Functions sentence example
It is worn by bishops, priests, deacons and subdeacons under the other eucharistic vestments, either at Mass or at functions connected with it.
On 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.
There is also an official performing the functions of a director at Bastia, in Corsica.
The 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.
The first was the appointment of a grand council with supreme judicial and financial functions P 7 ?Advertisement
It 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.
If 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.
It 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.
In 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.
Besides its legislative functions, the senate is the highest court of justice in the case of political offences or the impeachment of ministers.Advertisement
There 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.
In the provincial administrations the functions of the prefects have been curtailed.
Besides 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.
Early 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.
The 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.Advertisement
The 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.
Of 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.
As 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.
The sub-umbral cavity (s.c.) functions as a brood-space for the developing embryos, which are set free by rupture of the wall.
The appendages show various types of form and structure corresponding to different functions.Advertisement
The 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.
The 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.
It 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.
Deposition 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.
The Assembly appoints a commission to exercise some of its functions during the intervals of its session.Advertisement
This 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.
In 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.
The 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.
The leaves, again, have special histological features adapted to the performance of their special functions.
But hairs have a variety of other functions.
Wood 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.
The 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.
We come, accordingly, to regard it as practically an exoskeleton, and its functions as distinctly subordinate to those of the protoplasm which it clothes.
Every 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.
They 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.
It 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.
It 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.
Tissues.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.
The 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.
The 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.
It 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.
Accepting 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.
The 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.
If 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.
The 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.
One 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.
The 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.
Admission to military command was won first, then admission to civil jurisdiction; a share in religious functions was won last of all.
But in the more strictly crown, even if of quite humble origin, are "commanded" to court functions with their husbands.
The 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.
Its functions, which are exceedingly various, are carried out by the different departments into which it is divided.
Above all, there was no dividing 'line between the judiciary and the administrative functions.
From 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.
The confusion of the judicial and administrative functions was introduced again by the appointment of officials as judges.
In 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.
The state commissions, since their establishment in the 'seventies and the 'eighties, had increased their functions and influence.
Grindal remonstrated, claiming some voice for the Church, and in June 1577 was suspended from his jurisdictional, though not his spiritual, functions for disobedience.
And it is also probable that certain persons combined in their own individuality the functions of magician and sacrificer as well as soothsayer.
Their 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.
When 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.
Worship 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.
Their functions were political rather than religious, though their influence was by no means purely secular.
Thus the Babylonian academies combined the functions of specialist law-schools, universities and popular parliaments.
A supreme court of appeal, which also discharges the functions of a court of cassation, sits at Canea.
It is clear that the rulers, as so P p commonly in ancient states, fulfilled priestly as well as royal functions.
Indeed his functions as Labour adviser so occupied his time and attention, that it was thought desirable to relieve him in Aug.
The 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.
Guided 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.
They 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.
One 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.
Specially 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.
But 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.
After 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.
In 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.
In 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.
The functions of the archdeacon are in the present day ancillary in a general way to those of the bishop of the diocese.
In the colonies there are two or more archdeacons in each diocese, and their functions correspond to those of English archdeacons.
After a successful trial of a few years this association was dissolved, transferring its functions to the Highland and Agricultural Society.
Even 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.
In 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.
In 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.
The 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.
It 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.
The 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.
The extension of government functions appeared much more likely to continue than to be checked.
The development of the powers of the central government has been less than that of the functions of local governing authorities.
Local 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.
Great 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.
The 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.
As 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.
In 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.
Less specialized in their functions are many of the figures of modern folklore, some of whom have perhaps replaced some ancient goddess, e.g.
The 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.
It 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."
The 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.
Hence Caesar seems to assign more extensive functions to the Druids than they actually possessed.
In 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.
The 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.
In 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.
His 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.
This 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.
In 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.
Whether 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.
Four 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).
At 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."
Three 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.
To 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.
By 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.
A 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.
An 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.
The position of a Friend did not carry with it necessarily any functions; it was in itself purely honorary.
From 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.
Both kinds of functions were discharged by slaves, not only at Rome, but in the rural and provincial municipalities.
Nominally 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.
Joachim, 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.
The prefect has to a certain extent a double character and two series of functions.
In 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.
The character of this body was altered in 1890, and in 1898, in which latter year its functions were reduced to the essentially judicial.
A similar theory holds for relations which arise from the consideration of propositional functions with two or more variable arguments.
Under 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.
Functions, operations, transformations, substitutions, correspondences, are but names for various types of relations.
Until 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.
All 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.
Regulations 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.
After 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.
It is used not only at pontifical High Mass but at all solemn pontifical functions, e.g.
In addition to their functions as guardians of the poor, the parish members have to investigate crimes and punish misdemeanours, settle litigations and divide inheritances.
As 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.
It 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.
Liturgical vestments, as their name implies, are those which are especially associated with the various functions of the liturgy.
By the 4th century the garments worn at liturgical functions had been separated from those in ordinary use, though still identical in form.
Hitherto 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.
However, 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.
They 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.
The administration is in the hands of a council of chiefs which exercises legislative, executive and, to some extent, judicial functions.
But 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.
The 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.
The 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.
P. 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.
There is no difficulty in expressing the resultant by the method of symmetric functions.
Now 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.
A separation is the symbolic representation of a product of monomial symmetric functions.
The 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!
By solving k linear equations we similarly express the latter functions as linear functions of the former, and this table will also be symmetrical.
For 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,...
The important result is that the theory of invariants is from a certain point of view coincident with the theory of non-unitary symmetric functions.
Denote by brackets () and  symmetric functions of the quantities p and a respectively.
All 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.
The 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.
The 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.
Since 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.
In the theory of forms we seek functions of the coefficients and variables of the original quantic which, save as to a power of the modulus of transformation, are equal to the like functions of the coefficients and variables of the transformed quantic. We may have such a function which does not involve the variables, viz.
Invariantive forms will be found to be homogeneous functions alike of the coefficients and of the variables.
There 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.
If, 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.
Hesse's canonical form shows at once that there cannot be more than two independent invariants; for if there were three we could, by elimination of the modulus of transformation, obtain two functions of the coefficients equal to functions of m, and thus, by elimination of m, obtain a relation between the coefficients, showing them not to be independent, which is contrary to the hypothesis.
The general equation of degree 5 cannot be solved algebraically, but the roots can be expressed by means of elliptic modular functions.
The 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.
Remark, 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.
Lead generally functions as a divalent element of distinctly metallic character, yielding a definite series of salts derived from the oxide PbO.
It forms crystallizable salts with potassium and calcium hydrates, and functions as a weak acid forming salts named plumbates.
Rowland,' 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.
Certain religious functions of minor importance were also reserved for them.
The functions of these gastric diverticula have never been carefully investigated.
It 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.
This essentially simplifies the treatment of such functions.
After 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.
The 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.
Under 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.
His 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).
The functions of the council were among the most important in the state.
The 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.
As 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.
The 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.
In 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.
The 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.
Stockdale 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.
The 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.
According to some, ordination simply entitles a man to hold an office and perform its functions.
Aquinas 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.
The introduction of hyperbolic functions into trigonometry was also due to him.
Moreover, 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.
Hungary 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.
Laplace 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.
In 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.
The 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.
We cannot, for instance, say that the fraction C _2 I is arithmetically equal to x+I when x= I, as well as for other values of x; but we can say that the limit of the ratio of x 2 - I to x - I when x becomes indefinitely nearly equal to I is the same as the limit of x+ On the other hand, if f(y) has a definite and finite value for y = x, it must not be supposed that this is necessarily the same as the limit which f (y) approaches when y approaches the value x, though this is the case with the functions with which we are usually concerned.
Pierpont, Lectures on the Theory of Functions of Real Variables (1905).
His 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.
The whole issue had, in fact, become confused with the confusion of functions of the Church and State.
Owing 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.
A 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.
Analogous 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.
Homologous 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."
The integrals are then properly functions of the direction in which the light is to be estimated.
The maximum brightnesses and the places at which they occur are easily determined with the aid of certain properties of the Bessel's functions.
The 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.
The 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.
School boards and district committees are formed, but their functions are almost entirely advisory.
His 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.
It 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).
Where 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.
In 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.
In 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.
Much light has been thrown upon the functions and diseases of the blood-forming tissues.
The 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.
The 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.
There 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.
Or 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.
The functions of the thymus gland begin to cease after the second year from birth.
As 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.
Oertel thinks that in man we have these two different functions carried on by the one nucleus containing both chromatin orders.
The structure and functions of the body form the subject of Anatomy (q.v.) and Physiology.
There 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.
The tendency of the school was to explain the actions and functions of the body on physical, and especially on mechanical, principles.
James 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.
For 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.
Thus 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.
The 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.
Nor was the theory of reflex action confined to the more "mechanical" functions.
These 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.
In general criticism and miscellaneous writing Voltaire is not inferior to himself in any of his other functions.
The 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.
His 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.
Originally intended as assistants to the tribunes, they exercised certain police functions, were empowered to inflict fines and managed the plebeian and Roman games.
But although the curule aediles always ranked higher than the plebeian, their functions gradually approximated and became practically identical.
Under 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.
Its vice-governor-general exercised all the executive functions of the governor-general and corresponded directly with Brussels.
In 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.
In 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.
For these opinions he was in 1728 suspended from the discharge of ministerial functions, and finally deposed in 1730.
The 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.
For 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.
The 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.
Toulmin Smith (1816-1869), and they throw much light on the functions of the gilds.
While 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.
It 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.
In these two ways then the feudal system was formed, and took possession of the state territorially, and of its functions in government.
The 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.
As yet, however, the jurisdiction and functions of these courts have not been ascertained.
The people were to be directly governed by their native chiefs, whose duty was to collect the tribute and exercise magisterial functions.
His TEXvrt rypaµµanKii, which we possess (though probably not in its original form), begins with the definition of grammar and its functions.
From 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.
Under 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.
For further details of his mathematical investigations see the articles Theory of groups, and Functions Of Complex Variables.
In 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.
To fit them for their functions, they are to be raised above material cares, and they are to be thoroughly educated.
Meanwhile 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.
Martineau's mental qualities fitted him to fulfil these high interpretative functions.
From 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.
In 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.
The 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.
His 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.
The 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.
Nothing 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.
The 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.
Since 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.
The 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.
These 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.
The 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.
The 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'.
The 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.
He may, however, exercise his purely episcopal functions.
In 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.
In 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.
Organization 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.
In 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.
It functions chiefly as an acidic oxide, 'being less basic than aluminium oxide, and forming no stable oxy-salts.
There was also a lord warden, who was usually a nobleman and performed no judicial functions.
This 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.
In the meanwhile Bolivar had accepted the presidency, and resumed the functions belonging to his official position.
At 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.
His functions are rather more extensive than those of the average American executive.
Re-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.
Ow 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.
His researches on elliptic functions are of considerable elegance, but their great merit lies in the stimulating effect which they had on later mathematicians.
Further, 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.
On 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.
Meanwhile the Conqueror had invested him with important judicial functions.
The Church was not only organized like a modern bureaucracy, but performed many of the functions of a modern State.
The civil tribunals, however, practically assumed the functions of regular inquisitorial courts, in spite of the objections urged by the ecclesiastical courts.
Another 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.
The 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.
The 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.
Opposite each carpel is a small scale which functions as a nectary.
The 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.
In 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.
The 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.
In 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.
For 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.
The provincial councils had been swept away in 1876, and their functions divided between the central authority and small powerless local bodies.
In 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.
Angels 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.
The 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.
In the middle ages, however, it was the custom to wear it at nearly all liturgical functions.
The 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.
The 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.
Several doctrines extracted from his works were condemned at Rome, and he was suspended from his priestly functions, spending the rest of his life in literary work.
This commission was admitted to have exercised its functions with impartiality as a matter of fact; but as a matter of form it stood on a weak foundation.
The lowest administrative unit was the political sub-district (Bezirk) under an official (Bezirkshauptmann), who united nearly all the administrative functions which were divided among the various ministries according to their attributions.
Since the functions of these organizations were commercial, for which the regular Government officials were unsuited, they were established as commercial joint-stock companies under peculiar conditions adapting them to the service of the state.
These boards also undertook other functions, such as introducing new methods of manufacture and supplying the workers in the munition factories with beer.
A second peculiarity of Ultramontanism is its confusion of religion with politics; it claims for the Roman Catholic Church the functions of a political power, and asserts that it is the duty of the secular state to carry out its instructions and wishes.
A somewhat different conception of the sovereign's functions was that of Disraeli's great rival, Gladstone, who, though his respect for the person and office of the sovereign was unbounded, not only expected all people, the queen included, to agree with him when he changed his mind, but to become suddenly enthusiastic about his new ideas.
It embraces many duties - some of them menial and disagreeable - besides the purely medical and surgical functions.
The numerous palace and other functions make some demand upon ministers' time, and, as the king transacts most of his affairs at night, high officials usually keep late office hours.
Epiphanius says quite distinctly that they were woman-elders and not priestesses in any sense of the term, and that their mission was not to interfere with the functions allotted to priests but simply to perform certain offices in connexion with the care of women.
One of the early supporters of this natural method was Augustin Pyramus de Candolle (1778-1841), who in 1813 published his Theorie elementaire de la botanique, in which he showed that the affinities of plants are to be sought by the comparative study of the form and development of organs (morphology), not of their functions (physiology).
Nehemiah Grew seems to have been the first to describe, in a paper on the Anatomy of Plants, read before the Royal Society in November 1676, the functions of the stamens and pistils.
He was one of the first to point out the existence of certain cellular bodies in these plants which appeared to perform the functions of reproductive organs, and to them the names of antheridia and pistillidia were given.
Apostleship, prophecy and teaching were only functions, whose frequent or regular exercise by one or another, under the inspiration of the Spirit, led his brethern to call him an apostle, prophet or teacher.
This enlargement of the bishop's parish and multiplication of the chuches under his care led to a change in the functions of the presbyterate.
For the importance of Charlemagne's work, from the point of view of the Church, consists also, not so much in the fact that, by his conversion of the Saxons, the Avars and the Wends in the eastern Alps, he substantially extended the Church's dominions, as in his having led back the Frankish Church to the fulfilment of her functions as a religious and civilizing agent.
Hence the king starting as a magician tends gradually to exchange the practice of magic for the functions of prayer and sacrifice."
The question, however, remains to be settled how far the officials and their functions, which in the much more highly developed civilization of Babylonia came to be differentiated and specialized, can be strictly included under the functions of priesthood.
Here we find magic and soothsaying closely intertwined with priestly functions as, we shall see, was the case in early Hebrew pre-exilian days with the Kohen.
Tallquist's edition of the Maklu series of incantations and his explanations of the ritual, and also the publications by Zimmern of the Surpu series of tablets in his Beitreige have rendered us familiar with the functions of the asipu.
I do not deny that the same man might unite all three functions in one person.
Thus we find that the Egyptian monarch was empowered to exercise priestly functions before all the gods.
The king, in fact, both in Greece and in Rome, was the acting head of the state religion, and when the regal power came to an end his sacred functions were not transferred to the ordinary priests, but either they were distributed among high officers of state, as archons and prytanes, or the title of " king " was still preserved as that of a religious functionary, as in the case of the rex sacrorum at Rome and the archon basileus at Athens.
It was open to any one to obtain entrance into the priesthood, while on the other hand it was only as a priest that he could exercise sacerdotal functions, for these were strictly reserved to priests.
With the beginning of a settled state the sanctuaries must rise in importance and all the functions of revelation will gather round them.
According to Semitic ideas the declaration of law is quite a distinct function from the enforcing of it, and the royal executive came into no collision with the purely declaratory functions of the priests.
The ritual functions of the priesthood still appear in Deut.
It is instructive to observe how differently the prophets of the 8th century speak of the judicial or " teaching " functions of the priests and of the ritual of the great sanctuaries.
The bases of priestly power under this system are the unity of the altar, its inaccessibility to laymen and to the inferior ministers of the sanctuary, and the specific atoning functions of the blood of priestly sacrifices.
The High Priest at such a moment seemed to embody all the glory of the nation, as the kings had done of old, and when the time came to strike a successful blow for freedom it was a priestly house that led the nation to the victory which united in one person the functions of High Priest and prince.
The Exercices de calcul integral consist of three volumes, a great portion of the first and the whole of the third being devoted to elliptic functions.
It will thus be seen that Legendre's works have placed him in the very foremost rank in the widely distinct subjects of elliptic functions, theory of numbers, attractions, and geodesy, and have given him a conspicuous position in connexion with the integral calculus and other branches of mathematics.
The position and functions of the bishops vary in the different countries.