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This body delegated its functions to its secretary, W.

42Once in a while they did things together socially, usually to fulfill mandated coupling at necessary functions and seldom for the entertainment of either.

33On his release Consalvi hastened to his master's assistance; and he was soon after allowed to resume his functions under the restored pontificate at Rome.

33It is from such a living and assimilating cell, performing as it does all the vital functions of a green plant, that, according to current theory, all the different cell-forms of a higher plant have been differentiated in the course of descent.

33The central administration (administration centrale) comprises three classes of functions which together constitute Ia police.

10There is also an official performing the functions of a director at Bastia, in Corsica.

10The camping facilities were secondary to the main park functions, multiple ball fields, tennis courts and twenty-four horse shoe pits, for the serious pitcher.

11Besides its legislative functions the House prepares articles of impeachment and prosecutes the proceedings before the Senate.

11It is worn by bishops, priests, deacons and subdeacons under the other eucharistic vestments, either at Mass or at functions connected with it.

11The conseil colonial, besides its advisory functions, discusses and votes the budget, determines the nature of the taxes, has supreme control over the tariffs, and extensive powers in the administration of colonial domains.

11Soon his functions expanded.

11The theory of droit administratif lays down the principle that an agent of the government cannot be prosecuted or sued for acts relating to his administrative functions before the ordinary tribunals.

11If anything ail a man, so that he does not perform his functions, if he have a pain in his bowels even--for that is the seat of sympathy--he forthwith sets about reforming--the world.

11We are so degraded that we cannot speak simply of the necessary functions of human nature.

11Her life had no external aims--only a need to exercise her various functions and inclinations was apparent.

11He no longer pressed him to attend public functions.

00The Senate elects a president, confirms or rejects the nominations of the governor, and acts as a court of impeachment for the trial of public officers, besides sharing in legislative functions.

00The mayor of the palace, however, did not remain restricted to domestic functions; he had the discipline of the palace and tried persons who resided there.

00I Kings vi., vii.), the sacrifices and festivals and the functions of priests and prince are prescribed, a stream issuing from under the temple is to vivify the Dead Sea and fertilize the land (this is meant literally), the land is divided into parallel strips and assigned to the tribes.

00Archbishop Longley said in his opening address, however, that they had no desire to assume "the functions of a general synod of all the churches:in full communion with the Church of England," but merely to "discuss matters of practical interest, and pronounce what we deem expedient in resolutions which may serve as safe guides to future action."

00The ministry of war is divided into branches for infantry, cavalry, &c.and services for special subjects such as military law, explosives, health, &c. The general staff (stat major de larme) has its functions classed as follows: personnel; material and finance; 1st bureau (organization and mobilization), 2nd (intelligence), 3rd (military operations and training) and 4th (communications and transport); and the famous historical section.

00The naval prefect is assisted by a rearadmiral as chief of the staff (except at Lorient and Rochefort, where the office is filled by a captain), and a certain number of other officers, the special functions of the chief of the staff having relation principally to the efficien.cy and personnel of the fleet, while the major-general, who is usually a rear-admiral, is concerned chiefly with the materiel.

00in place of a council general, there is a colonial council which fulfils the functions of a council general.

00Leibnitz, in accord with the distinctive principle of his philosophy, affirmed the absolute independence of mind and body as distinct monads, the parallelism of their functions in life being due to the pre-established harmony.

00He was also the author of important papers in which he extended to complex quadratic forms many of Gauss's investigations relating to real quadratic forms. After 1864 he devoted himself chiefly to elliptic functions, and numerous papers on this subject were published by him in the Proc. Lond.

00At the time of his death he was engaged upon a memoir on the Theta and Omega Functions, which he left nearly complete.

00The three subjects to which Smith's writings relate are theory of numbers, elliptic functions and modern geometry; but in all that he wrote an "arithmetical" made of thought is apparent, his methods and processes being arithmetical as distinguished from algebraic. He had the most intense admiration of Gauss.

00The first was the appointment of a grand council with supreme judicial and financial functions P 7 ?

00It but remains to call attention to the fact that the earlier view of the liver as the seat of the soul gave way among many ancient nations to the theory which, reflecting the growth of anatomical knowledge, assigned that function to the heart, while, with the further change which led to placing the seat of soul-life in the brain, an attempt was made to partition the various functions of manifestations of personality among the three organs, brain, heart and liver, the intellectual activity being assigned to the first-named; the higher emotions, as love and courage, to the second; while the liver, once the master of the entire domain of soul-life as understood in antiquity, was degraded to serve as the seat of the lower emotions, such as jealousy, anger and the like.

00If the Rump or the Little Parliament had in a business-like spirit assumed and discharged the functions of a constituent assembly, such a foundation might have been provided.

005824) mentions a priest whose functions are characteristic of the place "L.

00It would be an anachronism to think of Francis as a philanthropist or a "social worker" or a revivalist preacher, though he fulfilled the best functions of all these.

00In the same city in which the administrative functions of the body politic are centralized there stifi exists the court of the spiritual potentate which in 1879 consisted of 1821 persons.

00Besides its legislative functions, the senate is the highest court of justice in the case of political offences or the impeachment of ministers.

00There is a council of state with advisory functions, which can also decide certain questions of administration, especially applications from local authorities and conflicts between ministries, and a court of accounts, which has the right of examining all details of state expenditure.

00In the provincial administrations the functions of the prefects have been curtailed.

00Besides possessing competence in regard to local government elections, which previously came within the jurisdiction of the provincial deputations, the provincial administrative juntas discharge magisterial functions in administrative affairs, and deal with appeals presented by private persons against acts of the communal and provincial administrations.

00Early in the 10th century the papacy fell into the hands of a noble family, known eventually as the counts of Tusculum, who almost succeeded in rendering the office hereditary, and in uniting the civil and ecclesiastical functions of the city under a single member of their house.

00The thirteen districts in their council nominated four caporioni, who acted in concert with a senator, appointed, like the podest of other cities, for supreme judicial functions.

00The proceeds of the sale of the suppressed convents and monasteries were partly converted into pensions for monks and nuns, and partly allotted to the municipal charity boards which had undertaken the educational and charitable functions formerly exercised by the religious orders.

00Of a far more complicated nature than these offerings are the Soma-sacrifices, which, besides the simpler ceremonies of this class, such as the Agnishtoma or "Praise of Agni," also include great state functions, such as the Rajasuya or consecration of a king, and the Asvamedha or horse-sacrifice, which, in addition to the sacrificial rites, have a considerable amount of extraneous, often highly interesting, ceremonial connected with them, which makes them seem to partake largely of the nature of public festivals.

00As in other cases where animal colonies are formed by organic union of separate individuals, there is ever a tendency for the polyp-colony as a whole to act as a single individual, and for the members to become subordinated to the needs of the colony and to undergo specialization for particular functions, with the result that they simulate organs and their individuality becomes masked to a greater or less degree.

00The sub-umbral cavity (s.c.) functions as a brood-space for the developing embryos, which are set free by rupture of the wall.

00The appendages show various types of form and structure corresponding to different functions.

00The observation that large groups of species of widely different habits present the same fundamental plan of structure; and that parts of the same animal or plant, the functions of which are very different, likewise exhibit modifications of a common plan.

00The observation of the existence of structures, in a rudimentary and apparently useless condition, in one species of a group, which are fully developed and have definite functions in other species of the same group.

00It is well known that Darwin was deeply impressed by differences in flora and fauna, which seemed to be functions of locality, and not the result of obvious dissimilarities of environment.

00Deposition of a bishop by a synod, or of a priest or deacon by his bishop, is to take effect even pending an appeal, and a cleric continuing his functions after sentence in first instance is to lose all right of appeal.

00The Assembly appoints a commission to exercise some of its functions during the intervals of its session.

00This was the centre of the life of the medieval city, the scene of all great public functions, such as the homage of the burghers to 1 Bavo, or Allowin (c. 589-c. 653), patron saint of Ghent, was a nobleman converted by St Amandus, the apostle of Flanders.

00In 1694 the apothecaries had increased from 114 to nearly 1000, and many of them, having acquired a knowledge of the uses of medicine, began to prescribe medicines for their customers and to assume the functions of the physician, who retorted in 1697 by establishing dispensaries, where medicines could be procured at their intrinsic value, or at cost price.

00The root differs from the shoot in the characters of its surface tissues, in the absence of the green assimilative pigment chlorophyll, in the arrangement of its vascular system and in the mode of growth at the apex, all features which are in direct relation to its normally subterranean life and its fixative and absorptive functions.

00The leaves, again, have special histological features adapted to the performance of their special functions.

00The remaining tissue of the plant-body, a tissue that we must regard phylogenetically as the remnant of the undifferentiated tissu~ of the primitive thallus, but which often undergoes further different,iation of its own, the better to fulfil its characteristically vital functions for the whole plant, is known, from its peripheral position in relation to the primitively central conducting tissue, as (3) the cortex.

00But hairs have a variety of other functions.

00bodies of the higher plants its functions are numerous.

00Wood thus altered is known as heart-wood, or duramen, as distinguished from the young sap-wood, or alburnum, which, forming a cylinder next the cambium, remains alive and carries on the active functions of the xylem, particularly the conduction of water.

00The whole of the writings of this time are dominated by a preoccupation with the functions of the different tissues, in itself an excellent standpoint for investigation, but frequently leading in the case of these early investigators to one-sided and distorted views of the facts of structure.

00We come, accordingly, to regard it as practically an exoskeleton, and its functions as distinctly subordinate to those of the protoplasm which it clothes.

00(5) There must be a supply of oxygen to the growing cell, for the protoplast is dependent upon this gas for the performance of its vital functions, and particularly for the liberation of the energy which is demanded in the constructive processes.

00Every 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.

00They may occur on all parts, buds, leaves, stems or roots, as shown by the numerous species of Cynips on oak, Phylloxera on vines, &c. The local damage is small, - but the general injury to assimilation, absorption and other functions, may be important if the numbers increase.

00It is sufficient to note here that cells were first of all discovered in various vegetable tissues by Robert Hooke in 1665 (Micrographia); Malpighi and Grew (1674-1682) gave the first clear indications of the importance of cells in the building up of plant tissues, but it was not until the beginning of the 19th century that any insight into the real nature of the cell and its functions was obtained.

00It is an integral part of an individual organization and as such the exercise of its functions must be governed by the organism as a whole.

00Tissues.The component parts of the tissues of which plants are composed may consist of but slightly modified cells with copious protoplasmic contents, or of cells which have been modified in various ways to perform their several functions.

00The term morphology, which was introduced into science by Goethe (1817), designates, in the first place, the study of the form and composition of the body and of the parts of which the body may consist; secondly, the relations of the parts of the same body; thirdly, the comparison of the bodies or parts of the bodies of plants of different kinds; fourthly, the study of the development of the body and of its parts (ontogeny); fifthly, the investigation of the historical origin and descent of the body and its parts (phylogeny); and, lastly, the consideration of the relation of the parts of the body to their various functions, a study that is known as organography.

00The earliest scientific result of the study of plants was the recognition of the fact that the various parts of the body are associated with the performance of different kinds of physiological work; that they are, in fact, organs discharging special functions.

00It is urged that the various parts are, as a matter of fact, organs; and that it is therefore inadmissible to ignore their functions, as is done in the foregoing definitions.

00Accepting this view of the phylogeny of the leaf, the perianthleaves (sepals and petals) and the foliage-leaves may be regarded as modified or metamorphosed sporophylls; that is, as leaves which are adapted to functions other than the bearing of spores.

00The sepals are generally organs for the protection of the flower-bud; the petals, for attracting insects by their conspicuous form and color; the foliage-leaves, for the assimilation of carbon dioxide and other associated functions.

00there is morphological differentiation, which can be traced in the distinction of the members of the body, root, stem, leaf, &c.; there is physiological differentiation, which can be traced in the adaptation of these members to various functions.

00If land forms may be compared to organs, the part they serve in the economy of the earth may, without straining the term, be characterized as functions.

00The narration of Herodotus is only a popular tradition which derives the origin of kingship from its judicial functions, considered as its principal and most beneficent aspect.

00One of the functions of this peculiar muscle (which is similarly developed in crocodiles, but absent, or not differentiated from the ilio-tibial and ilio-femoral mass, in other vertebrates) is that its contraction helps to close the second and third toes.

00The inconvenience of having a single ambo led to the substitution of two separate ambones, between which these various functions were divided, one on the south side of the chancel being for the reading of the gospel, and one on the north for reading the epistle.

00Admission to military command was won first, then admission to civil jurisdiction; a share in religious functions was won last of all.

00But in the more strictly crown, even if of quite humble origin, are "commanded" to court functions with their husbands.

00The judicial functions are discharged by four grades of officials - the local magistrates, the courts of common pleas, the quarterly courts (five in number) and the supreme court.

00Its functions, which are exceedingly various, are carried out by the different departments into which it is divided.

00- Alongside the local organs of the central government in Russia there are three classes of local elected bodies charged with administrative functions: (I) the peasant assemblies in the mir and the volost, ' From Catherine II.'s time to that of Alexander II.

00Above all, there was no dividing 'line between the judiciary and the administrative functions.

00p principles: the separation of the judicial and administrative functions, the independence of the judges and courts, the publicity of trials and oral procedure, the equality of all classes before the law.

00From the town The judge (ispravnik), who, in spite of the principle laid ordinary down in 1864, combines judicial and administrative functions, an appeal lies (as in the case of the justices of the peace) to an assembly of such judges; from these again there is an appeal to the district court (okrugniya sud), consisting of three judges; 4 from this to the court of appeal (sudebniya palata); while over this again is the senate, which, as the supreme court of cassation, can send a case for retrial for reason shown.

00The confusion of the judicial and administrative functions was introduced again by the appointment of officials as judges.

00In order to reply to accusations brought against them, or in order to be confirmed in their functions, they had to travel to the Golden Horde on the Volga or even to the camp of the grand khan in some distant part of Siberia, and the journey was considered so perilous that many of them, before setting out, made their last will and testament and wrote a parental admonition for the guidance of their children.

00the principle of the separation of the administrative and judicial functions) were largely curtailed.

00The state commissions, since their establishment in the 'seventies and the 'eighties, had increased their functions and influence.

00Grindal remonstrated, claiming some voice for the Church, and in June 1577 was suspended from his jurisdictional, though not his spiritual, functions for disobedience.

00And it is also probable that certain persons combined in their own individuality the functions of magician and sacrificer as well as soothsayer.

00The prince, it is true, has a central domain, but his functions are ecclesiastical and subordinate and his powers strictly limited (xlvi.

00Their functions are similar to those of the administrative officials in other states, with the exception that the governor does not possess the usual pardoning power but is ex officio a member of the pardoning board.

00When he realized the strength of the national reaction, he allowed the patriotic fascisti free rein to reestablish order and practically exercise many functions of Government, while he assumed an attitude of Olympic calm and posed as being au dessus de la melee, so as to avoid compromising himself with any party.

00Worship is simpler at the smaller shrines than at the more famous temples; and, as the rulers are the patrons of the religion and are brought into contact with the religious personnel, the character of the social organization leaves its mark upon those who hold religious and judicial functions alike.

00Their functions were political rather than religious, though their influence was by no means purely secular.

00Thus the Babylonian academies combined the functions of specialist law-schools, universities and popular parliaments.

00A supreme court of appeal, which also discharges the functions of a court of cassation, sits at Canea.

00It is clear that the rulers, as so P p commonly in ancient states, fulfilled priestly as well as royal functions.

00Indeed his functions as Labour adviser so occupied his time and attention, that it was thought desirable to relieve him in Aug.

00The ship which brought stamps and stamped paper to Wilmington in 1766 was not permitted to land, and the stampmaster was compelled by the people to take an oath that he would not exercise the functions of his office.

00Guided by such administrators as Warren Hastings, the East India Company had assumed more and more definitely the functions of government for a great part of India.

00They contrast with those of the Oligochaeta and Hirudinea by reason of their frequently close association with the gonads, the same organ sometimes serving the two functions of excretion and conveyance of the ova and spermatozoa out of the body.

00One of the functions of this official was to subsidize political pamphleteers, and Mirabeau had hoped to be so employed, but he ruined his chances by a series of writings on financial questions.

00Specially noteworthy in the Lezioni are the sections on human wants as the foundation of economical theory, on labour as the source of wealth, on personal services as economic factors, and on the united working of the great industrial functions.

00But the bulk of the work consists of problems leading to indeterminate equations of the second degree, and these universally take the form that one or two (and never more) linear or quadratic functions of one variable x are to be made rational square numbers by finding a suitable value for x.

00contains problems of finding rational right-angled triangles such that different functions of their parts (the sides and the area) are squares.

00After discussing the evolution of the different systems of cultivation, the nature of exchange and barter, money, and the functions of capital, he sets forth the theory of the impotunique, i.e.

00In the West, in the 6th and 7th centuries, besides the original functions of their office, archdeacons had certain well-defined rights of visitation and supervision, being responsible for the good order of the lower clergy, the upkeep of ecclesiastical buildings and the safe-guarding of the church furniture - functions which involved a considerable disciplinary power.

00In the Roman Church to-day the office of archdeacon is merely titular, his sole function being to present the candidates for ordination to the bishop. The title, indeed, hardly exists save in Italy, where the archdeacon is no more than a dignified member of a chapter, who takes rank after the bishop. The ancient functions of the archdeacon are exercised by the vicar-general.

00The functions of the archdeacon are in the present day ancillary in a general way to those of the bishop of the diocese.

00In the colonies there are two or more archdeacons in each diocese, and their functions correspond to those of English archdeacons.

00After a successful trial of a few years this association was dissolved, transferring its functions to the Highland and Agricultural Society.

00Even in the rural districts, manorial records reveal the existence of a great variety of classes and groups of persons engaged in the performance of economic functions.

00In the towns the division of labour had proceeded much further than in the rural districts, and there were in existence organized bodies, such as the Gild Merchant and the crafts, whose functions were primarily economic. But one of the most striking characteristics of town life in the middle ages was the manner in which municipal and industrial privileges and responsibilities were interwoven.

00In the middle ages this differentiation of the industrial, municipal and political life had not taken place, and in order to understand the working of at first sight purely economic regulations it is necessary to make a close study of the functions of local government.

00The community we are studying must have reached such a stage of development that its economic functions and those immediately cognate to them form a well-defined group, and adequate means must be available so that we can, as it were, watch the performance of these functions and test our hypotheses and conclusions by observation and experience.

00We can show, for example: (1) that the Statute of Apprenticeship did not stand alone; it was one of a long series of similar measures, beginning more than two centuries before, which in their turn join on to the municipal and gild regulations of the middle ages; one of an important group of statutes, more or less closely interwoven throughout their history, administered by local authorities whose functions had grown largely in connexion with this legislation and the gradual differentiation of the trades and callings to which it related.

00But these conclusions, after all, suggest more difficulties than they remove, for they show that our inquiry, instead of presenting certain well-marked features which can be readily dealt with, has to be split up into a number of highly specialized studies: the investigation of rates of wages, prices and the standard of comfort in different localities, bye-industries, regularity of employment, the organization of particular trades, the economic functions of local authorities, apprenticeship and a host of other subjects.

00It must be clearly recognized that the functions of economic science in the present requirements of the world cannot possibly be discharged by treatises on economic theory.

00The great increase in recent years in British military and naval expenditure, made necessary by the exceptional demands of a state of war and the great development of foreign powers, was partly responsible for the new difficulties; partly it was due to the great extension of the functions of the state during the latter part of the 19th century.

00The extension of government functions appeared much more likely to continue than to be checked.

00The development of the powers of the central government has been less than that of the functions of local governing authorities.

00Local governing authorities now discharge economic functions of enormous importance and complexity, involving sums of money larger than sufficed to run important states a generation ago.

00Herdman, " On the Structure and Functions of the Cerata or Dorsal Papillae in some Nudibranchiate Mollusca," Quart.

00The executive powers were placed almost entirely in his hands, as will be seen by the terms of article 41 which defined his functions: "The First Consul promulgates the laws; he appoints and dismisses at will the members of the Council of State, the ministers, the ambassadors and other leading agents serving abroad, the officers of the army and navy, the members of local administrative bodies and the commissioners of government attached to the tribunals.

00- As for the second and third consuls, their functions were almost entirely consultative and formal, their opposition being recorded, but having no further significance against the fiat of the First Consul.

00Great diversity exists in the texture and functions of fore and hind-wings in different in sects; these differences are discussed in the descriptions of the various orders.

00The secrecy of its deliberations and the rapidity with which it could act made it a useful adjunct to the constitution, and it gradually absorbed many of the more important functions of the state.

00As a result of the dispute between Governor Arthur St Clair and the Territorial legislature, the constitution of 1802 conferred nearly all of the ordinary executive functions on the legislature.

00In the parts of the state settled by people from New England township meetings were held in the early days, but their functions were gradually transferred, , to the trustees, and by 1820 the meetings had been given up almost entirely.

00Less specialized in their functions are many of the figures of modern folklore, some of whom have perhaps replaced some ancient goddess, e.g.

00The anatomy of the eye is next described; this is done well and evidently at first hand, though the functions of the parts are not given with complete accuracy.

00The capable performance of these functions, which often involved considerable pecuniary sacrifices, ensured public esteem, honorary inscriptions and statues; and to these honours the head of a great house was careful to add the glory of a splendid tomb, consecrated as the long home " (lit.

00It will no doubt aid the understanding of the functions of the latter if some explanation is offered of the needs met by the former, which are sometimes known technically as " deferred deliveries."

00The first is that spinners would be performing the two functions of industrial management and cotton buying (together with others perhaps), and that in consequence the best industrial men would not necessarily be able to maintain their position in the trade because as buyers of cotton they might be unfortunate.

00Hence Caesar seems to assign more extensive functions to the Druids than they actually possessed.

00In Irish literature, however, the Druids are frequently mentioned, and their functions in the island seem to correspond fairly well to those of their Gaulish brethren described by classical writers.

00The functions of Caesar's Druids we here find distributed amongst Druids, bards and poets (fili), but even in very early times the poet has usurped many of the duties of the Druid and finally supplants him with the spread of Christianity.

00In the religious system of Numa, Quirinus and Mars were both recognized as divine beings, distinct but of similar attributes and functions; thus, like Mars, Quirinus was at once a god of war and a nature god, the protector of fields and flocks.

00His functions were those of a secretary; and, though he profited by benefices conferred on him in lieu of salary, he remained a layman to the end of his life.

00This very effective and elaborate innervation, which has been directly traced to the brain, whence strong nerves (generally two) enter the proboscis, renders it exceedingly probable that the most important functions of the proboscis are of a sensiferous, tactile nature.

00In addition to the nerves starting from the brain-lobes just now especially mentioned, there is a double apparatus which can hardly be treated of in conjunction with the sense organs, because its sensory functions have not been sufficiently made out, and which will therefore rather be considered along with the brain and central nervous system.

00Whether in the Metanemertines, where the blood fluid is often provided with haemoglobiniferous disks, the chief functions of the side organs may not rather be a sensory one needs further investigation.

00Four legati juridici (or simply juridici) of consular rank were appointed for Italy, who took over certain important judicial functions formerly exercised by local magistrates (cases of fideicommissa, the nomination of guardians).

00At the Union, while the national functions of the lord high admiral were merged in the English office it was provided by the Act of Union that the Court of Admiralty in Scotland should be continued "for determination of all maritime cases relating to private rights in Scotland competent to the jurisdiction of the Admiralty Court."

003, 4a): the keepers of the house (the arms and hands) tremble, the strong men (the legs and perhaps the backbone) are bent, the grinding women (the teeth) cease to work, those that look out of the windows (the eyes) are darkened, the street-doors are shut, the sound of the mill being low (apparently a summary statement of the preceding details: communication with the outer world through the senses is cut off, the performance of bodily functions being feeble); the rest of v.

00Three years later he made the pilgrimage to Mecca, and on his return lived in retirement in the Fayum until 1399, when he was again called upon to resume his functions as cadi.

00To form a conception of this problem it is to be noted that since the position of the body in space can be computed from the six elements of the orbit at any time we may ideally conceive the coordinates of the body to be algebraically expressed as functions of the six elements and of the time.

00By cancelling the political power of the Areopagus and multiplying the functions of the popular law-courts, Ephialtes abolished the last checks upon the sovereignty of the commons.

00A physicist, however, does more than merely quantitatively determine specific properties of matter; he endeavours to establish mathematical laws which co-ordinate his observations, and in many cases the equations expressing such laws contain functions or terms which pertain solely to the chemical composition of matter.

00Two cases then arise: (I) the properties may be expressed as linear functions of the composition, the terminal values being identical with those obtained for the individual components, and there being a break in the curve corresponding to the absence of mixed crystals; or (2) similar to (I) except that different values must be assigned to the terminal values in order to preserve collinearity.

00An important distinction separates true mixed crystals and crystallized double salts, for in the latter the properties are not linear functions of the properties of the components; generally there is a contraction in /10.591 volume, while the re fractive indices and other physical properties do not, in general, obey the additive law.

00Sometimes the subordinate or joint kingship implies real functions.

00The position of a Friend did not carry with it necessarily any functions; it was in itself purely honorary.

0015) in conjunction with the functions of kingship and priesthood.

00appointed to it Cardinal Pecci, who thus returned to reside in Rome, with the prospect of having shortly responsible functions to perform during the vacancy of the Holy See, though the camerlengo was traditionally regarded as disqualified by his office from succeeding to the papal throne.

00From the former point of view the freeman, then essentially a warrior, and the slave were mutual auxiliaries, simultaneously exercising different and complementary functions - each necessary to the community.

00Both kinds of functions were discharged by slaves, not only at Rome, but in the rural and provincial municipalities.

00Nominally a free Greek city, Alexandria retained its senate to Roman times; and indeed the judicial functions of that body were restored by Septimius Severus, after temporary abolition by Augustus.

00Joachim, however, was unable to continue his abbatial functions in the midst of his labours in prophetic exegesis, and, moreover, his asceticism accommodated itself but ill with the somewhat lax discipline of Corazzo.

00The prefect has to a certain extent a double character and two series of functions.

00the intendants of provinces of the ancien regime were taken as a model, and there is a great resemblance between their respective functions.

00In 1801, when the audiencia - of which the captain-general was ex officio president - began its functions at that point, the governor of Santiago became subordinated in political matters as much as in military.

00The character of this body was altered in 1890, and in 1898, in which latter year its functions were reduced to the essentially judicial.

00A similar theory holds for relations which arise from the consideration of propositional functions with two or more variable arguments.

00Under the general heading "Analysis" occur the subheadings "Foundations of Analysis," with the topics theory of functions of real variables, series and other infinite processes, principles and elements of the differential and of the integral calculus, definite integrals, and calculus of variations; "Theory of Functions of Complex Variables," with the topics functions of one variable and of several variables; "Algebraic Functions and their Integrals," with the topics algebraic functions of one and of several variables, elliptic functions and single theta functions, Abelian integrals; "Other Special Functions," with the topics Euler's, Legendre's, Bessel's and automorphic functions; "Differential Equations," with the topics existence theorems, methods of solution, general theory; "Differential Forms and Differential Invariants," with the topics differential forms, including Pfaffians, transformation of differential forms, including tangential (or contact) transformations, differential invariants; "Analytical Methods connected with Physical Subjects," with the topics harmonic analysis, Fourier's series, the differential equations of applied mathematics, Dirichlet's problem; "Difference Equations and Functional Equations," with the topics recurring series, solution of equations of finite differences and functional equations.

00Functions, operations, transformations, substitutions, correspondences, are but names for various types of relations.

00Until quite recent times the conservative and fanatical spirit of the 'Ulema had been one of the greatest obstacles to progress and reform in a political system in which spiritual and temporal functions were intimately interwoven.

00All these officials unite in their own persons the judicial and executive functions, under the " Law of the Vilayets," which made its appearance in 1861, and purported, and was really intended by its framers, to confer on the provinces a large measure of self-government, in which both Mussulmans and non-Mussulmans should take part.

00Regulations prescribing the duties of the local governors and officials of all ranks were drawn up only in 1865 and 1870, but since Mahmud's time their functions were exclusively civil and administrative.

00After the promulgation of the reforms, the judicial duties of the Imperial Divan, which with other functions also exercised those of a kind of supreme court of appeal, were transferred to the Sheikh-ul-Islam.

00It is used not only at pontifical High Mass but at all solemn pontifical functions, e.g.

00In addition to their functions as guardians of the poor, the parish members have to investigate crimes and punish misdemeanours, settle litigations and divide inheritances.

00As in the rest of Indo-China, there is no hereditary nobility, but there exist castes founded on blood relationship - the members of the royal family within the fifth degree (the Brah-Vansa) those beyond the fifth degree (BrahVan), and the Bakou, who, as descendants of the ancient Brahmans, exercise certain official functions at the court.

00It seems likely that the coelomic pore-canals were originally excretory organs, but in the existing Enteropneusta the pore-canals (especially the collar canals) have, as we have seen, acquired new functions or become vestigial, and the function of excretion is now mainly accomplished by a structure peculiar to the Enteropneusta called the glomerulus, a vascular complex placed on either side of the anterior portion of the stomochord, projecting into the proboscis-coelom.

00Liturgical vestments, as their name implies, are those which are especially associated with the various functions of the liturgy.

00Of these again, according to the fully developed rules of the Catholic Church, there are three classes: (I) vestments worn only at the celebration of mass - chasuble, maniple, pontifical gloves, pontifical shoes, the pallium and the papal fanone and subcinctorium; (2) vestments never worn at mass, but at other liturgical functions, such as processions, administration of the sacraments, solemn choir services, i.e.

00By the 4th century the garments worn at liturgical functions had been separated from those in ordinary use, though still identical in form.

00Hitherto the chasuble had been worn indifferently by all ministers at the eucharist, even by the acolytes; it had been worn also at processions and other non-liturgical functions; it was now exalted into the mass vestment par excellence, worn by the celebrant only, or by his immediate assistants (deacon and subdeacon) only on very special occasions.

00of their use by the various orders of the clergy in the several liturgical functions, however, was established by the close of the 13th century and still continues in force.

00However, it is evident from the letters of appanage, dated April 1771, in favour of the count of Provence, how many functions of public authority an appanaged person still held.

00They are lined by cells charged with a yellow or brown pigment, and besides their excretory functions they act as ducts through which the reproductive cells leave the body.

00King makes out five pairs and an odd one, and individualizes their respective functions as follows: - Three pairs are lateral, having their members limited to the sides of the shell; one pair are transmedians, each member passing across the middle of the reverse side of the shell, while the odd muscle occupies the umbonal cavity.

00The administration is in the hands of a council of chiefs which exercises legislative, executive and, to some extent, judicial functions.

00But from 1600 onwards the same persons fulfilled both functions, the object being, by giving the assessors the duty of collecting the tax, to lead to a duster and more conscientious assessment.

00The theories of determinants and of symmetric functions and of the algebra of differential operations have an important bearing upon this comparatively new branch of mathematics.

00The investigation of the properties of these functions, as well for a single form as for a simultaneous set of forms, and as well for one as for many series of variables, is included in the theory of invariants.

00As far back as 1 773 Joseph Louis Lagrange, and later Carl Friedrich Gauss, had met with simple cases of such functions, George Boole, in 1841 (Camb.

00pp. 8 0 -94, 95112) showed by his calculus of hyper-determinants that an infinite series of such functions might be obtained systematically.

00P. Gordan first proved that for any system of forms there exists a finite number of covariants, in terms of which all others are expressible as rational and integral functions.

00+amam Expanding the right-hand side by the exponential theorem, and then expressing the symmetric functions of al, a2, ...a m, which arise, in terms of b1, b2, ...'

00b., we obtain by comparison with the middle series the symbolical representation of all symmetric functions in brackets () appertaining to the quantities p i, P2, P3,Ã¯Â¿Â½Ã¯Â¿Â½Ã¯Â¿Â½ To obtain particular theorems the quantities a l, a 2, a 3, ...a, n are auxiliaries which are at our entire disposal.

00We can eliminate the quantities S l, E2, Ã¯Â¿Â½Ã¯Â¿Â½Ã¯Â¿Â½ In and obtain n relations AbXi = (2B 11 - Ab)'Ã¯Â¿Â½k1 +2B21x2+2B31x3+Ã¯Â¿Â½Ã¯Â¿Â½Ã¯Â¿Â½, AbX2 = 2B12x1+ (2B22 - Ab) x2 +2B32x3+..., and from these another equivalent set Abx1 = (2B11 - X1 +2B12X2+2B13X3+Ã¯Â¿Â½Ã¯Â¿Â½Ã¯Â¿Â½, Abx2 = 2B21X1+(2B22 - Ab)X2+2B23X3+Ã¯Â¿Â½Ã¯Â¿Â½Ã¯Â¿Â½, and now writing 2Bii - Ab 2Bik - aii, Ob = aik, Ob we have a transformation which is orthogonal, because EX 2 = Ex2 and the elements aii, a ik are functions of the 2n(n- I) independent quantities b.

00x n / I l yl, y2,Ã¯Â¿Â½Ã¯Â¿Â½Ã¯Â¿Â½yn Theorem.-If the functions y 1, y2,Ã¯Â¿Â½Ã¯Â¿Â½Ã¯Â¿Â½ y n be not independent of one another the functional determinant vanishes, and conversely if the determinant vanishes, yl, Y2, ...y.

00are not independent functions of x1, x2,...xn.

00For if u, v, w be the polynomials of orders m, n, p respectively, the Jacobian is (u 1 v 2 w3), and by Euler's theorem of homogeneous functions xu i +yu 2 +zu 3 = mu xv1 +yv2 +zv3 = /IV xw 1+y w 2+ zw 3 = pw; denoting now the reciprocal determinant by (U 1 V2 W3) we obtain Jx =muUi+nvVi+pwWi; Jy=Ã¯Â¿Â½.., Jz=..., and it appears that the vanishing of u, v, and w implies the vanishing of J.

00There is no difficulty in expressing the resultant by the method of symmetric functions.

00Now by the theory of symmetric functions, any symmetric functions of the mn values which satisfy the two equations, can be expressed in terms of the coefficient of those equations.

00Similarly, if a form in k variables be expressible as a quadratic function of k -1, linear functions X1, X2, ...

00THE Theory Of Symmetric Functions Consider n quantities a l, a 21 a 3, ...

00+ax n, al, a2, ...an are called the elementary symmetric functions.

00p 1 p 2 ...p n), it is necessary to bring under view other functions associated with the same series of numbers: such, for example, as P P3 P2 P4 Pn -2 /, /, /, F i a i 1 a 2 Fi a 1 a 2 ...

00A separation is the symbolic representation of a product of monomial symmetric functions.

00the sum of the n th powers of the quantities, is expressible in terms of functions which are symbolized by separations of any partition (n"1n'2n'3...) 1 !

00) j1+j2+j3+..Ã¯Â¿Â½ (J1+ j2 +j3+...-1)!/T1)?1(J2)72 (J 3)/3..., j11j2!j3!... ?.1 for the expression of Za n in terms of products of symmetric functions symbolized by separations of (n 1 1n 2 2n 3 3) Let (n) a, (n) x, (n) X denote the sums of the n th powers of quantities whose elementary symmetric functions are a l, a 2, a31Ã¯Â¿Â½Ã¯Â¿Â½Ã¯Â¿Â½; x 1, x2, x31..; X1, X2, X3,...

00in terms of x 1, x2, x3,Ã¯Â¿Â½Ã¯Â¿Â½ The inverse question is the expression of any monomial symmetric function by means of the power functions (r) = sr. Theorem of Reciprocity.-If Ã¯Â¿Â½1 P2 "3 01 Q 2 7 3 Al A 2 A3 X m1 X m2 X m3 ...

00The elementary functions are denoted by (1), (12), (13), ...

00(0B) = (e), &c. The binomial coefficients appear, in fact, as symmetric functions, and this is frequently of importance.

00The sum of the monomial functions of a given weight is called the homogeneous-product-sum or complete symmetric function of that weight; it is denoted by h.; it is connected with the elementary functions by the formula 1 7r1l7r2!7r3!

00be any partitions of X, respectively, the function isexpressible by means of functions symbolized by separation of X1A2X 3.

00The law of reciprocity shows that p(s) = zti (m 1te2tmtL3t) t=1 st It 2t 3t viz.: a linear function of symmetric functions symbolized by the k specifications; and that () St =ti ts.

00A table may be formed expressing the k expressions Pa l), P(2),...P(1) as linear functions of the k expressions (m"`'smÃ¯Â¿Â½2smÃ¯Â¿Â½3s...), s =1, 2, ...k, and the numbers BSc occurring therein is 2s 3s possess row and column symmetry.

00By solving k linear equations we similarly express the latter functions as linear functions of the former, and this table will also be symmetrical.

00" The symmetric function (m Ã¯Â¿Â½8 m' 2s m Ã¯Â¿Â½3s ...) whose is 2s 3s partition is a specification of a separation of the function symbolized by (li'l2 2 l3 3 ...) is expressible as a linear function of symmetric functions symbolized by separations of (li 1 12 2 13 3 ...) and a symmetrical table may be thus formed."

00so that f (al, a 3, a3,.Ã¯Â¿Â½.an) =f, a rational integral function of the elementary functions, is converted into f(a1 +12, a2+ p a1,...

00It has been shown (vide " Memoir on Symmetric Functions of the Roots of Systems of Equations," Phil.

00by the same law which expresses the power function s, in terms of the elementary functions a 1, a2, a3,...

00a Product of Symmetric Functions.

00- Suppose f to be a product of symmetric functions f i f 2 ...f m .

00which is satisfied by every symmetric fraction whose partition contains no unit (called by Cayley non-unitary symmetric functions), is of particular importance in algebraic theories.

00= 0; and such functions satisfy the differential equation aoaa i +2a0a 2 +3a 2 aa 3 +...

00For such functions remain unaltered when each root receives the same infinitesimal increment h; but writing x-h for x causes ao, a1, a 2 a3,...

00The important result is that the theory of invariants is from a certain point of view coincident with the theory of non-unitary symmetric functions.

00=0, are non-unitary symmetric functions of the roots of a xn-a l xn 1 a2 x n-2 -...

00Denote by brackets () and [] symmetric functions of the quantities p and a respectively.

00Symmetric Functions Several Systems Quantities.

00All symmetric functions are expressible in terms of the quantities ap g in a rational integral form; from this property they are termed elementary functions; further they are said to be single-unitary since each part of the partition denoting ap q involves but a single unit.

00The partitions with one bipart correspond to the sums of powers in the single system or unipartite theory; they are readily expressed in terms of the elementary functions.

00in terms of elementary functions can be at once written down.

00which expresses the elementary functions in terms of the single bipart functions.

00The partitions being taken as denoting symmetric functions we have complete correspondence between the algebras of quantity and operation, and from any algebraic formula we can at once write down an operation formula.

00Since dp4+(-)P+T1(p +q qi 1)!dd4, the solutions of the partial differential equation d P4 =o are the single bipart forms, omitting s P4, and we have seen that the solutions of p4 = o are those monomial functions in which the part pq is absent.

001852; MacMahon, " Memoirs on a New Theory of Symmetric Functions," American 1 Phil.

001888-1890; " Memoir on Symmetric Functions of Roots of Systems of Equations," Phil.

00If the variables of the quantic f(x i, x 2) be subjected to the linear transformation x1 = a12Et2, x2 = a21E1+a2252, E1, being new variables replacing x1, x 2 and the coefficients an, all, a 21, a22, termed the coefficients of substitution (or of transformation), being constants, we arrive at a transformed quantic f% 1tn n n-1 n-2 52) = a S +(1)a11 E 2 + (2)a2E1 E 2 +Ã¯Â¿Â½Ã¯Â¿Â½Ã¯Â¿Â½ in the new variables which is of the same order as the original quantic; the new coefficients a, a, a'...a are linear functions 0 1 2 n of the original coefficients, and also linear functions of products, of the coefficients of substitution, of the nth degree.

00In 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.

00Invariantive forms will be found to be homogeneous functions alike of the coefficients and of the variables.

00in addition, and transform each pair to a new pair by substitutions, having the same coefficients a ll, a12, a 21, a 22 and arrive at functions of the original coefficients and variables (of one or more quantics) which possess the abovedefinied invariant property.

00If u, a quantic in x, y, z, ..., be expressed in terms of new variables X, Y, Z ...; and if, n,, ..., be quantities contragredient to x, y, z, ...; there are found to exist functions of, n, ?, ..., and of the coefficients in u, which need, at most, be multiplied by powers of the modulus to be made equal to the same functions of E, H, Z, ...

00of the transformed coefficients of u; such functions are called contravariants of u.

00There also exist functions, which involve both sets of variables as well as the coefficients of u, possessing a like property; such have been termed mixed concomitants, and they, like contravariants, may appertain as well to a system of forms as to a single form.

00If the invariants and covariants of this composite quantic be formed we obtain functions of X such that the coefficients of the various powers of X are simultaneous invariants of f and 4).

00If, however, we specify that all forms are to be rational, but not necessarily integral functions, a new system of forms arises which is easily obtainable.

00Another form, associated with the theory of elliptic functions, has been considered by Dingeldey (Math.

00Hesse'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.

00The general equation of degree 5 cannot be solved algebraically, but the roots can be expressed by means of elliptic modular functions.

00Enumerating Generating Functions Professor Michael Roberts (Quart.

00a little further progress has been made by Cayley, who established the two generating functions for the quintic 1 -a3s 11 -a8.1 a12.

00The extraordinary advantage of the transformation of S2 to association with non-unitary symmetric functions is now apparent; for we may take, as representative forms, the symmetric functions which are symbolically denoted by the partitions referred to.

00Remark, too, that we are in association with non-unitary symmetric functions of two systems of quantities which will be denoted by partitions in brackets ()a, ()b respectively.

00The number of different symbols a, b, c,...denotes the the covariants are homogeneous, but not in general isobaric functions, of the coefficients of the original form or forms. Of the above general form of covariant there are important transformations due to the symbolic identities: - (Ã¯Â¿Â½b) 2 2)2 = a b - a b; (xÃ¯Â¿Â½ = as a consequence any even power of a determinant factor may be expressed in terms of the other symbolic factors, and any uneven power may be expressed as the product of its first power and a function of the other symbolic factors.

00Lead generally functions as a divalent element of distinctly metallic character, yielding a definite series of salts derived from the oxide PbO.

00It forms crystallizable salts with potassium and calcium hydrates, and functions as a weak acid forming salts named plumbates.

00Rowland,' whose careful experiments led to general recognition of the fact previously ignored by nearly all investigators, that magnetic susceptibility and permeability are by no means constants (at least in the case of the ferromagnetic metals) but functions of the magnetizing force.

00Certain religious functions of minor importance were also reserved for them.

00The functions of these gastric diverticula have never been carefully investigated.

00It was during this period that he first formed those ideas on the theory of functions of a complex variable which led to most of his great discoveries.

00The fundamental method of research which Riemann employed has just been alluded to; the results will be best indicated in his own words: "The methods in use hitherto for treating functions of a complex variable always started from an expression for the function as its definition, whereby its value was given for every value of the argument; by our investigation it has been shown that, in consequence of the general character of a function of a complex variable, in a definition of this sort one part of the determining conditions is a consequence of the rest, and the extent of the determining conditions has been reduced to what is necessary to effect the determination.

00This essentially simplifies the treatment of such functions.

00liv., 1857) in which he applied the theory developed in his doctor's dissertation to the Abelian functions.

00After his fall he resumed his functions as president of the senate; but on the advent of the third Giolitti cabinet, he was not reappointed to that position.

00The leading idea of this work was contained in a paper published in the Berlin Memoirs for 1772.5 Its object was the elimination of the, to some minds, unsatisfactory conception of the infinite from the metaphysics of the higher mathematics, and the substitution for the differential and integral calculus of an analogous method depending wholly on the serial development of algebraical functions.

00His purely abstract mode of regarding functions, apart from any mechanical or geometrical considerations, led the way to a new and sharply characterized development of the higher analysis in the hands of A.

00The Theorie des fonctions is divided into three parts, of which the first explains the general doctrine of functions, the second deals with its 4 Notice by J.

00It is difficult to give briefly a clear idea of the functions of the three important officials comes sacrarum largitionum, comes rei privatae and comes sacri partrimonii; but the terms have been well translated by a German author as Finanzminister des Reichsschatzes (finance minister of the treasury of the Empire), F.

00Under the early Frankish kings some comites did not exercise any definite functions; they were merely attached to the king's person and executed his orders.

00His essential functions were judicial and executive, and in documents he is often described as the king's agent (agens publicus) or royal judge (judex publicus or fiscalis).

00ryepovaia, Doric ^yfpwia), the ancient council of elders at Sparta, corresponding in some of its functions to the Athenian Boule.

00The functions of the council were among the most important in the state.

00The judicial and financial functions in each province were vested in the Ouvidor, whose authority in the college of finance was second only to that of the governor.

00As early as 1618 a code of laws for the regulation of the mining industry had been drawn up by Philip III., the executive and judicial functions in the mining districts being vested in a provedor, and the fiscal in a treasurer, who received the royal fifths and superintended the weighing of all the gold, rendering a yearly account of all discoveries and produce.

00The bank, in addition to its private functions, farmed many of the regalia, and was in the practice of advancing large sums to the state, transactions which gave rise to extensive corruption, and terminated some years later in the breaking of the bank.

00In November of the same year, when Enfantin preached the enfranchisement of women and the functions of the couple-pretre, Leroux separated himself from the sect.

00The magnificent hall used for academic and public functions was the gift of William M ` Ewan, some time M.P. for the Central division of Edinburgh.

00Stockdale sued the Commons' publisher, and was met by the plea of parliamentary privilege, to which, however, the judges did not give effect, on the ground that they were entitled to define the privileges of the Commons, and that publication of papers was not essential to the functions of parliament.

00The imperial legates and the captains of the Spanish guard in Siena crushed both government and people by continual extortions and by undue interference with the functions of the balia.

00According to some, ordination simply entitles a man to hold an office and perform its functions.

00Aquinas regards the souls of men, like the angels, as immaterial forms; and he includes in the soul-unit, so to speak, not merely the anima rationalis of Aristotle, but also the vegetative, sensitive, appetitive and motive functions.

00The introduction of hyperbolic functions into trigonometry was also due to him.

00the decree ordering the demolition of the new castles, most of them little better than robber-strongholds; the decree compelling the great officers of state to suspend their functions during the session of the diet; the decree declaring illegal the new fashion of forming confederations on the Polish model, all of which measures were obviously directed against the tyranny and the lawlessness of the oligarchy.

00Moreover, to promote their own convenience, they readily allowed the king to assess as well as to collect the taxes, which consequently tended to become regular and permanent, while Matthias' reform of the treasury, which was now administered by specialists with separate functions, was economically of great benefit to the state.

00Hungary was declared to be a free, independent and unsubjected kingdom governed by its own laws and customs. The legislative functions were to be exercised by the king and the diet conjointly and by them alone.

00the king, weary of the tactics of a minority which for years had terrorized every majority and prevented the government from exercising its proper constitutional functions, had resolved to show the Magyars that he was prepared to rule unconstitu 1 The Austrian court resented especially the decree proclaiming national mourning for Louis Kossuth, though no minister was present at the funeral.

00Laplace published in 1779 the method of generating functions, the foundation of his theory of probabilities, and the first part of his Theorie analytique is devoted to the exposition of its principles, which in their simplest form consist in treating the successive values of any function as the coefficients in the expansion of another function with reference to a different variable.

00In particular, the equality or inequality of values of two functions is more readily grasped by comparison of the lengths of the ordinates of the graphs than by inspection of the relative positions of their bounding lines.

00Application of Binomial Theorem to Rational Integral Functions.

00derived functions of f(x).

00Algebraical division therefore has no definite meaning unless dividend and divisor are rational integral functions of some expression such as x which we regard as the root of the notation (Ã¯Â¿Â½ 28 (iv.)), and are arranged in descending or ascending powers of x.

00If P and M are rational integral functions of x, arranged in descending powers of x, the division of P by M is complete when we obtain a remainder R whose degree (Ã¯Â¿Â½ 45) is less than that of M.

00The highest common factor (or common factor of highest degree) of two rational integral functions of x is therefore found in the same way as the G.C.M.

00The generating functions of the two series, mentioned above, for example, are (I +x)-' n and (I+x) P / Q.

00The idea is utilized in the elementary consideration :of a differential coefficient; and its importation into the treatment of certain functions as continuous is therefore properly associated with the infinitesimal calculus.

00We 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.

00Pierpont, Lectures on the Theory of Functions of Real Variables (1905).

00His principal discovery is concerned with equations, which he showed to be derived from the continued multiplication of as many simple factors as the highest power of the unknown, and he was thus enabled to deduce relations between the coefficients and various functions of the roots.

00The whole issue had, in fact, become confused with the confusion of functions of the Church and State.

004 The superintendents (variously entitled also archpriests, deans, provosts, ephors) of the Evangelical (Lutheran) Church, as established in the several states of Germany and in Austria, are not bishops in any canonical sense, though their jurisdictions are known as dioceses and they exercise many episcopal functions.

00Owing to the connexion of medicine with these seats of learning, it was natural that the study of the structure and functions of the human body and of the animals nearest to man should take root there; the spirit of inquiry which now for the first time became general showed itself in the anatomical schools of the Italian universities of the 16th century, and spread fifty years later to Oxford.

00A subdivision of zoology which was at one time in favour is simply into morphology and physiology, the study of form and structure on the one hand, and the study of Scope the activities and functions of the forms and structures of zoo- on the other.

00No such distinction of mental activities as that involved in the division of the study of animal life into morphology and physiology has ever really existed: the investigator of animal forms has never entirely ignored the functions of the forms studied by him, and the experimental inquirer into the functions and properties of animal tissues and organs has always taken very careful account of the forms of those tissues and organs.

00Analogous structures in any two animals compared were by Owen defined as structures performing similar functions, but not necessarily derived from the modification of one and the same part in the " plan " or " archetype " according to which the two animals compared were supposed to be constructed.

00Homologous structures were such as, though greatly differing in appearance and detail from one another, and though performing widely different functions, yet were capable of being shown by adequate study of a series of intermediate forms to be derived from one and the same part or organ of the " plan-form " or " archetype."

00The integrals are then properly functions of the direction in which the light is to be estimated.

00Hamilton in a memoir on " Fluctuating Functions," Roy.

00The maximum brightnesses and the places at which they occur are easily determined with the aid of certain properties of the Bessel's functions.

00The accompanying table is given by Lommel, in which the first column gives the roots of J2(z) =o, and the second and third columns the corresponding values of the functions specified.

00A very thorough investigation of this and other related questions, accompanied by fully worked-out tables of the functions concerned, will be found in a paper by E.

00When the functions C and S have once been calculated, the discussion of various diffraction problems is much facilitated by the idea, due to M.

00The rest of the province is policed by the South African constabulary, a body 3700 strong, to which is also entrusted customs preventive work, fire brigade work and such like functions.

00School boards and district committees are formed, but their functions are almost entirely advisory.

00chemically; while, in most living things, mere hetero geneity is exchanged for a definite structure, whereby the body is distinguished into visibly different parts, which possess different powers or functions.

00His tastes were those of a student, and he did not disguise his dislike of public functions and the constant little journeys which take up so much of a bishop's time.

00It was a deep-seated belief that those who took part in religious functions were liable to communicate this " holiness " to others (compare the complex ideas associated with the Polynesian taboo).

00Where the king is the human representative of the Deity he is theoretically and officially the priesthood, although the priests carry on the ordinary subordinate functions.

00In the cinctus Gabinus, which was the fashion adopted in early times when fighting was in prospect, the end of the toga was drawn tightly round the waist and formed a kind of girdle; this was retained in certain official functions, such as the opening of the emple of Janus in historical times.

00In April 1895 the long-standing dispute as to the boundary between British Guiana and Venezuela was brought to a crisis by the action of the Venezuelan authorities in arresting Inspectors Barnes and Baker, of the British Guiana police, with a few of their subordinates, on the Cuyuni river, the charge being that they were illegally exercising the functions of British officials in Venezuelan territory.

00Much light has been thrown upon the functions and diseases of the blood-forming tissues.

00The use of the term " pathological physiology " may at first appear strange, for if we define physiology as the sum of the normal functions of the body or organism, it may be hard to see how there can be a physiology which is pathological.

00The difficulty, however, is more apparent than real, and in this sense, that if we start with a diseased organ as our subject of inquiry, we can quite properly, and without committing a solecism, treat of the functions of that organ in terms of its diseased state.

00There seems to be little doubt, notwithstanding, that one of the chief functions of the nerve cell is that of the propagation of a trophicj influence along its axon.

00Or again, in the case of paired organs, if one be removed by operation, or destroyed by disease, the other at once undertakes to carry on the functions of both.

00The functions of the thymus gland begin to cease after the second year from birth.

00As a result of these various degenerations the functions of the body deteriorate, the faculties become blunted, and the muscular energy of the body is below what it was in earlier life, while the secreting glands in certain instances become functionally obsolescent.

00Oertel thinks that in man we have these two different functions carried on by the one nucleus containing both chromatin orders.

00The structure and functions of the body form the subject of Anatomy (q.v.) and Physiology.

00There is no sign in the Homeric poems of the subordination of medicine to religion which is seen in ancient Egypt and India, nor are priests charged, as they were in those countries, with medical functions - all circumstances which throw grave doubts on the commonly received opinion that medicine derived its origin in all countries from religious observances.

00The tendency of the school was to explain the actions and functions of the body on physical, and especially on mechanical, principles.

00James Keill (1673-1719) applied Newtonian and mechanical principles to the explanation of bodily functions with still greater accuracy and completeness; but his researches have more importance for physiology than for practical medicine.

00For mechanical conceptions he substituted the theory of" animism "- attributing to the soul the functions of ordinary animal life in man, while the life of other creatures was left to mechanical laws.

00Thus the reciprocity of the various organs, maintained throughout the divisions of physiological labour, is not merely a mechanical stability; it is also a mutual equilibration in functions incessantly at work on chemical levels, and on those levels of still higher complexity which seem to rise as far beyond chemistry as chemistry beyond physics.

00The discoveries of the separate paths of sensory and motor impulses in the spinal cord, and consequently of the laws of reflex action, by Charles Bell and Marshall Hall respectively, in their illumination of the phenomena of nervous function, may be compared with the discovery in the region of the vascular system of the circulation of the blood; for therein a key to large classes of normal and aberrant functions and a fertile principle of interpretation were obtained.

00Nor was the theory of reflex action confined to the more "mechanical" functions.

00These events, although far more mischievous in the brain, the functions of which are far-reaching, and the collateral circulation of which is ill-provided, are seen very commonly in other parts.

00In general criticism and miscellaneous writing Voltaire is not inferior to himself in any of his other functions.

00The deputy commissioners perform the functions of district magistrates, district judges, collectors and registrars, besides the miscellaneous duties which fall to the principal district officer as representative of government.

00His pages abound in symbols representing unknown functions, the form of the function being left to be ascertained by observation of facts, which he does not regard as a part of his task, or only some known properties of the undetermined function being used as bases for deduction.

00Originally intended as assistants to the tribunes, they exercised certain police functions, were empowered to inflict fines and managed the plebeian and Roman games.

00But although the curule aediles always ranked higher than the plebeian, their functions gradually approximated and became practically identical.

00Under Augustus the office lost much of its importance, its juridical functions and the care of the games being transferred to the praetor, while its city responsibilities were limited by the appointment of a praefectus urbi.

00Its vice-governor-general exercised all the executive functions of the governor-general and corresponded directly with Brussels.

00But in the most general case it is possible to have three functions 0, ?, m of x, y, z, such that udx+vdy +wdz = -dcp-mdl,G, (I) as A.

00(22) Conjugate functions can be employed also for the motion of liquid in a thin sheet between two concentric spherical surfaces; the components of velocity along the meridian and parallel in colatitude 0 and longitude A can be written dÂ¢_ i _ d4, I dip _ dy (13) d8 sin - 0 dX' sin 0 dX de' and then = F (tan O.

00Uniplanar motion alone is so far amenable to analysis; the velocity function 4 and stream function 1G are given as conjugate functions of the coordinates x, y by w=f(z), where z= x +yi, w=4-Plg, and then dw dod,y az = dx + i ax - -u+vi; so that, with u = q cos B, v = q sin B, the function - Q dw u_vi=g22(u-}-vi) = Q(cos 8+i sin 8), gives f' as a vector representing the reciprocal of the velocity in direction and magnitude, in terms of some standard velocity Q.

00Beginning with a single body in liquid extending to infinity, and denoting by U, V, W, P, Q, R the components of linear and angular velocity with respect to axes fixed in the body, the velocity function takes the form = Ucb1+V42+W43+ P xi+Qx2+Rx3, (I) where the 0's and x's are functions of x, y, z depending on the shape of the body; interpreted dynamically, C -p0 represents the impulsive pressure required to stop the motion, or C +p4) to start it again from rest.

00In the arrangements of the commonwealth the clauses treating of royal privileges are more or less evenly distributed over all reigns, but the systematic development of police functions, especially in regard to responsibility for crimes, the catching of thieves, the suppression of lawlessness, is mainly the object of 10th and 11th century legislation.

00In course of time many of the high-priests assumed the functions and title of king; while retaining their priestly office they claimed at the same time to be supreme in the state in all secular concerns.

00For these opinions he was in 1728 suspended from the discharge of ministerial functions, and finally deposed in 1730.

00The last structural peculiarity of the group is the absence of the functions of regulation and reparation which are so highly developed in the more primitive Planarians.

00The absence of these functions of regeneration and of regulation affords, therefore, corroborative evidence of the highly specialized nature of the Cestode organization.

00The roots like all other parts of plants contain protoplasm or living material, which cannot carry on its functions unless it is supplied with an adequate amount of oxygen: hence the necessity for the continuous circulation of fresh air through the soil.

00For the carrying on of their functions they all need to be supplied with carbohydrates or other carbon compounds which they obtain ordinarily from humus and plant residues in the soil, or possibly in some instances from carbohydrates manufactured by minute green algae with which they live in close union.

00The last two categories, which do not become prominent anywhere in Europe until the 12th century, had, like all gilds, a religious tinge, but their aims were primarily worldly, and their functions were mainly of an economic character.

00Toulmin Smith (1816-1869), and they throw much light on the functions of the gilds.

00While in most towns the name and the old organization of the gild merchant thus disappeared and the institution was displaced by the aggregate of the crafts towards the close of the middle ages, in some places it survived long after the 15th century either as a religious fraternity, shorn of its old functions, or as a periodical feast, or as a vague term applied to the whole municipal corporation.

00It could most easily make them useful to gain the influence and support which it needed, and to provide for the public functions which fell to its share, by employing the precarium tenure.

00In these two ways then the feudal system was formed, and took possession of the state territorially, and of its functions in government.

00The early German governments whose chief functions, military, judicial, financial, legislative, were carried on by the freemen of the nation because they were members of the body politic, and were performed as duties owed to the community for its defence and sustenance, no longer existed.

00As yet, however, the jurisdiction and functions of these courts have not been ascertained.

00The people were to be directly governed by their native chiefs, whose duty was to collect the tribute and exercise magisterial functions.

00His TEXvrt rypaÂµÂµanKii, which we possess (though probably not in its original form), begins with the definition of grammar and its functions.

00From Berlin he passed to Freiberg, and here he made his brilliant researches in the theory of functions, elliptic, hyperelliptic and a new class known as Abelians being particularly studied.

00Under Abel's guidance, the prevailing obscurities of analysis began to be cleared, new fields were entered upon and the study of functions so advanced as to provide mathematicians with numerous ramifications along which progress could be made.

00For further details of his mathematical investigations see the articles Theory of groups, and Functions Of Complex Variables.

00In examining the conditions of a spiritual power properformodern times, he indicates in so many terms the presence in his mind of a direct analogy between his proposed spiritual power and the functions of the Catholic clergy at the time of its greatest vigour and most complete independence, - that is to say, from about the middle of the i i th century until towards the end of the 13th.

00To fit them for their functions, they are to be raised above material cares, and they are to be thoroughly educated.

00Meanwhile he had gained a high reputation as a preacher, and especially as the advocate of religious freedom; but his teaching became more and more offensive to the orthodox party, and on the appearance (1864) of his article on Renan's Vie de Jesus in the Nouvelle Revue de theologie he was forbidden by the Paris consistory to continue his ministerial functions.

00Martineau's mental qualities fitted him to fulfil these high interpretative functions.

00From that time until his death in 1881 the Aga Khan, while leading the life of a peaceful and peacemaking citizen, under the protection of British rule, continued to discharge his sacerdotal functions, not only among his followers in India, but towards the more numerous communities which acknowledged his religious sway in distant countries, such as Afghanistan, Khorasan, Persia, Arabia, Central Asia, and even distant Syria and Morocco.

00In 1901, Professor Max Neuberger of Vienna called attention to certain anticipations of modern views made by Swedenborg in relation to the functions of the brain.

00The cerebral cortex, and, more definitely, the cortical elements (nerve cells), formed the seat of the activity of the soul, and were ordered into departments according to various functions.

00His views as to the physiological functions of the spinal cord are also in agreement with recent research, and he anticipated many of the pre-eminent offices of the ductless glands which students of the present time are only beginning to discover.

00ETIQUETTE, a term for ceremonial usage, the rules of behaviour observed in society, more particularly the formal rules of ceremony to be observed at court functions, &c., the procedure, especially with regard to precedence and promotions in an organized body or society.

00The cobra venom is supposed to extinguish the functions of the various nerve-centres of the cerebro-spinal system, the paralysation extending from below upwards, and it has a special affinity for the respiratory centre.

00Nothing now remains of the former French predominance in the Lebanon, except a certain influence exerted by the fact that the railway is French, and by the precedence in ecclesiastical functions still accorded by the Maronites to official representatives of France.

00The continued use of large doses of alcohol produces chronic gastritis, in which the continued irritation has led to overgrowth of connective tissue, atrophy of the gastric glands and permanent cessation of the gastric functions.

00Since 1887 the management of the town had been entrusted to a nominated sanitary board, under the chairmanship of the mining commissioner appointed by the South African Republic. In 1890 elected members had been admitted to this board, but at the end of 1897 an elective stadsraad (town council) was constituted, though its functions were strictly limited.

00(9) These must be expressed as functions of v and 0, which is theoretically possible if the values of s, p, and dp/do are known.

00The energy E and the total heat F are functions of the temperature only, by equations (9) and (I I), and their variations take the form dE = sdO, d F = Sd0.

00These functions do not, however, represent energy existing in the substance, like the intrinsic energy; but the increment of 90 represents heat supplied to, and the decrement of (E-04) represents work obtainable from, the substance when the temperature is kept constant.

00The direct integration of this equation requires that L and v" - v' should be known as functions of p and 0, and cannot generally be performed.

00Writing formulae (3r) and (33) for the energy and entropy with indeterminate constants A and B, instead of taking them between limits, we obtain the following expressions for the thermodynamic functions in the case of the vapour: " =Solog e 0 - R log e p - ncp/D+A".

00The values of the corresponding functions for the liquid or solid cannot be accurately expressed, as the theoretical variation of the specific heat is unknown, but if we take the specific heat at constant pressure s to be approximately constant, and observe the small residual variation dh of the total heat, we may write F'=s'D+dh+B'.

00G, J, Thermodynamic potential functions.

00The functions of the archbishop, as at present exercised, developed out of those of the metropolitan; though the title of archbishop, when it first appeared, implied no metropolitan jurisdiction.

00He may, however, exercise his purely episcopal functions.

00In the Orthodox and other churches of the East the title of archbishop is of far more common occurrence than in the West, and is less consistently associated with metropolitan functions.

00In much the same manner Mars takes on in addition to his agricultural character the functions of war-god, which in time completely superseded the earlier idea.

00Organization showed itself in the fixing of the annual calendar and the development of the character and functions of the priesthood, and as we should expect, in a new conception of the legal relation of the gods to the state.

00In the foregoing account only those particulars which bear directly on Villehardouin himself have been detailed; but the chronicle is as far as possible from being an autobiography, and the displays of the writer's personality, numerous as they are, are quite involuntary, and consist merely in his way of handling the subject, not in the references (as brief as his functions as chronicler will admit) to his own proceedings.

00It functions chiefly as an acidic oxide, 'being less basic than aluminium oxide, and forming no stable oxy-salts.

00There was also a lord warden, who was usually a nobleman and performed no judicial functions.

00This view being confirmed by a resolution of congress, although it was not a unanimous one, Bolivar decided to resume his functions, and he repaired to Bogota to take the oaths.

00In the meanwhile Bolivar had accepted the presidency, and resumed the functions belonging to his official position.

00At the beginning of the 13th century there existed a tour des pairs which exercised judicial functions and dated possibly from the 11th century, but their prerogatives at the beginning of the 14th century appear to have been mainly ceremonial and decorative.

00ap par wxos, a man-at-arms), the name given to some Greeks who discharged certain military and police functions under the Turkish government.

00His functions are rather more extensive than those of the average American executive.

00Re-elected in the municipal elections of the 2nd of December 1792, he was soon charged with the functions of procurator of the Commune, and contributed with success to the enrolments of volunteers by his appeals to the populace.

00Ow are any momenta or functions of the co-ordinates and momenta or co-ordinates alone which are subject only to the condition that they do not enter into the coefficients a 1, a 2, &c.

00In the time of Ezra the Jewish "magistrates and judges" among their ecclesiastico-civil functions have the right of pronouncing sentence whether it be unto death, or to "rooting out," or to confiscation of goods, or to imprisonment (Ezra vii.

00His researches on elliptic functions are of considerable elegance, but their great merit lies in the stimulating effect which they had on later mathematicians.

00Further, the praetorian praefect acquired, in addition to his military functions, a criminal jurisdiction, which he exercised not as the delegate but as the representative of the emperor, and hence it was decreed by Constantine (331) that from the sentence of the praetorian praefect there should be no appeal.

00On his return he removed to Berlin, where he lived as a royal pensioner till his death, which occurred on the 18th of February 18 His investigations in elliptic functions, the theory of which he established upon quite a new basis, and more particularly his development of the theta-function, as given in his great treatise Fundamenta nova theoriae functionum ellipticarum (Konigsberg, 1829), and in later papers in Crelle's Journal, constitute his grandest analytical discoveries.

001875); Examples of Analytical Geometry of Three Dimensions (1858, 3rd ed., 1873); Mechanics (1867), History of the Mathematical Theory of Probability from the Time of Pascal to that of Lagrange (1865); Researches in the Calculus of Variations (1871); History of the Mathematical Theories of Attraction and Figure of the Earth from Newton to Laplace (1873); Elementary Treatise on Laplace's, Lame's and Bessel's Functions (1875); Natural Philosophy for Beginners (1877).

00Meanwhile the Conqueror had invested him with important judicial functions.

00The Church was not only organized like a modern bureaucracy, but performed many of the functions of a modern State.

00It included: the number, character and nationality of the cardinals, the abuse of the " reservations " made by the apostolic see, the annates, the collation to benefices, expectative favours, cases to be brought before the papal Curia (including appeals), functions of the papal chancery and penitentiary, benefices in commendam, confirmation of elections, income during vacancies, indulgences, tenths, for what reasons and how is a pope to be corrected or deposed.

00The civil tribunals, however, practically assumed the functions of regular inquisitorial courts, in spite of the objections urged by the ecclesiastical courts.

00Another almost equally exceptional feature is the persistence of the colonial executive council, consisting of members chosen to represent divisions of the state, who assist the governor in his executive functions.

00The popular reforms of Solon (594 B.C.), so far as they were carried into effect, tended practically tolimit the Council of the Areopagus, though constitutionally it retained all its earlier powers and functions, augmented by the right to try persons accused of conspiracy against the state (Arist.

00Following up this advantage, Ephialtes (462 B.C.), and less prominently Archestratus and Pericles, proposed and carried measures for the transfer of most of its functions to the Council of Five Hundred, the Ecclesia, and the popular courts of law (Arist.

00Under Roman supremacy in addition to earlier functions they had jurisdiction in cases of forgery, tampering with the standard measures, and probably other high crimes, the supervision of buildings, and the care of religion and of education (Cic. Fain.

00The uses to which the textiles were put were for clothing, furniture for the house, utensils for a thousand industries, fine arts, social functions and worship.

00DRAGOMAN (from the Arabic terjuman, an interpreter or translator; the same root occurs in the Hebrew word targum signifying translation, the title of the Chaldaean translation of the Bible), a comprehensive designation applied to all who act as intermediaries between Europeans and Orientals, from the hotel tout or travellers' guide, hired at a few shillings a day, to the chief dragoman of a foreign embassy whose functions include the carrying on of the most important political negotiations with the Ottoman government, or the dragoman of the imperial divan (the grand master of the ceremonies).

00The functions of the first dragoman are mainly political; he accompanies the ambassador or minister at his audiences of the sultan and usually of the ministers, and it is he who is charged with the bulk of diplomatic negotiations at the palace or the Porte.

00Opposite each carpel is a small scale which functions as a nectary.

00The Levites, who were exempted from military duties, were separately enumerated from the age of thirty upwards, and a similar process was ordained subsequently by Solomon, in order to distribute amongst them the functions assigned to the priestly body in connexion with the temple.

00(B) Mensuration of Graphs of Algebraical Functions.

00For functions of higher degrees in x the formulae become more complicated.

00In the case of mathematical functions certain conditions of continuity are satisfied, and the extent to which the value given by any particular formula differs from the true value may be estimated within certain limits; the main inaccuracy, in favourable cases, being due to the fact that the numerical data are not absolutely exact.

00The fusion into one office of the functions of " elders " and " deacons " (still distinguished in the Savoy Declaration of 1658) was partly at least a symptom of the decay of the church-idea in its original fulness, a decay itself connected with the general decline in spiritual intensity which maiked 18th-century religion, after the overstrain of the preceding age.

00Later it was found that an assembly so constituted combined the incompatible functions of a council for the transaction of business and a congress for shaping or expressing common opinion: and its constitution was modified so as to secure the latter object only.

00In the meantime the functions of the university had been extended to include an oversight of the professional, scientific and technical schools, the administration of laws relating to admission to the professions, the charge of the State Library at Albany, the supervision of local libraries, the custody of the State Museum and the direction of all scientific work prosecuted by the state.

00For twenty years thereafter the political history of the colony consisted of two long, intermittent struggles - one constitutional between the central government (first seated in Auckland, but after 1864 in Wellington) and the powerful provincial councils, of which there were nine charged with important functions and endowed with the land revenues and certain rating powers.

00The provincial councils had been swept away in 1876, and their functions divided between the central authority and small powerless local bodies.

00In the New Testament angels appear frequently as the ministers of God and the agents of revelation"; and Our Lord speaks of angels as fulfilling such functions', implying in one saying that they neither marry nor are given in marriage.

00Angels are constantly spoken of as " men," and, including even the Angel of Yahweh, are spoken of as discharging the various functions of human life; they eat and drink 12, walk 13 and speak 14.

00The new constitution provided for a boule whose members were to be recruited by lot from all citizens above thirty; the functions of this body to be exercised by four sections succeeding one another by yearly rotation and serving without pay; all high officials to be chosen by it out of its own members.

00In the middle ages, however, it was the custom to wear it at nearly all liturgical functions.

00It even seems possible from a close attention to descriptions of sacred ordinances to conclude that his special: interests are those of a common Levite rather than of a priest,, and that of all Levitical functions he is most partial to those of the singers, a member of whose guild he may have been.

00The judex and the arbiter had the same functions, and apparently the only express basis for the distinction between the two words is that there might be several arbitri but never more than one judex in a cause.

00When a building contract provides that a certificate of the architect, showing the final balance due to the contractor, shall be conclusive evidence of the works having been duly completed, the architect occupies the position of an arbitrator, and enjoys the same immunity from liability for negligence in the discharge of his functions (Chambers v.

00The procedure was gratuitous and voluntary; and the functions of the arbitrator were not judicial; he merely recorded the arrangement arrived at, or the refusal of conciliation.

00Several doctrines extracted from his works were condemned at Rome, and he was suspended from his priestly functions, spending