyes," the black tips of crabs' claws, Oriental pearls, Oriental bezoar and white coral, and was administered in jelly made of hart's horn, but was prescribed by physicians chiefly for wealthy people, as it cost about forty shillings per ounce.The subject of patent medicines is but little understood by the general public. Any medicine, the composition of which is kept secret, but which is advertised on the label for the cure of diseases, must in Great Britain bear a patent medicine stamp

**equal**to about one-ninth of its face value.These show that a definite intake of carbon dioxide is always accompanied by an exhalation of an

**equal**volume of oxygen.AdvertisementEvery plant is constrained to carry Out its functions of germination, growth, nutrition, reproduction, &c., between certain limits of temperature, and somewhere between the extremes of these limits each function finds ao optimum temperature at which the working of the living machinery is at its best, and, other things being

**equal**, any great departure from this may induce pathological conditions; and many disasters are due to the failure to provide such suitable temperaturese.g.As they pass into this position they undergo a longitudinal splitting by which the chromatin in each chromosome becomes divided into

**equal**halves.It is clear, however, that an

**equal**quantitative division and distribution of the chromatin to the daughter cells is brought about; and if, as has been suggested, the chromatin consists of minute particles or units which are the carriers of the hereditary characteristics, the nuclear division also probably results in the**equal**division and distribution of one half of each of these units to each daughter cell.Buffon remarked that the same temperature might have been expected, all other circumstances being

**equal**, to produce the same beings in different parts of the globe, both in the animal and vegetable kingdoms. Yet lawns in the United States are destitute of the common English daisy, the wild hyacinth of the woods of the United Kingdom is absent from Germany, and the foxglove from Switzerland.The Ptolemies in Egypt showed

**equal**anxiety to extend the bounds of geographical knowledge.This value, although considerably in excess of that previously found by different methods, was held by Airy, from the care and completeness with which the observations were carried out and discussed, to be "entitled to compete with the others on, at least,

**equal**terms."The plebs, like the English commons, contained families differing widely in rank and social position, among them those families which, as soon as an artificial barrier broke down, joined with the patricians to form the new older settlement, a nobility which had once been the whole people, was gradually shorn of all exclusive privilege, and driven to share

**equal**rights with a new people which had grown up around it.It is customary to ascribe to Offa a policy of limited scope, namely the establishment of Mercia in a position

**equal**to that of Wessex and of Northumbria.This area is divided into nearly two

**equal**parts - one, the Lei river coal-fields, yielding anthracite, and the other the Siang river coal-fields, yielding bituminous coal.The total valuation is then divided into three

**equal**parts, representing three groups of electors very unequal in number, each of which elects an**equal**number of delegates to the municipal duma.In several university towns there are free teaching establishments for women, supported by subscription, with programmes and examinations

**equal**to those of the universities.But the breeding of horses and sheep is of

**equal**importance with agriculture.Napoleon, who could brook no

**equal**, was nourishing the secret hope that his confederate might be used as a docile subordinate in the realization of his own plans, and the confederate soon came to suspect that he was being duped.The first efforts at railway legislation were governed by the

**equal**mileage principle; that is, the attempt was made to make rates proportionate to the distance.This rate increases as the distance increases, but not in

**equal**proportion; while the rates from large trade centres to other trade centres at a great distance are not higher than those to intermediate points somewhat less remote; if the law permits, there is a tendency to make them actually a little lower.Its primary purpose was to embody in statutory form the commonlaw principle of

**equal**treatment under like circumstances, and to provide machinery for enforcement.The number killed (811) is

**equal**to about three in every thousand trainmen employed.The number of passengers (36) killed in train accidents in 1907 was

**equal**to o 0759 per million passengers carried and o o024 per million kilometres travelled by passengers, or 0.1503 per million kilometres travelled by trains.Other things being

**equal**, that route is best which will serve the district most conveniently and secure the highest revenue; and the most favourable combination of curves and gradients is that by which the annual cost of conveying the traffic which the line will be called on to carry, added to the annual interest on the capital expended in construction, will be made a minimum.An endeavour is made so to plan the works of a railway that the quantity of earth excavated in cuttings shall be

**equal**to the quantity required for the embankments; but this is not always practicable, and it is sometimes advantageous to obtain the earth from some source close to the embankment rather than incur the expense of hauling it from a distant cutting.Assuming the wheels to roll along the rail without slipping, this couple will be equivalent to the couple formed by the

**equal**opposite and parallel forces, F 1 acting in the direction shown, from the axle-box on to the frame, and F 1 =µ0, acting along the rail.The couple T is necessarily accompanied by an

**equal**and opposite couple acting on the frame, which couple endeavours to turn the frame in the opposite direction to that in which the axle rotates.If there are several driving-axles in a train, the product Tw must be estimated for each separately; then the sum of the products will be

**equal**to RV.The value of is variable, but is between 7 and 8, and for approximate calculations may be taken

**equal**to unity.This high mean pressure cannot be maintained for long, because as the speed increases the demand for steam per unit of time increases, so that cut-off must take place earlier and earlier in the stroke, the limiting steady speed being attained when the rate at which steam is supplied to the cylinders is adjusted by the cut-off to be

**equal**to the maximum rate at which the boiler can produce steam, which depends upon the maximum rate at which coal can be burnt per square foot of grate.If h is the water heat at the lower temperature, h l the water heat at the higher temperature, and L the latent heat at the higher temperature, the heat supply per pound of steam is

**equal**to h1 - h2+L1, which, from the steam tables, with the values of the temperatures given, is**equal**to 1013 B.Th.U.That is to say, the engine actually utilized 61% of the energy which it was possible to utilize by means of a perfect engine working with the same initial pressure against a back pressure

**equal**to;the atmosphere.In numerous instances clear evidence of recent movements along the fault planes has been discovered; and frequent earthquakes testify with

**equal**force to the present uplift of the mountain blocks.It is doubtful whether even the genius of Valdemar would have proved

**equal**to such a stupendous task.The wood, which in Indian temples is burnt as incense, is yellowish-red, close-grained, tough, hard, readily worked, durable, and

**equal**in quality to that of the deodar.It was a characteristic of

**equal**importance that Dr Lightfoot, like Dr Westcott, never discussed these subjects in the mere spirit of controversy.The combined areas of the valleys and the area occupied by the mountains are about

**equal**.It can be shown that unless a quantity of meteors in collective mass

**equal**to our moon were to plunge into the sun every year the supply of heat could not be sustained from this source.Experiments in protection on a larger scale, and under more ordinary conditions, have been carried out with

**equal**success by Professor Celli and other Italian authorities.The number of persons was 78, and they were divided into two

**equal**groups of 39 each.Power by its very nature belongs to no one man but to a multitude of men; and the reason is obvious, since all men are born

**equal**.Acting on the queen's explicit instructions, Essex, after some ill-managed operations, had a meeting with Tyrone at a ford on the Lagan on th 7th of September 1599, when a truce was arranged; but Elizabeth was displeased by the favourable conditions allowed to the O'Neill and by Essex's treatment of him as an

**equal**.Berthelot, on the other hand, assumed that the heat-capacity of an aqueous solution is

**equal**to that of an**equal**volume of water, and calculated his results on this assumption, which involves much the same uncertainty as that of Thomsen.The oxygen contained in the compound was deducted, together with the equivalent amount of hydrogen, and the heat of combustion of the compound was then taken to be

**equal**to the heats of combustion of the elements in the residue.The arguments of conservative writers involve concessions which, though often overlooked by their readers, are very detrimental to the position they endeavour to support, and the objections they bring against the theory of the introduction of new law-books (under a Josiah or an Ezra) apply with

**equal**force to the promulgation of Mosaic teaching which had been admittedly ignored or forgotten.Under Vespasian and Titus the Jews enjoyed freedom of conscience and

**equal**political rights with non-Jewish subjects of Rome.Already under Charlemagne this development is noticeable; in his generous treatment of the Jews this Christian emperor stood in marked contrast to his contemporary the caliph Harun al-Rashid, who persecuted Jews and Christians with

**equal**vigour.In the 17th century a considerable number of Jews had made a home in the English colonies, where from the first they enjoyed practically

**equal**Tights with the Christian settlers.The Leibzoll (body-tax) was also abolished, in addition to the special law-taxes, the passport duty, the nightduty and all similiar imposts which had stamped the Jews as outcast, for they were now (Dec. 19) to have

**equal**rights with the Christian inhabitants."In Australia the Jews from the first were welcomed on perfectly

**equal**terms. The oldest congregation is that of Sydney (1817); the Melbourne community dates from 1844.In 1904 the financial and legal administration was put into the hands of the British High Commissioner for the Western Pacific. The native king is assisted by a legislative assembly consisting, in

**equal**numbers, of hereditary nobles and popular (elected) representatives.Its territory is divided into two nearly

**equal**parts by the eastern branch of the Sierra Madre Occidental, the northern part belonging to the great central plateau region, and the southern to an extremely broken region formed by the diverging branches of the Sierra Madre, with their wooded terraces and slopes and highly fertile valleys.There are separate schools for whites and blacks, and the equipment and service are approximately

**equal**, although the whites pay about nine-tenths of the school taxes.This sudden development of the Japanese is perhaps the most important event of the second half of the 19th century, since it marks the rise of an Asiatic power capable of competing with Europe on

**equal**terms. Their history is so different from that of the rest of Asia that it is not surprising if the result is different.Much the same, however, might have been said of Europe until two centuries ago, and the scientific knowledge of the Arabs under the earlier Caliphates was

**equal**or superior to that of any of their contemporaries.As a piece of writing the vivid narratives are without an

**equal**.His character is perhaps best described by a writer who says "his strength was not

**equal**to his goodness."The interior of the province is also thickly sprinkled with lakes, the combined area of which is

**equal**to about one-twentieth of the entire surface.The government of the church is chiefly according to the congregational principle, and the women have an

**equal**voice with the men; but annual meetings, attended by the bishops, teachers and other delegates from the several congregations are held, and at these sessions the larger questions involving church polity are considered and decided by a committee of five bishops.He surrendered all his offices and all his preferments except the archbishopric of York, receiving in return a pension of 1000 marks (

**equal**to six or seven thousand pounds a year in modern currency) from the bishopric of Winchester, and retired to his see, which he had never before visited.But his pride was

**equal**to his abilities.What is called Atholl brose is a compound, in

**equal**parts, of whisky and honey (or oatmeal), which was first commonly used in the district for hoarseness and sore throat.In general his object is to reduce the final equation to a simple one by making such an assumption for the side of the square or cube to which the expression in x is to be

**equal**as will make the necessary number of coefficients vanish.The forests of Peak and Duffield had their separate courts and officers, the justice seat of the former being in an extra-parochial part at

**equal**distances from Castleton, Tideswell and Bowden, while the pleas of Duffield Forest were held at Tutbury.The Mosaic Institute contained an agrarian law, based upon an

**equal**division of the soil amongst the adult males, a census of whom was taken just before their entrance into Canaan.The year 1903 was memorable for a very heavy rainfall, comparable though not

**equal**in its disastrous effects to that of 1879.Accordingly it is more susceptible to exhaustion of surface soil as to its nitrogenous, and especially as to its mineral supplies; and in the common practice of agriculture it is found to be more benefited by direct mineral manures, especially phosphatic manures, than is wheat when sown under

**equal**soil conditions.Among other things, he made a more thorough study of socialist writers, with the result that, though he was not converted to any of their schemes as being immediately practicable, he began to look upon some more

**equal**distribution of the produce of labour as a practicability of the remote future, and to dwell upon the prospect of such changes in human character as might render a stable society possible without the institution of private property.Leaving Edward, now his only brother in blood and almost his

**equal**in arms, in Galloway, he suddenly transferred his own operations to Aberdeenshire.The bark and young cones afford a tanning material, inferior indeed to oakbark, and hardly

**equal**to that of the larch, but of value in countries where substances more rich in tannin are not abundant.Still De Blainville made some advance in a right direction, as for instance by elevating the parrots' and the pigeons as " Ordres,"

**equal**in rank to that of the birds of prey and some others.Marsh states that he had fully satisfied himself that Archaeopteryx belonged to the Odontornithes, which he thought it advisable for the present to regard as a subclass, separated into three orders - Odontolcae, Odontotormae and Saururae - all well marked, but evidently not of

**equal**rank, the last being clearly much more widely distinguished from the first two than they are from one another.The whole site of Venice is dominated by the existence of one great main canal, the Grand Canal, which, winding through the town in the shape of the letter S, divides it into two

**equal**parts.The powers of the two houses are

**equal**in every respect except that the Senate passes upon the governor's appointments and tries impeachment cases brought before it by the House of Representatives.The property rights of husband and wife are nearly

**equal**; a wife may hold her property the same as if single, and a widower or a widow is entitled to the use for life of one-third of the real estate of which his or her deceased consort was seized at the time of his or her death.This anonymous writer,' he says, acquired his learning by teaching others, and adopted a dogmatic tone, which has caused him to be received at Paris with applause as the

**equal**of Aristotle, Avicenna, or Averroes.The lemniscate of Bernoulli may be defined as the locus of a point which moves so that the product of its distances from two fixed points is constant and is

**equal**to the square of half the distance between these points.They are in almost continuous motion, their power of endurance being

**equal**to the rapidity of their motions.As a commercial product spider-silk has been found to be

**equal**, if not superior, to the best silk spun by lepidopterous larvae; but the cannibalistic propensities of spiders, making it impossible to keep more than one in a single receptacle, coupled with the difficulty of getting them to spin freely in a confined space, have hitherto prevented the silk being used on any extensive scale for textile fabrics.Though it is probably destined to be used even more extensively as a fertilizer before the demand for it as a feeding stuff becomes

**equal**to the supply, practically all the cotton seed meal of the south will ultimately be used for feeding.For a long time these shells or hulls, as they are called, were burned at oil mills for fuel, 22 tons being held

**equal**to a cord of wood, and 43 tons to a ton of coal.They are adapted to special conditions which are lacking in their new surroundings, but a few will probably do fairly well the first year, and the seeds from these probably rather better the next, and so on, so that in a few years' time a strain may be available which is

**equal**or even superior to the original one introduced.Other things being

**equal**, the broker would be better off if he could hedge with**equal**ease against all his risks.In this table measurements of price movements stated both absolutely and as percentages of price levels are given, because authorities have expressed doubts as to whether the former or the latter might be expected to remain constant, other things being

**equal**, when price rose.The calyx is a long tube, or a series of connected tubes, situated above the core barrel, to which it is

**equal**in diameter.To the same great rendezvous other leaders also gathered, some of higher rank than Godfrey or Raymund or Bohemund, but none destined to exercise an

**equal**influence on the fate of the Crusade.In diplomacy he proved fully the

**equal**of all - white or black - with whom he had to deal, while he ruled with a rare combination of vigour and moderation over the nation which he had created.The extinction of the western caliphate and the dispersion of the once noble heritage of the Ommayads into numerous petty independent states, had taken place some thirty years previously, so that Castilian and Moslem were once again upon

**equal**terms, the country being almost equally divided between them.Whatever were his qualities as a fighter, the Cid was but indifferent material out of which to make a saint, - a man who battled against Christian and against Moslem with

**equal**zeal, who burnt churches and mosques with**equal**zest, who ravaged, plundered and slew as much for a livelihood as for any patriotic or religious purpose, and was in truth almost as much of a Mussulman as a Christian in his habits and his character.When a current is passed through the wire, continuous or alternating, it creates heat, which expands the air in the bulb and forces the liquid up one side of the U-tube to a certain position in which the rate of loss of heat by the air is

**equal**to the rate at which it is gaining heat.In painting, sculpture and music he considered himself the

**equal**of specialists.He was a man of considerable intellectual attainments, of prodigious memory, master of both Latin and Greek, and wrote prose and verse with

**equal**facility.An ardent Liberal, he took an active part in party struggles under the Restoration, while throwing himself with

**equal**vigour into the great work of historical regeneration which was going on at that period.There was a strong demand for the removal of these Creek Indians, known as Seminoles, and by treaties at Payne's Landing in 1832 and Fort Gibson in 1833 the Indian chiefs agreed to exchange their Florida lands for

**equal**territory in the western part of the United States.Many ingenious devices for forming bars have been produced; but generally a strong frame is used, across which steel wires are stretched at distances

**equal**to the size of the bars to be made, the blocks being first cut into slabs and then into bars.Glycerin soap ordinarily consists of about

**equal**parts of pure hard soap and glycerin (the latter valuable for its emollient properties).By Kepler's second law the radius vector, FP, sweeps over

**equal**areas in**equal**times.He discharged the duties of an envoy with

**equal**magnificence and dexterity; the treaty of May r r60, which put an end to the war, was of his making.But Henry and Marguerite still continued friends; she still bore the title of queen; she visited Marie de' Medici on

**equal**terms; and the king frequently consulted her on important affairs, though his somewhat parsimonious spirit was grieved by her extravagance.He burned phosphorus in air standing over mercury, and showed that (1) there was a limit to the amount of phosphorus which could be burned in the confined air, (2) that when no more phosphorus could be burned, one-fifth of the air had disappeared, (3) that the weight of the air lost was nearly

**equal**to the difference in the weights of the white solid produced and the phosphorus burned, (4) that the density of the residual air was less than that of ordinary air.He also showed that on heating mercury calx alone an " air " was liberated which differed from other " airs," and was slightly heavier than ordinary air; moreover, the weight of the " air " set free from a given weight of the calx was

**equal**to the weight taken up in forming the calx from mercury, and if the calx be heated with charcoal, the metal was recovered and a gas named " fixed air," the modern carbon dioxide, was formed.Matter can neither be created nor destroyed; however a chemical system be changed, the weights before and after are

**equal**.'At a later date Berzelius denoted an oxide by dots,

**equal**in number to the number of oxygen atoms present, placed over the element; this notation survived longest in mineralogy.Of great importance is the chemical identity of the diamond, graphite and charcoal, a fact demonstrated in part by Lavoisier in 1773, Smithson Tennant in 1796, and by Sir George Steuart-Mackenzie (1780-1848), who showed that

**equal**weights.In methane and ethane the hydrogen atoms are of

**equal**value, and no matter which one may be substituted by another element or group the same compound will result.These three acids yield on heating phenol, identical with the substance started with, and since in the three oxybenzoic acids the hydroxyl groups must occupy positions other than I, it follows that four hydrogen atoms are

**equal**in value.Obviously, isomeric ring-systems are possible, since the carbon atoms in the original rings are not all of

**equal**value.Unfortunately, the term normal is sometimes given to solutions which are strictly decinormal; for example, iodine, sodium thiosulphate, &c. In technical analysis, where a solution is used for one process only, it may be prepared so that I cc. is

**equal**to.According to the law of Avogadro,

**equal**volumes of different gases under the same conditions of temperature and pressure contain**equal**numbers of molecules; therefore, since the density depends upon the number of molecules present in unit volume, it follows that for a comparison of the densities of gases, the determinations must be made under coincident conditions, or the observations reduced or re-computed for coincident conditions.It is found that isomers have nearly the same critical volume, and that

**equal**differences in molecular content occasion**equal**differences in critical volume.Recent researches have shown that the law originally proposed by Kopp - " That the specific volume of a liquid compound (molecular volume) at its boiling-point is

**equal**to the sum of the specific volumes of its constituents (atomic volumes), and that every element has a definite atomic value in its compounds " - is by no means exact, for isomers have different specific volumes, and the volume for an increment of CH 2 in different homologous series is by no means constant; for example, the difference among the esters of the fatty acids is about 57, whereas for the aliphatic aldehydes it is 49.Obviously, therefore, liquids are comparable when the pressures, volumes and temperatures are

**equal**fractions of the critical constants.It follows that the thermal effects stated above must be

**equal**, i.e.It is remarkable that the position of the halogen in the molecule has no effect on the heat of formation; for example, chlorpropylene and allylchloride, and also ethylene dichloride and ethylidene dichloride, have

**equal**heats of formation.To reduce these figures to a common standard, so that the volumes shall contain

**equal**numbers of molecules, the notion of molecular volumes is introduced, the arbitrary values of the crystallographic axes (a, b, c) being replaced by the topic parameters' (x, ?i, w), which are such that, combined with the axial angles, they enclose volumes which contain**equal**numbers of molecules.The piece was warmly received at Dresden on the 2nd of January 1843; but its success was by no means

**equal**to that of Rienzi.Fears were entertained, and even the friends of the viceroy to some extent shared them, that he was not

**equal**to the crisis.In the male the right tooth usually remains similarly concealed, but the left is immensely developed, attaining a length

**equal**to more than half that of the entire animal.It will suffice therefore to point out that the ordinary needs of the cartographer can be met by conical projections, and, in the case of maps covering a wide area, by Lambert's

**equal**area projection.The Measurement of Areas is easily effected if the map at our disposal is drawn on an

**equal**area projection.The parallels or climata 2 drawn through places, of which the longest day is of

**equal**length and the decimation (distance) from the equator is the same, he maintained, ought to have been inserted at**equal**intervals, say of half an hour, and the meridians inserted on a like principle.Dorpfeld, we assume an Attic stadium of 200 steps (500 ft.) to be

**equal**to 164 metres, a degree of 700 stad.The errors due to an exaggeration of distances were still further increased on account of his assuming a degree to be

**equal**to Soo stadia, as determined by Posidonius, instead of accepting the 700 stadia of Eratosthenes.The seven climates adopted by Idrisi are erroneously supposed to be

**equal**in latitudinal extent.The charts in use of the medieval navigators of the Indian Ocean - Arabs, Persians or Dravidas - were

**equal**in value if not superior to the charts of the Mediterranean.Nor can Idrisi's map of the world, were drawn at intervals of zams, supposed to be

**equal**to three hours' sail.These miles, however, were not the ordinary Roman miles of l000 paces or 5000 ft., but smaller miles of Greek or Oriental origin, of which six were

**equal**to five Roman miles, and as the latter were**equal**to 1480 metres, the Portolano miles had a length of only 1233 metres, and 75 2 of the former, and 90 3 of the latter were**equal**to a degree.A trigonometrical survey of British India was begun in 1800 and the country can now boast of a survey which in most respects is

**equal**to those of most European states.He became the leader and spokesman of the democratic party in the Connexion which claimed for the laity the free election of class-leaders and stewards, and

**equal**representation with ministers at Conference.To the student of ecclesiastical history it is remarkable as exhibiting a form of Christianity widely divergent from the prevalent types, being a religious fellowship which has no formulated creed demanding definite subscription, and no liturgy, priesthood or outward sacrament, and which gives to women an

**equal**place with men in church organization.Several imprisonments, including that of George Fox at Derby in 1650-1651, were brought about under the Blasphemy Act of 1650, which inflicted penalties on any one who asserted himself to be very God or

**equal**with God, a charge to which the Friends were peculiarly liable owing to their doctrine of perfection.Of late years the meetings have been, for the most part, held jointly, with

**equal**liberty for all men and women to state their opinions, and to serve on all committees and other appointments.Dividing the province into three

**equal**parts of 250 m.Without taking any step to verify it, Emmet put on a green and white uniform and placed himself at the head of some eighty men, who marched towards the castle, being joined in the streets by a second body of about

**equal**strength.The largest of all, however, was the macrocollon, probably of good quality and

**equal**to the hieratic, and a cubit or nearly 18 in.After a month's vigorous drilling Hicks led 5000 of his men against an

**equal**force of dervishes in Sennar, whom he defeated, and cleared the country between the towns of Sennar and Khartum of rebels.The majority of plant specimens are most suitably fastened on paper by a mixture of

**equal**parts of gum tragacanth and gum arabic made into a thick paste with water.Subsequent popes manifested

**equal**ardour, with the same damaging results, in the repair and adornment of the catacombs, and many of the paintings covering their walls, which have been assigned to the period of their original construction, are really the work of these later times.In North America the earliest representative of the group is Systemodon of the Lower Eocene, in which all the upper premolars are quite simple; while the molars are of a type which would readily develop into that of the modern tapirs, both outer columns being conical and of

**equal**size.Of the upper incisors the first and second are nearly

**equal**, with short, broad crowns, the third is large and conical, considerably larger than the canine, which is separated from it by an interval.Hind-feet with the typical perissodactyle arrangement of three toes - the middle one being the largest, the two others nearly

**equal**.The skull is high, with -the facial and cranial portions approximately

**equal**.The post-glenoid process is small, and the facial and cranial portions of the skull are approximately of

**equal**length.It divides the entrance to the Gulf of Mexico into two passages of nearly

**equal**width, - the Strait of Florida, about I io m.Gradual abolition of slavery was declared by a law of the 13th of February 1880; definitive abolition in 1886; and in 1893 the

**equal**civil status of blacks and whites in all respects was proclaimed by General Calleja.Note that the class of rationals less than or

**equal**tò 2r is not a real number.The patriarch of Constantinople is the nominal head of the Orthodox priesthood; but by an arrangement concluded in 1879, his authority was delegated to the Austrian emperor, in exchange for a revenue

**equal**to the tribute previously paid by the clergy of the provinces; and his nominations for the metropolitanate of Serajevo, and the bishoprics of Dolnja Tuzla, Banjaluka and Mostar require the imperial assent.The Serb and Moslem delegates, who had started on the same day for Budapest, to present their petition to the emperor, learned from the rescript that the government intended to concede to their compatriots "a share in the legislation and administration of provincial affairs, and

**equal**protection for all religious beliefs, languages and racial distinctions."From Emerson he gained more than from any man, alive or dead; and, though the older philosopher both enjoyed and learned from the association with the younger, it cannot be said that the gain was

**equal**.In this work Law shows himself at least 'the

**equal**of the ablest champion of Deism.So also did the " Midhat Constitution " promulgated by Abd-ul-Hamid almost immediately after his accession to the throne, owing largely to the reactionary spirit at that time of the' Ulema and of the sultan's immediate advisers, but almost, if not quite, in

**equal**measure to the scornful reception of the Constitution by the European powers.All Turkish subjects, of whatever race or religion, have

**equal**juridical and political rights and obligations, and all discrimination as to military service has been abolished.The copper money was in pieces of a nominal value of 40, 20, TO, 5 and i paras, 40 paras being

**equal**to 1 piastre.The division of the line into

**equal**sections of 200 kilometres apiece produced at once a somewhat ridiculous result.On the 2nd of June 1908 a fresh convention was signed between the government and the Bagdad Railway Company providing, on the same financial basis, for the extension of the line from Bulgurlu to Helif and of the construction of a branch from Tel-Habesh to Aleppo, covering a total aggregate length of approximately 840 kilometres, The principle of

**equal**sections of 200 kilometres was thus set on one side.Admitted on

**equal**terms to the European family of nations, the Ottoman government had given a solemn guarantee of its intention to make the long-promised reforms a reality.He himself tried versification, and some of his lines which have come down to us appear quite

**equal**to the average work of his contemporaries.His hasidas are almost

**equal**to his ghazels; for, while they rival those of Nef `i in brilliancy, they surpass them in beauty of diction, and are not so artificial and dependent on fantastic and farfetched conceits.In cavalry they were weak, for the Russian does not take kindly to equitation and the horses were not

**equal**to the accepted European standard of weight, while the Cossack was only formidable to stragglers and wounded.Had the Austrians possessed mobility

**equal**to that of the French the latter should have been overwhelmed in detail, but whilst the French covered 17 and 19 m.In his position at Bautzen he felt himself

**equal**to all his enemy's combinations.The actual working of his mind towards that strategic and tactical ascendancy that rendered his presence on the battlefield, according to the testimony of his opponents,

**equal**to a reinforcement of 40,000 men, was entirely undiscernible.But it probably pleased Mme de Stael to quite an

**equal**degree that Napoleon should apparently put forth his power to crush her and fail.Arabian authors already had found three square numbers of

**equal**difference, but the difference itself had not been assigned in proposing the question.The flowers have a hollow tube at the base bearing at its free edge five sepals, an

**equal**number of petals, usually concave or spoon-shaped, pink or white, and a great number of stamens.Under the laws of the state the legal existence and legal personality of a woman are not affected by marriage, and the property rights of a husband and wife are nearly

**equal**.The head is rather large, and is furnished at first with five simple eyes of nearly

**equal**size; but as it increases in size the homologues of the facetted eyes of the imago become larger, whereas those equivalent to the ocelli remain small.The natural harbour, which, with a depth diminishing from 70 to 30 fathoms, strikes in from the northwest so as to cut the island into two fairly

**equal**portions, with an isthmus not more than 14 m.The mean result of the best determinations shows that when a current of one ampere is passed for one second, a mass of silver is deposited

**equal**to o ooi i 18 gramme.If the ions move at

**equal**rates, the salt which is decomposed to supply the ions liberated must be taken equally from the neighbourhood of the two electrodes.If we assume that no other cause is at work, it is easy to prove that, with non-dissolvable electrodes, the ratio of salt lost at the anode to the salt lost at the cathode must be

**equal**to the ratio of the velocity of the cation to the velocity of the anion.The velocity with which the ions move past each other is

**equal**to the sum of their individual velocities, which can therefore be calculated.A series of equivalent solutions all containing the same coloured ion have absorption spectra which, when photographed, show identical absorption bands of

**equal**intensity.If a certain minimum charge must be collected in order to start coagulation, it will need the conjunction of 6n monovalent, or 3n divalent, to

**equal**the effect of 2n trivalent ions.In the case of substances like ammonia and acetic acid, where the dissociation is very small, I - a is nearly

**equal**to unity, and only varies slowly with dilution.An explanation of the failure of the usual dilution law in these cases may be given if we remember that, while the electric forces between bodies like undissociated molecules, each associated with

**equal**and opposite charges, will vary inversely as the fourth power of the distance, the forces between dissociated ions, each carrying one charge only, will be inversely proportional to the square of the distance.Zinc dissolves at the anode, an

**equal**amount of zinc replaces an equivalent amount of copper on the other side of the porous partition, and the same amount of copper is deposited on the cathode.Let an electromotive force exactly

**equal**to that of the cell be applied to it in the reverse direction.Thus, if L denote the heat corresponding with the chemical changes associated with unit electric transfer, Le will be the heat corresponding with an electric transfer e, and will also be

**equal**to the change in internal energy of the cell.Again, we may calculate the osmotic work done, and, if the whole cycle of operations be supposed to occur at the same temperature, the osmotic work must be

**equal**and opposite to the electrical work of the first operation.If secondary effects are eliminated, the deposition of metals also is a reversible process; the decomposition voltage is

**equal**to the electromotive force which the metal itself gives when going into solution.We can therefore calculate the rate at which the salt as a whole will diffuse by examining the conditions for a steady transfer, in which the ions diffuse at an

**equal**rate, the faster one being restrained and the slower one urged forward by the electric forces.Their representation in the states general was exactly

**equal**to that of the Dutch, though their population was in the proportion of seven to five.It is therefore to be expected that as time goes on the quality of " plantation " rubber will improve, and there would seem to be no reason why it should not eventually be fully

**equal**to that of the " wild " rubber.A partial exception may perhaps be made in the case of cocoa, when the two plants are placed not too closely in about

**equal**numbers.Proselytes are still not allowed, in Orthodox circles, to become the wives of reputed descendants of the priestly families, but otherwise marriage with proselytes is altogether

**equal**to marriage between born Jews.Its principal courts are constituted of an

**equal**number of ministers and laymen.In 1824 the settled indigenes had to pay the very heavy rate of 11 roubles (about 1) per head, and the arrears, which soon became

**equal**to the sums levied, were rigorously exacted.In the same address he called attention to the conditions of the world's food supply, urging that with the low yield at present realized per acre the supply of wheat would within a comparatively short time cease to be

**equal**to the demand caused by increasing population, and that since nitrogenous manures are essential for an increase in the yield, the hope of averting starvation, as regards those races for whom wheat is a staple food, depended on the ability of the chemist to find an artificial method for fixing the nitrogen of the air.It was not an

**equal**tax falling on all landowners, but the question as to whether a certain estate was to be taxed or not was decided according to the quality of the property, and not that of the owner.Similarly, by putting one or more of the deleted rows or columns

**equal**to rows or columns which are not deleted, we obtain, with Laplace, a number of identities between products of determinants of complementary orders.Hence the product determinant has the principal diagonal elements each

**equal**to A and the remaining elements zero.It is the resultant of k polynomials each of degree m-I, and thus contains the coefficients of each form to the degree (m-I)'-1; hence the total degrees in the coefficients of the k forms is, by addition, k (m - 1) k - 1; it may further be shown that the weight of each term of the resultant is constant and

**equal**to m(m-I) - (Salmon, l.c. p. loo).The discriminant of the product of two forms is

**equal**to the product of their discriminants multiplied by the square of their resultant.In the theory of forms we seek functions of the coefficients and variables of the original quantic which, save as to a power of the modulus of transformation, are

**equal**to the like functions of the coefficients and variables of the transformed quantic. We may have such a function which does not involve the variables, viz.This will be recognized as the resultant of the two linear forms. If the two linear forms be identical, the umbral sets a l, a2; b l, b 2 are alternative, are ultimately put

**equal**to one another and (ab) vanishes.The vanishing of this invariant is the condition for

**equal**roots.This method of solution fails when the discriminant R vanishes, for then the Hessian has

**equal**roots, as also the cubic f.The discriminant, whose vanishing is the condition that f may possess two

**equal**roots, has the expression j 2 - 6 i 3; it is nine times the discriminant of the cubic resolvent k 3 - 2 ik- 3j, and has also the expression 4(1, t') 6 .The quartic has four

**equal**roots, that is to say, is a perfect fourth power, when the Hessian vanishes identically; and conversely.The vanishing of the invariants i and j is the necessary and sufficient condition to ensure the quartic having three

**equal**roots.The quartic will have two pairs of

**equal**roots, that is, will be a perfect square, if it and its Hessian merely differ by a numerical factor.Hesse's canonical form shows at once that there cannot be more than two independent invariants; for if there were three we could, by elimination of the modulus of transformation, obtain two functions of the coefficients

**equal**to functions of m, and thus, by elimination of m, obtain a relation between the coefficients, showing them not to be independent, which is contrary to the hypothesis.Putting n

**equal**to co, in a generating function obtained above, we find that the function, which enumerates the asyzvgetic seminvariants of degree 0, is 1 1-z2.1-z3.1-z4....1-z0 that is to say, of the weight w, we have one form corresponding to each non-unitary partition of w into the parts 2, 3, 4,...0.In practice this oxidation process is continued until the whole of the oxygen is as nearly as possible

**equal**in weight to the sulphur present as sulphide or as sulphate, i.e.In the 14th century Gerson (1363-1429) seems to have been the earliest divine who composed and preached in French, but his example was not followed by any man of

**equal**genius.His part in the later phases of the Russo-Turkish struggle has never been fully explained, for with

**equal**wisdom and generosity he declined to gratify public curiosity at the cost of some of his colleagues.The retirement of the timid primate left him without an

**equal**in the Estate of Clergy, and it was very largely due to his co-operation that the king was able to carry through the famous "Act of Unity and Security" which converted Sweden from a constitutional into a semi-absolute monarchy.Its weak point is that the apparent motion of the node depends partly upon the motion of the ecliptic, which cannot be determined with

**equal**precision.At shorter distances the magnetism induced in the weaker magnet will be stronger than its permanent magnetism, and there will be attraction; two magnets with their like poles in actual contact will always cling together unless the like poles are of exactly

**equal**strength.But no magnet can have a single pole; if there is one, there must also be at least a second, of the opposite sign and of exactly

**equal**strength.In every magnet the strength of the south pole is exactly

**equal**to that of the north pole, the action of the same magnetic force upon the two poles being**equal**and oppositely directed.A unit magnetic pole is that which acts on an

**equal**pole at a distance of one centimetre with a force of one dyne.The opposite and parallel forces acting on the poles are always

**equal**, a fact which is sometimes expressed by the statement that the total magnetism of a magnet is zero.The line through the given point along which the potential decreases most rapidly is the direction of the resultant magnetic force, and the rate of decrease of the potential in any direction is

**equal**to the component of the force in that direction.If the constituent solenoids are parallel and of

**equal**strength, the magnet is also uniformly magnetized.A magnet consisting of a series of plane shells of

**equal**strength arranged at right angles to the direction of magnetization will be uniformly magnetized.The cases of greatest practical importance are those of a sphere (which is an ellipsoid with three

**equal**axes) and an ovoid or prolate ellipsoid of revolution.Since 7ra'I is the moment of the sphere (=volume X magnetization), it appears from (10) that the magnetized sphere produces the same external effect as a very small magnet of

**equal**moment placed at its centre and magnetized in the same direction; the resultant force therefore is the same as in (14).If the magnetization is parallel to the major axis, and the lengths of the major and minor axes are 2a and 2C, the poles are situated at a distance

**equal**to 3a from the centre, and the magnet will behave externally like a simple solenoid of length 3a.The internal force F is opposite to the direction of the magnetization, and

**equal**to NI, where N is a coefficient depending only on the ratio of the axes.Kohlrausch 2 the distance between the poles of a cylindrical magnet the length of which is from io to 30 times the diameter, is sensibly

**equal**to five-sixths of the length of the bar.Hence if the induction per square centimetre at any point is denoted by B, then in empty space B is numerically

**equal**to H; moreover in isotropic media both have the same direction, and for these reasons it is often said that in empty space (and practically in air and other nonmagnetic substances) B and H are identical.Suppose the whole space in which induction exists to be divided up into unit tubes, such that the surface integral of the induction over any cross-section of a tube is

**equal**to unity, and along the axis of each tube let a line of induction be drawn.The distance between the poles may with sufficient accuracy for a rough determination be assumed to be

**equal**to five-sixths of the length of the magnet.This extraneous influence may, however, be eliminated by surrounding the rod with a coil of wire carrying a current such as will produce in the interior a magnetic field

**equal**and opposite to the vertical component of the earth's field.If the distance of the mirror from the scale is

**equal**to n scale divisions, and if a deflection 0 of the needle causes, the reflected spot of light to move over s scale divisions, we shall have s/n = tan 20 exactly, s/2n = tan 0 approximately.Therefore and m = v I - 'm of d22 (47) constant cell B21 its object is to produce inside the tube a magnetic field

**equal**and opposite to that due to the earth's magnetism.Since the induction B is

**equal**to H 47rI, it is easy from the results of experiments such as that just described to deduce the relation between B and H; a curve indicating such relation is called a curve of induction.Then if a known change of induction SB a inside the standard coil is found to cause a throw of d scale-divisions, any change of induction SB through the experimental coil will be numerically

**equal**to the corresponding throw D multiplied by snRBa/SNrd.When a current of strength i is suddenly interrupted in the primary, the increment of induction through the secondary is sensibly

**equal**to 47rin/l units.Suppose the switches to be adjusted so that the effective number of turns in the variable coil is loo; the magnetizing forces in the two coils will then be

**equal**, and if the test rod is of the same quality as the standard, the flow of induction will be confined entirely to the iron circuit, the two yokes will be at the same magnetic potential, and the compass needle will not be affected.Further, Maxwell's stress is a tension along the lines of force, and is

**equal**to B 2 /87r only when B = H, and there is no magnetization.Heusler 2 in 1903 that certain alloys of the non-magnetic metal manganese with other non-magnetic substances were strongly magnetizable, their susceptibility being in some cases

**equal**to that of cast iron.The tranverse electromotive force is

**equal**to KCH/D, where C is the current, H the strength of the field, D the thickness of the metal, and K a constant which has been termed the rotatory power, or rotational coefficient.If V is the volume of a ball, H the strength of the field at its centre, and re its apparent susceptibility, the force in the direction x is f= K'VH X dH/dx; and if K',, and are the apparent susceptibilities of the same ball in air and in liquid oxygen, K' Q -K'o is

**equal**to the difference between the susceptibilities of the two media.Meyer thinks that the susceptibilities of the metals praseodymium, neodymium, ytterbium, samarium, gadolinium, and erbium, when obtained in a pure form, will be found to

**equal**or even exceed those of the well-known ferromagnetic metals.The strength of the induced current is - HScosO/L, where 0 is the inclination of the axis of the circuit to the direction of the field, and L the coefficient of self-induction; the resolved part of the magnetic moment in the direction of the field is

**equal**to - HS 2 cos 2 6/L, and if there are n molecules in a unit of volume, their axes being distributed indifferently in all directions, the magnetization of the substance will be-3nHS 2 /L, and its susceptibility - 3S 2 /L (Maxwell, Electricity and Magnetism, § 838).In particular, he found that the calculated velocity with which it transmitted electromagnetic disturbances was

**equal**to the observed velocity of light; hence he was led to believe, not only that his medium and the ether were one and the same, but, further, that light itself was an electromagnetic phenomenon.A portion of the estate,

**equal**in size to the average holdings, is left to the owner, without, however, the proviso that this portion must necessarily coincide with the administrative centre, the manor or family house.He was lodged in `the Louvre, received the grant of an income

**equal**to that he had hitherto enjoyed, and, with the title of "veteran pensioner" in lieu of that of "foreign associate" (conferred in 1772), the right of voting at the deliberations of the Academy.Latreille,2 rightly estimating the value of these differences, though he was not an original worker in the field of vertebrate zoology, proposed to separate Brongniart's Batrachia from the class of Reptilia proper, as a group of

**equal**value, for which he retained the Linnaean name of Amphibia.Both religions were of Oriental origin; they were propagated about the same time, and spread with

**equal**rapidity on account of the same causes, viz.It is

**equal**to the actual diameter of the cylinder of rays admitted by a telescope.No other country has been able to

**equal**Brazil in the production of coffee, and under better labour conditions the country might compete with the foremost in the production of cane sugar, cotton and tobacco.The policy of many of Pombal's measures is more than questionable; but his admission of all races to

**equal**rights in the eye of the law, his abolition of feudal privileges, and the firmer organization of the powers of the land which he introduced, powerfully co-operated towards the development of the capabilities of Brazil.Sustained by their enthusiasm, however, the recruits displayed

**equal**courage, and, at the end of four hours' stubborn fighting, their defence was still intact.And, their own predominance being assured by their numerical strength and influence, they accorded

**equal**shares of power to the other monti.The literary history of Siena, while recording no gifts to the world

**equal**to those bequeathed by Florence, and without the power and originality by which the latter became the centre of Italian culture, can nevertheless boast of some illustrious names.The subdeacons, no doubt, became a necessity when the deacons, whose number was limited to seven in memory of their original institution, were no longer

**equal**to their duties in the " regions " of the imperial city, and left their lower work, such as preparation of the sacred vessels, to their subordinates.Pre-Cape Rocks principally the yellow wood (Podocarpus)., sneezewood (Pteroxylon utile), stinkwood (Oreodaphne bullata), black ironwood (Olea laurifolia), white ironwood (Vepris lanceolata), and umtomboti (Exoecaria africana); all are very useful woods, and the yellow wood, sneezewood, stinkwood and ironwood when polished have grain and colour

**equal**to maple, walnut and ebony.Colenso's bold advocacy of the cause of the natives - which he maintained with vigour until his death (in 1883) - attracted almost

**equal**attention.The Bulgarians, Serbs, Croats and Avars in the southern provinces were subdued with

**equal**ease.The alliance was cemented in July by a military demonstration, of which Jellachich was the hero, at Vienna; as the result of which the government mustered up courage to declare publicly that the basis of the Austrian state was " the recognition of the

**equal**rights of all nationalities."Its object was to resist the anti-clerical tendencies of the Liberals, and for this purpose it appealed to the " nationalities " against the dominant Magyar parties, the due enforcement of the Law of

**Equal**Rights of Nationalities (1868) forming a main item of its programme.In their efforts to establish Hungarian independence on the firm basis of national efficiency they had succeeded in changing their country from one of very backward economic conditions into one which promised to be in a position to hold its own on

**equal**terms with any in the world.Since its reorganization in 1869 the academy has, however, paid

**equal**attention to the various departments of history, archaeology, national economy and the physical sciences.It would be difficult, in the whole range of scientific literature, to point to a memoir of

**equal**brilliancy with that published (divided into three parts) in the volumes of the Academy for 1784, 1 785 and 1786.Seven flous are regarded as

**equal**to the French five-centime piece.But it does not mean the same as 5 X 21 lb, though the two are

**equal**, i.e.The important thing to notice is that where, in any of these five cases, one statement is followed by another, the second is not to be regarded as obtained from the first by logical reasoning involving such general axioms as that " if equals are taken from equals the remainders are

**equal**"; the fact being that the two statements are merely different ways of expressing the same relation.To say, for instance, that X is

**equal**to A -B, is the same thing as to say that X is a quantity such that X and B, when added, make up A; and the above five statements of necessary connexion between two statements of equality are in fact nothing more than definitions of the symbols -, m of, =,, and loga.We can, however, find a number whose square shall be as nearly

**equal**to 5 as we please, and it is this number that we treat arithmetically as 1 15.We find that the product a m X a m X a m is

**equal**to a im; and, by definition, the product; /a X -la X Ala is**equal**to a, which is a'.One of the relations most commonly illustrated in this way is the time-relation; the passage of time being associated with the passage of a point along a straight line, so that

**equal**intervals of time are represented by**equal**lengths.We take a fixed line OX, usually drawn horizontally; for each value of X we measure a length or abscissa ON

**equal**to x.L, and draw an ordinate NP at right angles to OX and**equal**to the corresponding value of y .Graphic representation thus rests on the principle that

**equal**numerical quantities may be represented by**equal**lengths, and that a quantity mA may be represented by a length mL, where A and L are the respective units; and the science of graphics rests on the converse property that the quantity represented by pL is pA, i.e.Here each member is a number, and the equation may, by the commutative law for multiplication, be written 2(x+I) - 4(x-2) This means that, whatever unit A we take, 2(x+ I) A Sand 5 4(x-2) A are

**equal**.Thus, if we have an equation P=Q, where P and Q are numbers involving fractions, we can clear of fractions, not by multiplying P and Q by a number m, but by applying the

**equal**multiples P and Q to a number m as unit.Thus 2 x 2 - I - x +x-2 (x _ I) (x+2) is

**equal**to x + 2 q, except when x=1.We know that (A+a)" is

**equal**to a multinomial of n+I terms with unknown coefficients, and we require to find these coefficients.In this table any number is

**equal**to the sum of the numbers which lie horizontally above it in the preceding Column, And The Difference Of Any Two Numbers In A Column Is**Equal**To The Sum Of The Numbers Horizontally Between Them In The Preceding Column.The consideration of cases where two roots are

**equal**belongs to the theory of equations (see Equation).We want to express (23/13) 3 in the form a l b, where b is nearly

**equal**to i.We cannot, for instance, say that the fraction C _2 I is arithmetically

**equal**to x+I when x= I, as well as for other values of x; but we can say that the limit of the ratio of x 2 - I to x - I when x becomes indefinitely nearly**equal**to I is the same as the limit of x+ On the other hand, if f(y) has a definite and finite value for y = x, it must not be supposed that this is necessarily the same as the limit which f (y) approaches when y approaches the value x, though this is the case with the functions with which we are usually concerned.We know that log l oN(I+9) = log l oN+log 10 (I+0), and inspection of a table of logarithms shows that, when 0 is small, log 10 (I+B);s approximately

**equal**to X0, where X is a certain constant, whose value is.Every unit of the rth species which does not vanish is the product of r different units of the first species; two such units are independent unless they are permutations of the same set of primary units e i, in which case they are

**equal**or opposite according to the usual rule employed in determinants.The full title is ilm al jebr wa'l-mugabala, which contains the ideas of restitution and comparison, or opposition and comparison, or resolution and equation, jebr being derived from the verb jabara, to reunite, and mugabala, from gabala, to make

**equal**.The rateable value of the parish being known, so much on each pound of the rateable value as will

**equal**the amount required to be raised is levied, and is known as the "rate."After affirming that the Serbs, Croats and Slovenes constitute a single nation and appealing to the right of self-determination, it declared in favour of complete national unity under the Karagjorgjevic dynasty, " a constitutional democratic and parliamentary monarchy, equality of the three national names and flags, of the Cyrilline and Latin alphabets, and of the Orthodox Catholic and Mussulman religions,

**equal**rights for all citizens, universal suffrage in parliamentary and municipal life, and the freedom of the Adriatic to all nations."Fortunately the new Giolitti and Vesnie Cabinets showed

**equal**moderation and skill in restraining the hotheads on both sides, and the new Foreign Minister, Count Sforza, was assisted by a personal knowledge of Serbian and Balkan problems all too rare among western statesmen.He was the sole male survivor of the ancient royal line; his valour and ability were universally recognized, and in Absalon, elected bishop of Roskilde in 1158, he possessed a minister of

**equal**genius and patriotism.We imagine a wave-front divided o x Q into elementary rings or zones - often named after Huygens, but better after Fresnelby spheres described round P (the point at which the aggregate effect is to be estimated), the first sphere, touching the plane at 0, with a radius

**equal**to PO, and the succeeding spheres with radii increasing at each step by IX.The component vibrations at P due to the successive zones are thus nearly

**equal**in amplitude and opposite in phase (the phase of each corresponding to that of the infinitesimal circle midway between the boundaries), and the series which we have to sum is one in which the terms are alternately opposite in sign and, while at first nearly constant in numerical magnitude, gradually diminish to zero.For the aggregate effect of the secondary waves is the half of that of the first Fresnel zone, and it is the central element only of that zone for which the distance to be travelled is

**equal**to r.Let us conceive the zone in question to be divided into infinitesimal rings of

**equal**area.The effects due to each of these rings are

**equal**in amplitude and of phase ranging uniformly over half a complete period.The middle element alone contributes without deduction; the effect of every other must be found by introduction of a resolving factor,

**equal**to cos 0, if 0 represent the difference of phase between this element and the resultant.In the applications with which we are concerned, t, n are very small quantities; and we may take P = x yn - At the same time dS may be identified with dxdy, and in the de nominator p may be treated as constant and

**equal**to f.By the principle of energy the illumination over the entire focal plane must be

**equal**to that over the diffracting area; and thus, in accordance with the suppositions by which (3) was obtained, its value when integrated from E= co to = -1-x, and from n = - oo to n = -1-oo should be**equal**to ab.Let us now consider the distribution of brightness in the image of a double line whose components are of

**equal**strength, and at such an angular interval that the central line in the image of one coincides with the first zero of brightness in the image of the other.We conclude that a double line cannot be fairly resolved unless its components subtend an angle exceeding that subtended by the wave-length of light at a distance

**equal**to the horizontal aperture.If the angle subtended by the components of a double line be twice that subtended by the wave-length at a distance

**equal**to the horizontal aperture, the central bands are just clear of one another, and there is a line of absolute blackness in the middle of the combined images.One of these, of width

**equal**, say, to one-tenth of an inch, is inserted in front of the object-glass, and the telescope, carefully focused all the while, is drawn gradually back from the grating until the lines are no longer seen.It has already been suggested that the principle of energy requires that the general expression for I 2 in (2) when integrated over the whole of the plane, n should be

**equal**to A, where A is the area of the aperture.Analytically expressed ff+ co x I 2 d dn=ff dxdy= A (9) We have seen that Io (the intensity at the focal point) was

**equal**to A 2 /X 2 f2.When z is great, the descending series (io) gives i 2J 1 (z) = 2 sin (z1 7r) 22; (2) z so that the places of maxima and minima occur at

**equal**intervals.If the angular interval between the components of a double star were

**equal**to twice that expressed in equation (15) above, the central disks of the diffraction patterns would be just in contact.Foucault, who employed a scale of

**equal**bright and dark alternate parts; it was found to be proportional to the aperture and independent of the focal length.Verdet has compared Foucault's results with theory, and has drawn the conclusion that the radius of the visible part of the image of a luminous point was

**equal**to half the radius of the first dark ring.When the interval is very small the discrepancy, though mathematically existent, produces no practical effect, and the illumination at B due to P is as important as that due to A, the intensities of the two luminous sources being supposed

**equal**.Observed through this the structure of some wire gauze just disappeared at a distance from the eye

**equal**to 17 in., the gauze containing 46 meshes to the inch.Since at these points the resultant due to the whole aperture is zero, any two portions into which the whole may be divided must give

**equal**and opposite resultants.In the case of a single lens of glass with the most favourable curvatures, Sf is about

**equal**to a 2 f, so that a 4 must not exceed off.When parallel rays fall directly upon a spherical mirror the longitudinal aberration is only about one-eighth as great as for the most favourably shaped single lens of

**equal**focal length and aperture.We conclude that, with a grating composed of transparent and opaque parts, the utmost light obtainable in any one spectrum is in the first, and there amounts to I/wr 2, or about 6, and that for this purpose W a and d must be

**equal**.For the alteration of wave-length entails, at the two limits of a diffracted wave-front, a relative retardation

**equal**to mndX.In the present application 4' is not necessarily

**equal**to; but if P correspond to a line upon the grating, the difference of retardations for consecutive positions of P, so far as expressed by the term of the first order, will be**equal**to mX (m integral), and therefore without influence, provided v (sin 0-sin0') = nzX (11), where a denotes the constant interval between the planes containing the lines.