solution+method

A function of class $C^1$

We call $C$ a module of ellipticity

The natural definition of addition and multiplication

A type of convergence

A problem of uniqueness

The condition of ellipticity

The hypothesis of positivity

The method of proof

The point of increase (decrease)

A polynomial of degree $n$

A circle of radius $n$

A matrix of order $n$

An algebraic equation of degree $n$ (of first (second, third) degree)

A differential equation of order $n$ (of first (second, third) order; но an integral equation of the first (second) kind)

A manifold of dimension $n$

A function of bounded variation

The (an) equation of motion

The (a) velocity of propagation

An element of finite order

A solution of polynomial growth

A ball of radius $r$

A function of norm $p$

A matrix of full rank

Однако: (the) elements of the form $a=b+c$ (of the form (1))

A function of class $C^1$

We call $C$ a module of ellipticity

The natural definition of addition and multiplication

A type of convergence

A problem of uniqueness

The condition of ellipticity

The hypothesis of positivity

The method of proof

The point of increase (decrease)

A polynomial of degree $n$

A circle of radius $n$

A matrix of order $n$

An algebraic equation of degree $n$ (of first (second, third) degree)

A differential equation of order $n$ (of first (second, third) order; но an integral equation of the first (second) kind)

A manifold of dimension $n$

A function of bounded variation

The (an) equation of motion

The (a) velocity of propagation

An element of finite order

A solution of polynomial growth

A ball of radius $r$

A function of norm $p$

A matrix of full rank

Однако: (the) elements of the form $a=b+c$ (of the form (1))

exĭtus, a, um, Part., from exeo, II.

exĭtus, ūs, m. [exeo], a going out or forth, egress, departure (class., esp. in the trop. signif.).

Lit.:

reditum mihi gloriosum injuria tua dedit, non exitum calamitosum,

Cic. Par. 4, 29:

omni exitu et pabulatione interclusi,

Caes. B. G. 7, 44 fin.:

exitum sibi parere,

id. B. C. 3, 69, 3.—In plur.:

singulorum hominum occultos exitus asservare,

Caes. B. C. 1, 21, 4; 1, 25, 4. —Of things:

introitusque elementis redditus exstat,

Lucr. 6, 494:

exitus ut classi felix faustusque daretur,

a setting sail, departure, id. 1, 100:

amnis,

a flowing out, discharge, id. 6, 727: animaï (i. e. venti), a bursting or rushing out, id. 6, 586; cf. Quint. 1, 11, 7.—

Transf., concr., way of egress, outlet, passage:

exitum non habent, ac pervium non est,

Varr. L. L. 5, § 145 Müll.:

cum angusto portarum exitu se ipsi premerent,

Caes. B. G. 7, 28, 3:

in exitu paludis,

mouth, Plin. 2, 103, 106, § 226:

cibi,

vent, id. 11, 34, 40, § 116 et saep.:

si de multis nullus placet exitus,

Juv. 6, 33.—In plur.:

insula undique exitus maritimos habet,

Cic. Verr. 2, 2, 75, § 185:

septem exitus e domo fecerat,

Liv. 39, 51, 5; Col. 6, 30, 8:

alvorum,

Plin. 21, 14, 48, § 82 et saep.

Trop.

A way out, an end, close, conclusion, termination (syn.: eventus, eventum).

In gen.:

hujus orationis difficilius est exitum quam principium invenire,

end, close, Cic. de Imp. Pomp. 1, 3; cf.: quemadmodum expediam exitum hujus institutae orationis, non reperio, id. Fam. 3, 12, 2:

exitus fuit orationis,

Caes. B. G. 4, 8, 1:

ut tragici poëtae, cum explicare argumenti exitum non potestis, confugitis ad deum,

Cic. N. D. 1, 20, 53:

adducta ad exitum quaestio est,

id. Tusc. 5, 6, 15; cf.:

ad exitum pervenire,

id. Fam. 10, 22, 2; id. Or. 33, 116:

ita magnarum initia rerum celerem et facilem exitum habuerunt,

Caes. B. C. 3, 22 fin.:

verba quae casus habent in exitu similes,

at the end, Cic. Or. 49, 164; cf.

in the foll.: fugam quaerebamus omnes, quae ipsa exitum non habebat,

end, aim, id. Phil. 5, 16, 42:

hinc omne principium, huc refer exitum,

Hor. C. 3, 6, 6 et saep.:

in exitu est meus consulatus,

Cic. Mur. 37, 80; cf.: in exitu jam annus erat Liv. 35, 10, 1:

superioris anni,

id. 30, 26, 2:

veris,

Plin. 17, 22, 35, § 170:

oppugnationis,

Caes. B. C. 3, 9, 8:

mimi, fabulae,

the catastrophe, conclusion, Cic. Cael. 27, 65:

vitae,

end of life, latter end, Nep. Eum. 13; cf.:

vitae mortisque,

Vell. 2, 7, 1.—In plur.:

tristes exitus habuit consulatus,

Cic. Brut. 34, 128: eae causae sunt plenissimae, quae plurimos exitus dant ad ejusmodi degressionem, outlets, i. e. opportunities, id. de Or. 2, 77, 312: habent exitus aut in a aut in e, etc., Varr. L. L. 10, § 62 Müll.—

In partic., end of life, end, death:

natura ad humanum exitum (Romulum) abripuit,

Cic. Rep. 1, 16 fin.:

duravere usque ad Sejani exitum,

Plin. 8, 58, 74, § 197; Amm. 14, 11:

exitus in dubio est,

Ov. M. 12, 522:

Thrasymachi,

Juv. 7, 204:

saevus et illum exitus eripuit,

id. 10, 127; 271.—In plur.:

nonnumquam bonos exitus habent boni,

Cic. N. D. 3, 37, 89:

non igitur fatales exitus habuerunt,

id. Div. 2, 9, 24.—

A means, method, way, device, solution of a difficulty:

cum autem exitus ab utroque datur conturbato errantique regi,

Cic. Fin. 5, 22, 63:

non solum viam quaestus invenerunt, verum etiam exitum ac rationem defensionis,

id. Verr. 2, 3, 82, § 190:

jam nullum fortunis communibus exitum reperietis,

id. Dom. 47, 123.—

Issue, result, event, i. q. eventus:

si mihi alterutrum de eventu atque exitu rerum promittendum est,

Cic. Fam. 6, 1, 5:

in unum exitum spectare,

id. de Or. 1, 20, 92:

videtur ad exitum venisse quaestio,

id. Tusc. 5, 7, 18; id. Fin. 2, 1, 3:

neque exitum legis esse in meretrice publicanda,

i. e. the law would be without proper effect, id. Inv. 2, 40, 118, v. the context:

de exitu rerum sentire,

Caes. B. G. 7, 52 fin.:

incerto etiam nunc exitu victoriae,

id. ib. 7, 62, 6:

de exitu fortunarum suarum consultabant,

id. ib. 7, 77, 1; cf. id. ib. 3, 8, 3; and:

prudens futuri temporis exitum Caliginosa nocte premit deus,

events, Hor. C. 3, 29, 29: ut quae rei publicae polliceremur, exitu praestaremus, Planc. ap. Cic. Fam. 10, 8, 3:

exitum rei imponere,

Liv. 37, 19, 1:

quaestiones ad exitum perductae,

id. 40, 19, 10:

ad exitum spei pervenire,

accomplishment, id. 5, 12, 4; so,

serae exitum spei exspectare,

id. 5, 6, 2:

sine exitu esse,

without result, id. 32, 40, 3.—In plur.:

fortasse haec omnia meliores habebunt exitus,

Cic. Fam. 2, 16, 6:

quae (responsa haruspicum) aut nullos habuerint exitus aut contrarios,

id. Div. 2, 24, 52:

Liber vota bonos ducit ad exitus,

Hor. C. 4, 8, 34; cf.:

(fortuna) Belli secundos reddidit exitus,

id. ib. 4, 14, 38.—Prov.:

exitus acta probat,

the event justifies the deed, Ov. H. 2, 85.

a statistical theory and method for drawing conclusions about the future occurrence of a given parameter of a statistical distribution by calculating from prior data on its frequency of occurrence. The theory is useful in the solution of theoretical and applied problems in science, industry, and government, for example, in econometrics and finance.

SUBJECT AREA: Metallurgy

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His father was a mason and brickmaker but, anxious to improve himself, Cort set up in London in 1765 as a navy agent, said to have been a profitable business. He recognized that, at that time, the conversion of pig iron to malleable or wrought iron, which was needed in increasing quantities as developments in industry and mechanical engineering gathered pace, presented a bottleneck in the ironmaking process. The finery hearth was still in use, slow and inefficient and requiring the scarce charcoal as fuel. To tackle this problem, Cort gave up his business and acquired a furnace and slitting mill at Fontley, near Fareham in Hampshire. In 1784 he patented his puddling process, by which molten pig iron on the bed of a reverberatory furnace was stirred with an iron bar and, by the action of the flame and the oxygen in the air, the carbon in the pig iron was oxidized, leaving nearly pure iron, which could be forged to remove slag. In this type of furnace, the fuel and the molten iron were separated, so that the cheaper coal could be used as fuel. It was the stirring action with the iron bar that gave the name "puddling" to the process. Others had realized the problem and reached a similar solution, notably the brothers Thomas and George Cranage, but only Cort succeeded in developing a commercially viable process. The laborious hammering of the ball of iron thus produced was much reduced by an invention of the previous year, 1783. This too was patented. The iron was passed between grooved rollers to form it into bars. Cort entered into an agreement with Samuel Jellico to set up an ironworks at Gosport to exploit his inventions. Samuel's father Adam, Deputy Paymaster of the Navy, advanced capital for this venture, Cort having expended much of his own resources in the experimental work that preceded his inventions. However, it transpired that Jellico senior had, unknown to Cort, used public money to advance the capital; the Admiralty acted to recover the money and Cort lost heavily, including the benefits from his patents. Rival ironmasters were quick to pillage the patents. In 1790, and again the following year, Cort offered unsuccessfully to work for the military. Finally, in 1794, at the instigation of the Prime Minister, William Pitt the Younger, Cort was paid a pension of £200 per year in recognition of the value of his improvements in the technology of ironmaking, although this was reduced by deductions to £160. After his death, the pension to his widow was halved, while some of his children received a pittance. Without the advances made by Cort, however, the iron trade could not have met the rapidly increasing demand for iron during the industrial revolution.

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SUBJECT AREA: Metallurgy

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He was apprenticed to his uncles, makers of metalware, in 1815 and showed such aptitude for business that he was taken into partnership. On their deaths, Elkington assumed sole ownership of the business. In conjunction with his cousin Henry (1810–52), by unrelenting enterprise, he established an industry for electroplating and electrogilding. Up until c.1840, silver-plated goods were produced by rolling or soldering thin sheets of silver to a base metal, such as copper. Back in 1801, the English chemist William Wollaston had deposited one metal upon another by means of an electric current generated from a voltaic pile or battery. In the 1830s, certain inventors, such as Bessemer used this result to produce plated articles and these efforts in turn induced the Elkingtons to apply the method in their trade. In 1836 and 1837 they took out patents for "mercurial gilding", and one patent of 1838 refers to a separate electric current. In 1840 they bought from John Wright, a Birmingham surgeon, his discovery of what proved to be the best electroplating solution: namely, solutions of cyanides of gold and silver in potassium cyanide. They also purchased rights to use the electric machine invented by J.S. Woolrich. Armed with these techniques, the Elkingtons produced in their large new works in Newhall Street a wide range of gold-and silver-plated decorative and artistic ware. Henry was particularly active on the artistic side of the business, as was their employee Alexander Parkes. For some twenty-five years, Britain enjoyed a virtual monopoly of this kind of ware, due largely to the enterprise of the Elkingtons, although by the end of the century rising tariffs had closed many foreign markets and the lead had passed to Germany. George spent all his working life in Birmingham, taking some part in the public life of the city. He was a governor of King Edward's Grammar School and a borough magistrate. He was also a caring employer, setting up houses and schools for his workers.

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SUBJECT AREA: Textiles

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Early drawing and spinning processes failed to give linen yarn the requisite fineness and homogeneity. In 1810 Napoleon offered a prize of a million francs for a successful flax-spinning machine as part of his policy of stimulating the French textile industries. Spurred on by this offer, Girard suggested three improvements. He was too late to win the prize, but his ideas were patented in England in 1814, although not under his own name. He proposed that the fibres should be soaked in a very hot alkaline solution both before drawing and immediately before they went to the spindles. The actual drawing was to be done by passing the dried material through combs or gills that moved alternately; gill drawing was taken up in England in 1816. His method of wet spinning was never a commercial success, but his processes were adopted in part and developed in Britain and spread to Austria, Poland and France, for his ideas were essentially good and produced a superior product. The successful power-spinning of linen thread from flax depended primarily upon the initial processes of heckling and drawing. The heckling of the bundles or stricks of flax, so as to separate the long fibres of "line" from the shorter ones of "tow", was extremely difficult to mechanize, for each strick had to be combed on both sides in turn and then in the reverse direction. It was to this problem that Girard next turned his attention, inventing a successful machine in 1832 that subsequently was improved in England. The strick was placed between two vertical sheets of combs that moved opposite to each other, depositing the tow upon a revolving cylinder covered with a brush at the bottom of the machine, while the holder from which the strick was suspended moved up and down so as to help the teeth to penetrate deeper into the flax. The tow was removed from the cylinder at the bottom of the machine and taken away to be spun like cotton. The long line fibres were removed from the top of the machine and required further processing if the yarn was to be uniform.

When N.L.Sadi Carnot's book Réflexions sur la puissance motrice du feu, was published in 1824, Girard made a favourable report on it.

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SUBJECT AREA: Metallurgy

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The son of a Congregationalist minister, Hall was educated at Oberlin College. There he was instructed in chemistry by Professor F.F.Jewett, a former student of the German chemist Friedrich Wöhler, who encouraged Hall to believe that there was a need for a cheap process for the manufacture of aluminium. After graduating in 1885, Hall set to work in his private laboratory exploring the method of fused salt electrolysis. On Wednesday 10 February 1886 he found that alumina dissolved in fused cryolite "like sugar in water", and that the bath so produced was a good conductor of electricity. He contained the solution in a pure graphite crucible which also acted as an efficient cathode, and by 16 February 1886 had produced the first globules of metallic aluminium. With two backers, Hall was able to complete his experiments and establish a small pilot plant in Boston, but they withdrew after the US Patent Examiners reported that Hall's invention had been anticipated by a French patent, filed by Paul Toussaint Héroult in April 1886. Although Hall had not filed until July 1886, he was permitted to testify that his invention had been completed by 16 February 1886 and on 2 April 1889 he was granted a seventeen-year monopoly in the United States. Hall now had the support of Captain A.E. Hunt of the Pittsburgh Testing Institute who provided the capital for establishing the Pittsburgh Reduction Company, which by 1889 was selling aluminium at $1 per pound compared to the $15 for sodium-reduced aluminium. Further capital was provided by the banker Andrew Mellon (1855–1937). Hall then turned his attention to Britain and began negotiations with Johnson Matthey, who provided land on a site at Patricroft near Manchester. Here the Aluminium Syndicate, owned by the Pittsburgh Reduction Company, began to produce aluminium in July 1890. By this time the validity of Hall's patent was being strongly contested by Héroult and also by the Cowles brothers, who attempted to operate the Hall process in the United States. Hall successfully sued them for infringement, and was confirmed in his patent rights by the celebrated ruling in 1893 of William Howard Taft, subsequently President of the USA. In 1895 Hall's company changed its name to the Pittsburgh Aluminium Company and moved to Niagara Falls, where cheap electrical power was available. In 1903 a legal compromise ended the litigation between the Hall and Héroult organizations. The American rights in the invention were awarded to Hall, and the European to Héroult. The Pittsburgh Aluminium Company became the Aluminium Company of America on 1 January 1907. On his death he left his estate, worth about $45 million, for the advancement of education.

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SUBJECT AREA: Horology, Ports and shipping

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John Harrison was the son of a carpenter and was brought up to that trade. He was largely self-taught and learned mechanics from a copy of Nicholas Saunderson's lectures that had been lent to him. With the assistance of his younger brother, James, he built a series of unconventional clocks, mainly of wood. He was always concerned to reduce friction, without using oil, and this influenced the design of his "grasshopper" escapement. He also invented the "gridiron" compensation pendulum, which depended on the differential expansion of brass and steel. The excellent performance of his regulator clocks, which incorporated these devices, convinced him that they could also be used in a sea dock to compete for the longitude prize. In 1714 the Government had offered a prize of £20,000 for a method of determining longitude at sea to within half a degree after a voyage to the West Indies. In theory the longitude could be found by carrying an accurate timepiece that would indicate the time at a known longitude, but the requirements of the Act were very exacting. The timepiece would have to have a cumulative error of no more than two minutes after a voyage lasting six weeks.

In 1730 Harrison went to London with his proposal for a sea clock, supported by examples of his grasshopper escapement and his gridiron pendulum. His proposal received sufficient encouragement and financial support, from George Graham and others, to enable him to return to Barrow and construct his first sea clock, which he completed five years later. This was a large and complicated machine that was made out of brass but retained the wooden wheelwork and the grasshopper escapement of the regulator clocks. The two balances were interlinked to counteract the rolling of the vessel and were controlled by helical springs operating in tension. It was the first timepiece with a balance to have temperature compensation. The effect of temperature change on the timekeeping of a balance is more pronounced than it is for a pendulum, as two effects are involved: the change in the size of the balance; and the change in the elasticity of the balance spring. Harrison compensated for both effects by using a gridiron arrangement to alter the tension in the springs. This timekeeper performed creditably when it was tested on a voyage to Lisbon, and the Board of Longitude agreed to finance improved models. Harrison's second timekeeper dispensed with the use of wood and had the added refinement of a remontoire, but even before it was tested he had embarked on a third machine. The balance of this machine was controlled by a spiral spring whose effective length was altered by a bimetallic strip to compensate for changes in temperature. In 1753 Harrison commissioned a London watchmaker, John Jefferys, to make a watch for his own personal use, with a similar form of temperature compensation and a modified verge escapement that was intended to compensate for the lack of isochronism of the balance spring. The time-keeping of this watch was surprisingly good and Harrison proceeded to build a larger and more sophisticated version, with a remontoire. This timekeeper was completed in 1759 and its performance was so remarkable that Harrison decided to enter it for the longitude prize in place of his third machine. It was tested on two voyages to the West Indies and on both occasions it met the requirements of the Act, but the Board of Longitude withheld half the prize money until they had proof that the timekeeper could be duplicated. Copies were made by Harrison and by Larcum Kendall, but the Board still continued to prevaricate and Harrison received the full amount of the prize in 1773 only after George III had intervened on his behalf.

Although Harrison had shown that it was possible to construct a timepiece of sufficient accuracy to determine longitude at sea, his solution was too complex and costly to be produced in quantity. It had, for example, taken Larcum Kendall two years to produce his copy of Harrison's fourth timekeeper, but Harrison had overcome the psychological barrier and opened the door for others to produce chronometers in quantity at an affordable price. This was achieved before the end of the century by Arnold and Earnshaw, but they used an entirely different design that owed more to Le Roy than it did to Harrison and which only retained Harrison's maintaining power.

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See also: Burgi, Jost; Huygens, Christiaan