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function

But that, too, is a function of present technology.

333133In the sense he can function but the memory is probably gone for good.

285203She didn't understand how humans could feel so much and still function when she was overwhelmed.

14288Exportable technology can function around the world.

11775Her mind was beginning to function somewhere near normal.

9968Tests revealed normal brain function, and there was no indication of any permanent damage.

8623Unable to function from shock, he sat down on a rock overlooking the lake.

8269And yes, everything and everyone in my life has a function and purpose.

6138It was then that my slowly reacting brain, flowing like cold molasses began to function, more than a gerbil driven wheel.

5638"Loss of cognitive function is a sign, yes," he replied.

5241He was performing the same function, but meeting him didn't darken Dean's mood appreciably.

4635To this high conception of a preacher's function the prophet was faithful throughout his career.

4325The first is " the law that thought is a function of the brain."

4221Based on her medical records, the tumor didn't grow until she hit puberty and didn't interfere with her ability to function before a few years ago, Wynn started.

4229In many sees there were no vidames, their function being exercised by viscounts or chatelains.

3214Free and peaceful societies function best when government is transparent and open.

2917That said, if I had to pick one function I think the Internet will turn out to "be," it is this: The Internet will become a repository and a set of applications for storing the sum total of all life experiences of all people on earth.

2718Dean guessed she had been pampered and protected all of her life and was now ill equipped to function independently.

2637When Ethel did the inviting, the function was nearly always out of town.

2018Depending on function, robots can come in all shapes and sizes, and I see no compelling reason to make them like humans.

187In the first place, with a given size of particles, the direction of complete polarization indicated by (23) is a function of the colour of the light, the value of 0 being 3 or 4 times as large for the violet as for the red end of the spectrum.

1717The maxillae are not piercing organs, and their function is to protect the mandibles and labrum and separate the hairs or feathers of the host.

1619Accepting the law he distinguishes productive from permissive or transmissive function (p. 32), and, rejecting the view that brain produces thought, he recognizes that in our present condition brain transmits thought, thought needs brain for its organ of expression; but this does not exclude the possibility of a condition in which thought will be no longer so dependent on brain.

159Of late years the function of the collector is discharged in some forms of apparatus by a salt of radium.

1516They perform a useful function in protecting their clients from the cruel usury which prevails, especially in the south.

1517Some of it is known, but the function of each of the thirty thousand genes has to be figured out one at a time.

1518The operator, whose attention is thus attracted, inserts a peg in the jack, then throws over the speaking key of the cord circuit, and having ascertained particulars of the requirement places the other peg of the pair in the nearest multiple jack of the wanted subscriber, whom she proceeds to ring up. In the meantime the callinglamp has darkened; and each subscriber's line being equipped with a cut-off relay whose function it is to disconnect tl, e calling apparatus while the circuit is in use, the insertion o r a peg is immediately followed by the disappearance of the calling signal.

1521Simspon concluded that for a given wind velocity dissipation is practically a linear function of ionization.

1410Its function is to examine criminal cases and to decide whether they shall be referred for trial to the lower courts or the cours dassises.

136One of the chief names for the priest was baru - literally the "inspector" - which was given to him because of the prominence of his function as an inspector of livers for the purpose of divining the intention of the gods.

139An instance of this characteristic function, specially remarkable because the composer has taken exceptional measures for it, is Beethoven's overture to Fidelio.

1310All access to the Deity is restricted to the one priesthood and to the one sanctuary at Jerusalem; the worshipping subject is the nation of Israel as a unity, and the function of worship is discharged on its behalf by divinely chosen priests.

1312The most important change introduced by the new law has been the creation in every province of a provincial administrative junta entrusted with the supervision of communal administrations, a function previously discharged by the provincial deputation.

126The polyps are all non-sexual individuals whose function is purely nutritive.

127Bach's conception of the function of an instrument is that it holds a regular part in a polyphonic scheme; and his blending of tones is like the blending of colours in a purely decorative design.

128If Ritschl had clearly shown that judgments of value enfold and transform other types of knowledge, just as the "spiritual man" includes and transfigures but does not annihilate the "natural man," then within the compass of this spiritually conditioned knowledge all other knowledge would be seen to have a function and a home.

129As it is with mechanical improvements, so is it to a still greater degree with changes in the function of timbre in art.

117Mention may also be made of the Tribunal des Conflits, a special court whose function it is to decide which is the competent tribunal when an administration and a judicial court both claim or refuse to deal with a given case.

119The principal function of these courts is the hearing of appeals both civil and criminal from the courts of first instance; only in.

119It 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.

1118Early in the 19th century the venue of the dinner, which had now become a ministerial function, was transferred to Greenwich, and though at first not always held here, was later celebrated regularly at the "Ship," an hotel of ancient foundation, closed in 1908.

106And yet, we know of no cases of mass "left behind-ness," of people unable to learn how to function in this environment.

1014He should therefore in all such passages play extremely lightly, so as to give the violin and 'cello the function of drawing the main outline.

813And indeed, whilst in theoretic theology Brahma has retained his traditional place and function down to our own days, his practical cult has at all times remained extremely limited, the only temple dedicated to the worship of this god being found at Pushkar (Pokhar) near Ajmir in Rajputana.

813With the judges were associated a body of elders, who shared in the decision, but whose exact function is not yet clear.

815These instruments thus produced, in Haydn's and Beethoven's times, a very remarkable but closely limited series of effects, which, as Sir George Macfarren pointed out in the article "Music" in the 9th edition of the Encyclopaedia Britannica, gave them a peculiar character and function in strongly asserting the main notes of the key.

825As the direction and intensity of this induced current are a function of the position of the second coil in its field, and as this position is determined by its mechanical connexion with the recorder coil, it is evident that, by a suitable choice of the electrical elements of the second coil and its alternating field, the indications on the siphon recorder can be magnified to any reasonable extent.

710The function of the " combiner " in each receiving instrument is so to group the received combination of positive and negative currents that they operate polarized relays in such a manner that the position of the tongues corresponds with the operation of the levers on the transmitter.

616And we got them all, more or less, by trade and the wealth generated by our work doing some function for which we are trained.

633In this way the hydroid colony becomes composed of two portions of different function, the nutritive " trophosome," composed of non-sexual polyps, and the reproductive " gonosome," composed of sexual medusaindividuals, which never exercise a nutritive function while attached to the colony.

514Their function is not quite clear.

30Laplace is due the theoretical proof that this function is independent of temperature and pressure, and apparent experimental confirmation was provided by Biot and Arago's, and by Dulong's observations on gases and vapours.

20As, however, the temperature developed is a function of the time needed to complete the action, the degree of heat attained varies with every form of generator, and while the water in one form may never reach the boiling-point, the carbide in another may become red-hot and give a temperature of over 800° C. Heating in a generator is not only a source of danger, but also lessens the yield of gas and deteriorates its quality.

20In 1408, however, the clergy of the city and archiepiscopal diocese of Prague laid before the archbishop a formal complaint against Huss, arising out of strong expressions with regard to clerical abuses of which he had made use in his public discourses; and the result was that, having been first deprived of his appointment as synodal preacher, he was, after a vain attempt to defend himself in writing, publicly forbidden the exercise of any priestly function throughout the diocese.

20If the system is supposed to obey the conservation of energy and to move solely under its own internal forces, the changes in the co-ordinates and momenta can be found from the Hamiltonian equations aE aE qr = 49 - 1 57., gr where q r denotes dg r ldt, &c., and E is the total energy expressed as a function of pi, qi,.

20The special function of the so-called "Socratic dialectic" was to show the inadequacy of popular beliefs.

20Finally the theory defined is opposed to all forms of realism, whether in the older form which sought to reduce mind to a function of matter, or in any of the newer forms which seek for the ultimate essence of both mind and matter in some unknown force or energy which, while in itself it is neither, yet contains the potentiality of both.

20Perhaps the earliest of such specializations is connected with the reproductive function.

10Their chief function is to ordain, and to act as "intercessors."

10i, and the plane surface performs the function of the mirror.

10This latter function, however, is one unknown to the English common law.

10There's no reason to feel embarrassed about a normal body function.

00As your husband, what will he consider his most important function?

00She tried to grip the necklace around her neck with hands still too cold to function.

00The only thing which the " Ideas " of " Reason " can do for theoretic knowledge is to exert a " regulative " function.

00(After Allman.) sensory, contractile and pro tective in function.

00function of circulating and distributing nutriment through the colony.

00Brooks regards these organs as sensory, serving for the sense of balance, and representing a primitive stage of the tentaculocysts of Trachylinae; Linko, on the other hand, finding no nerve-elements connected with them, regards them as digestive (?) in function.

00- p elagic floating Hydrozoa with great differentiation of parts, each performing a special function; generally regarded as colonies showing differentiation of individuals in correspondence with a physiological division of labour.

00In the Disconectae the coenosarc forms a spongy mass, the " centradenia," which is partly hepatic in function, forming the socalled liver, and partly excretory.

00The cormus is always differentiated into two parts; an upper portion termed the nectosome, in which the appendages are locomotor or hydrostatic in function, that is to say, serve for swimming or floating; and a lower portion termed the siphosome, bearing appendages which are nutritive, reproductive or simply protective in function.

0068, k), absent in Chondrophorida and Cystophorida; they are contractile and resemble, both in appearance, structure and function, the umbrella of a medusa, with radial canals, ring-canal and velum; but they are without manubrium, tentacles or sense-organs, and are always bilaterally symmetrical, a peculiarity of form related with the fact that they are attached on one side to the stem.

00In Velella the pneumatophore becomes of complex structure and sends air-tubes, lined by a chitin and resembling tracheae, down into the compact coenosarc, thus evidently serving a respiratory as well as a hydrostatic function.

00If a distal pore or aperture is present, it is excretory in function; suck varieties have been termed " cystons " by Haeckel.

00If sterile they remain attached and locomotor in function, forming the nectosome, the pneumatophore and swimming-bells.

00In the lower jaw the incisors and canines are directed straight forwards, and are of small size and nearly similar form; the function of the canine being discharged by the first premolar, which is larger than the other teeth of the same series.

00Many forms, even when multicellular, have all their cells identical in structure and function, and are often spoken of as physiologically unicellular.

00The surface-layer of the body in the massive Fungi differs in character according, to its function, which is not constant throughout the class, as in the Algae, because of the very various conditions of life to which different Fungi are exposed.

00The formation of a massive body naturally involves the localization of the absorptive region, and the function of absorption (which in the simpler forms is carried out by the whole of the vegetative part of the mycelium penetrating a solid or immersed in a liquid substratum) is subserved by the outgrowth of the hyphae of the surface-layer of that region into rhizoids, which, like those of the Algae living on soil, resemble the root-hairs of the higher plants.

00some bulky Brown Seaweeds, where assimilation is strongly localized, some of the deep cells are highly specialized for the latter function.

00Besides absorption, assimilation, conduction and protection there is another very important function for which provision has to be made in any plant-body of considerable size, especially when raised into the air, that of support.

00In relation to its characteristic function of protection, the epidermis, which, as above defined, consists of a single layer of cells has typically thickened and cuticularized outer walls.

00A very common function of hairs is to diminish transpiration, by creating a still atmosphere between them, as in the case of the sunk stomata already mentioned.

00In correspondence with its water-absorbing epidermis function it is not cuticularized, but remains usually thinof Root, walled; the absorbing surface is increased by its cell~

00It corresponds in function with the rhizoid of i Bryophyte.

00In all green plants which have a special protective epidermis, the cortex of the shoot has to perform the primitive fundamental function of carbon assimilation.

00In the leafy shoot this function is mainly localized in the cortical tissue of the leaves, known as mesophyll, Mesophyli.

00The cortex of a young stem is usually green, and plays a more or less important part in the assimilative function.

00When the sieve-tube has ceased to function and the protoplasm, slime strings, and callose have disappeared, the perforations through which the slime strings passed are left as relatively large holes, easily visible in some cases with low powers of the microscope, piercing the sieve.plate.

00The sieve-tubes differ, however, from the tracheids in being immediately associated, apparently constantly, not with starchy parenchyma, but with parenchymatous cells, containing particularly abundant proteid contents, which seem to have a function intimately connected with the conducting function of the sieve-tubes, and which we may call proteid-cells.

00The relation ~ of the laticiferous tissue to the assimi I lating cells under which they often end, and the fact that where this tissue is / richly developed the conducting paren ~ chyma of the bundles, and sometimes also 4 the sieve-tubes, are poorly developed, as well as various other facts, point to the conclusion that the laticiferous system has an important function in conducting plastic substances, in addition to acting as an excretory reservoir.

00As a secondary function we may recognize, in certain cases, the power of closing wounds, which results from the rapid coagulation of exuded latex in contact with the air.

00The lenticels of the stem are usually formed beneath stomata, whose function they take up after the stomata have been ruptured and cast off with the rest of the epidermis.

00The requirements of the several protoplasts must be met by supplies from without, and, as many of them are deep seated, varieties of need arise, so that various members of the colony are set apart for special duties, masses of them being devoted to the discharge of one function, others to that of another, and so on.

00The latter function has been found to be of extreme importance in the case of plants exposed to the direct access of the suns rays, the heat of which would rapidly cause the death of the protoplasts were it not employed in the evaporation of the water.

00Another enzyme, rennet, which in the animal body is proteolytic, is frequently met with in plants, but its function has not been ascertained.

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.

00It is true that in the unicellular plants all the vital activities are performed by a single cell, but in the multicellular plants there is a more or less highly developed differentiation of physiological activity giving rise to different tissues or groups of cells, each with a special function.

00The special function of this organ has been a source of controversy during the past few years, and much uncertainty still exists as to its true nature.

00The communication with the atrium is guarded by a valvula cardiaca dextra, which only in function represents the mammalian tricuspid; it consists of an oblique reduplication of the muscular fibres together with the endocardiac lining of the right ventricle, while the opposite wall is convex and forms neither a velum nor papillary muscles, nor chordae tendineae.

00The crop is modified into a large and very rugose triturating apparatus, while the gizzard, thereby relieved of its function, is reduced to the utmost.

00It is most developed in the young of both sexes, is of unknown function, and becomes more or less obliterated in the adult.

00In more primitive ritual, however, set forms of prayer are the rule, and their function is mainly to accompany and support a ceremony the nerve of which consists in action rather than speech.

00In general, these performed very much the same function as the lives of saints in the early and medieval church.

00The function of the stridulating organs just described is presumably to afford means of recognition by sound.

00Lastly, it examines into registers and promulgates new laws, a function which, in theory, gives it a power, akin to that of the Supreme Court of the United States, of rejecting measures not in accordance with the fundamental laws.

002 Finally, in the towns every house is provided with a detective policeman in the person of the porter (dvornik), who is charged with the duty of reporting to the police the presence of any suspicious characters or anything else that may interest them .3 In addition to the above there is also a police organization, in direct subordination to the ministry of the interior, of which the principal function is the discovery, pre vention and extirpation of political sedition.

00The draw-bar pull for a given load is a function of the speed of the train, and numerous experiments have been made to find the relation connecting the pull with the speed under various conditions.

00For a stated value of the boiler pressure and the cut-off the mean pressure p is a function of the piston speed v.

00A like function belonged to the Babylonian Ishtar.

00Prophetic personality now moved in a larger sphere than that of divination, important though that function be in the social life of the ancient state as instrumental in declaring the will of the deity when any enterprise was on foot.

00For the prophet's function became in an increasing degree a function of mind, and not merely of traditional routine or mechanical technique, like that of the diviner with his arrows or his lots which he cast in the presence of the ephod or plated Yahweh image.

00The new name nabhi' became necessary to express this function of more exalted significance, in which human personality played its larger role.

00Even as early as the time of David it would seem that Nathan assumed this more developed function as interpreter of Yahweh's righteous will to David.

00To this is united the noble ideal of the suffering servant, which serves both as a contribution to the great problem of suffering as purifying and vicarious and as the interpretation to the mind of the nation itself of that nation's true function in the future, a lesson which the actual future showed that Israel was slow to receive.

00Wing-covers and hind-wings are alike absent, and the latter are represented by a pair of little knobbed organs, the halteres or balancers, which have a controlling and directing function in flight.

00This function of the office, although it may not have been contemplated at first, is attested by the internal history of Rome.

00These dictators appointed for minor purposes were expected to retire from office as soon as their function was completed.

00These segments are abundantly supplied with elongate tooth-like projections connected with nerve-endings probably olfactory in function.

00Apart from conquest their most important function has been to keep up communications in central Asia, and to transport religions and civilizations from one region to another.

00The setae are organs of locomotion, though their large size and occasionally jagged edges in some of the Polychaeta suggest an aggressive function.

00Finally, in function there are some points of likeness.

00In the case of many Oligochaeta where there is no vascular network surrounding the nephridium, this function must be the chief one of those glands, the more elaborate process of excretion taking place in the case of nephridia surrounded by a rich plexus of blood capillaries.

00This differentiation is not, however, peculiar to the Polychaetes; for in several Oligochaetes the anterior nephridia are of large size, and opening as they do into the buccal cavity clearly play a different function to those which follow.

00The oesophagus is provided often with caeca which in Syllids and Hesionidae have been found to contain air, and possibly therefore perform the function of the fish's air-bladder.

00These sacs are dealt with later under the description of the spermathecae, which function they appear to perform.

00(as it appears) of the epidermis, and that it performs the function of a spermatheca is shown by its containing spermatozoa, or, in Stuhlmannia, a spermatophore.

00We have thus the replacement of a spermatheca, corresponding to those of the remaining families of Oligochaeta, and derived, as is believed, from the epidermis, by a structure performing the same function, but derived from the mesoblastic tissues, and with a cavity which is coelom.

00We have here clearly a case of a true coelomoduct performing the function of an oviduct in both leeches and Eudrilidae.

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 existence of the root nodules had long been recognized, but hitherto no adequate explanation had been afforded as to their function.

00In the sheep section of the Smithfield show the classes for ewes were finally abolished in 1898, and the classes restricted to wethers and wether lambs, whose function is exclusively the production of meat.

00In reasserting and amplifying the empirical conclusions of his predecessors, especially in the sphere of ethics, Mill's chief function was the introduction of the humanist element.

00In these, as in Patella, the typical ctenidia are aborted, and the branchial function is assumed by close-set lamelliform processes arranged in a series beneath the mantle-skirt on either side of the foot.

00When (he held) we have discovered what man in himself is, and what his relation to his environment, we shall then know his function - what he is fitted to do.

00Self-reflection gradually reveals to us human capacity, human function, with, consequently, human responsibility.

00In the relative development and shape of the various segments of the leg there is almost endless variety, dependent on the order to which the insect belongs, and the special function - walking, running, climbing, digging or swimming - for which the limb is adapted.

009, E) that cover the body of an insect have a tactile function.

00Organs of similar type on the maxillae and epipharynx appear to exercise the function of taste.

0012, c) and a proventriculus or " gizzard," whose function is to strain the foodsubstances before they pass on into the tubular stomach, which has no chitinous lining.

0012, k), which have a renal function.

00Nevertheless, the function of reproduction is occasionally exercised by larvae.

00Chladni's experiment of strewing a vibrating bell with flour, investigated the nature of sound and the function of the air in respiration and combustion, and originated the idea of using the pendulum as a measure of gravity.

00Besides the distinctions of human and nonhuman, hostile and friendly, the demons in which the lower races believe are classified by them according to function, each class with a distinctive name, with extraordinary minuteness, the list in the case of the Malays running to several score.

00SPIDERS, the common English name of Arachnida of the order Araneae, resembling the Pedipalpi in many structural points, but differing from them as well as from all other Arachnida in retaining short abdominal appendages known from their silk-manipulating function as spinnerets or spinning mamillae, with which are associated silk glands.

00The primary function of this poison is to kill the prey upon which they feed, its action being very rapid upon insects.

00So far as is known, however, only the large spiders belonging to this group possess this special means of defence, and in many other species this is accompanied by highly-developed stridulating organs resembling those of rattlesnakes and scorpions in function.

00The roots are prevented from fulfilling their function of taking up water and salts from the soil; the leaves accordingly droop, and the whole plant wilts and in bad attacks dies.

00William James distinguishes the transmissive function of the brain from the productive in relation to thought, and admits only the former, and not the latter (Human Immortality, p. 32).

00Their function is still little understood.

00Its function is less that of respiration than of FIG.

00In this last case the shunt need not be contained in the instrument itself but may be at a considerable distance, wires being brought from the shunt which carries the main current to the movable coil ammeter itself, which performs the function simply of an indicator, 3.

00The judge advocate of the fleet is a practising barrister whose function it is to advise the admiralty on all matters connected with courts-martial.

00The tips of these tufts enclose a flamecell similar to those found in Platyhelminths, &c., and these probably function as excretory organs.

00Also communicating with this pouch is a pair of long slender flagelliform tubes, of which the function is unknown.

00In 1513 the arrival of its first bishop, who later also exercised the function of general inquisitor, added one more to the discordant elements ruling the island.

00Instances had already been recorded of cases where a halogen element replaced hydrogen with the production of a closely allied substance: Gay Lussac had prepared cyanogen chloride from hydrocyanic acid; Faraday, hexachlorethane from ethylene dichloride, &c. Here the electronegative halogens exercised a function similar to electro-positive hydrogen.

00Claus maintained that Baeyer's view was identical with his own, for as in Baeyer's formula, the fourth valencies have a different function from the peripheral valencies, being united at the centre in a form of potential union.

00He introduced the idea of comparing the refractivity of equimolecular quantities of different substances by multiplying the function (n-1)/d by the molecular weight (M) of the substance, and investigated the relations of chemical grouping to refractivity.

00Although establishing certain general relations between atomic and molecular refractions, the results were somewhat vitiated by the inadequacy of the empirical function which he employed, since it was by no means a constant which depended only on the actual composition of the substance and was independent of its physical condition.

00It is found experimentally that the Lorenz and Lorentz function holds fairly well, and better than the Gladstone and Dale formula.

00If, on the other hand, the general laws are regarded as intuitive, then the discernment of them may be taken as the true function of conscience.

00This usage derives from the last function of conscience mentioned above.

00In the latter respect the army came regularly into function under Alexander, and in the wars which followed his death (Diod.

00He had received from his father the smatterings of a liberal education, but until the outbreak of the Revolution he was a domestic servant, and from 1785 occupied the invidious office of cornmissaire a terrier, his function being to assist the nobles and priests in the assertion of their feudal rights as against the unfortunate peasants.

00In Homer, where they appear indifferently under the name of apirveac and 615EXAca, their function is to carry off those whose sudden disappearance is desired by the gods.

00Their function of snatching away mortals to the other world brings them into connexion with the Erinyes, with whom they are often confounded.

00Their monotheism remains Semitic - even in their conception of the cosmogonic and illuminating function of Wisdom they regard God as standing outside the world of physical nature and man, and do not grasp or accept the idea of the identity of the human and the divine; there is thus a sharp distinction between their general theistic position and that of Greek philosophy.

00They were intended to be the chief organs of internal administration, and have, in fact, discharged this function, especially under the First and Second Empire, surviving, though with diminished importance, under the other forms of government which modern France has seen.

00This function was most useful in the year VIII., when communications were difficult, even within a department, but nowadays it only leads to complications.

00Call it a propositional function; and, if 4)x be a propositional function, the undetermined variable x is the argument.

00Now consider a propositional function Fx in which the variable argument x is itself a propositional function.

00For the subjects of this heading see the articles DIFFERENTIAL EQUATIONS; FOURIER'S SERIES; CONTINUED FRACTIONS; FUNCTION; FUNCTION OF REAL VARIABLES; FUNCTION COMPLEX; GROUPS, THEORY OF; INFINITESIMAL CALCULUS; MAXIMA AND MINIMA; SERIES; SPHERICAL HARMONICS; TRIGONOMETRY; VARIATIONS, CALCULUS OF.

00The function of the British navy in the long conflict with Napoleon was of the first importance, and its services were rendered in every sea, but their very number, extent and complexity render it impossible here to record them in detail.

00Though the giving of blessings as a sacerdotal function is proper to the whole order of priests, particular benedictions have, by ecclesiastical authority, been reserved for the bishops, who may, however, delegate some of them; i.e.

00The collar-pores are remarkable for their constancy; this is probably owing to the fact that they have become adapted to a special function, the inhalation of water to render the collar turgid during progression.

00The respiratory current of water is therefore conducted to the exterior by different means from that adopted by Amphioxus, and this difference is so great that the theory which seeks to explain it has to postulate radical changes of structure, function and topography.

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.

00Interchanges must be supposed to go on whether a current passes or not, the function of the electric forces in electrolysis being merely to determine in what direction the parts of the molecules shall work their way through the liquid and to effect actual separation of these parts (or their secondary products) at the electrodes.

00This result again indicates that the parts of the molecules are effectively separate from each other, the function of the electric forces being merely directive.

00Clausius extended to electrolysis the chemical ideas which looked on the opposite parts of the molecule as always changing partners independently of any electric force, and regarded the function of the current as merely directive.

00Its exact function is not fully understood.

00The function of this pair of muscles is the closing of the valves.

00Two other pairs have been termed divaricators by Hancock, or cardinal muscles (" muscles diducteurs " of Gratiolet), and have for function the opening of the valves.

00The function of these muscles, according to the same authority, is not only that of erecting the shell; they serve also to attach the peduncle to the shell, and thus effect the steadying of it upon the peduncle.

00Those who have not seen the animal in life, or who did not believe in the possibility of the valves crossing each other with a slight obliquity, would not consent to appropriating any of its muscles to that purpose, and consequently attributed to all the lateral muscles the simple function of keeping the valves in an opposite position, or holding them adjusted.

00the right to present to a benefice in a newly appointed bishop's patronage at the option of the archbishop. By canon 40 of the canons of 1603 an oath against simony was to be administered to every person admitted to any spiritual or ecclesiastical function, dignity or benefice.

00Suppose n dependent variables yl, y2,ï¿½ï¿½ï¿½yn, each of which is a function of n independent variables x1, x2 i ï¿½ï¿½ï¿½xn, so that y s = f s (x i, x 2, ...x n).

00R is a function of the coefficients which is called the " resultant " or " eliminant " of the k equations, and the process by which it is obtained is termed " elimination."

00If al, a2, ...a, n be the roots of f=o, (1, R2, -Ai the roots of 0=o, the condition that some root of 0 =o may qq cause f to vanish is clearly R s, 5 =f (01)f (N2) ï¿½ ï¿½;f (Nn) = 0; so that Rf,q5 is the resultant of f and and expressed as a function of the roots, it is of degree m in each root 13, and of degree n in each root a, and also a symmetric function alike of the roots a and of the roots 1 3; hence, expressed in terms of the coefficients, it is homogeneous and of degree n in the coefficients of f, and homogeneous and of degree m in the coefficients of 4..

00= 0, we find that, eliminating x, the resultant is a homogeneous function of y and z of degree mn; equating this to zero and solving for the ratio of y to z we obtain mn solutions; if values of y and z, given by any solution, be substituted in each of the two equations, they will possess a common factor which gives a value of x which, corn bined with the chosen values of y and z, yields a system of values which satisfies both equations.

00Hence in all there are mn such systems. If, therefore, we have a third equation, and we substitute each system of values in it successively and form the product of the mn expressions thus formed, we obtain a function which vanishes if any one system of values, common to the first two equations, also satisfies the third.

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

00n be permuted, is a rational integral symmetric function of the quantities.

00The function Zap 1 a 2 P2 ...an n being as above denoted by a partition of the weight, viz.

00Theorem.- The function symbolized by (n), viz.

00in terms of x 1, x2, x3,ï¿½ï¿½ The inverse question is the expression of any monomial symmetric function by means of the power functions (r) = sr. Theorem of Reciprocity.-If ï¿½1 P2 "3 01 Q 2 7 3 Al A 2 A3 X m1 X m2 X m3 ...

00The general monomial symmetric function is a P1 a P2 a P3.

00aPn 1 2 3 ' the summation being for all permutations of the indices which result in different terms. The function is written (plp2p3ï¿½4n) for brevity, and repetitions of numbers in the bracket are indicated by exponents, so that (p1p1p2) is written (p1p 2).

00The weight of the function is the sum of the numbers in the bracket, and the degree the highest of those numbers.

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!

00- " If a symmetric function be symboilized by (Aï¿½v...) and (X1X2X3..ï¿½), (ï¿½i/-12ï¿½3ï¿½ï¿½ï¿½), (v1v2v3...)...

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

00ï¿½ ï¿½ P1 v2 v3...) ï¿½ For, writing as before, Xm 'Xm 2 Xm '= zzo(SQls:2s73...) xi'x12x13..., 1 2 3" 1231 2 3 = EPxi l x A2 x A3, P is a linear function of separations of(/ 1 / 2 A2 / 4 3 3 ...) of specification (m"`1mï¿½2m"`3...), and if X; 1 X 3 2X8 3 ' ..

00function of separations of (li'12 2 13 3 ...) of specification (si 1 s 22 s 33) Suppose the separations of (11 1 13 2 1 3 3 ...) to involve k different specifications and form the k identities ï¿½1s ï¿½ s Al A 2 A3 ..

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.

00" The symmetric function (m ï¿½8 m' 2s m ï¿½3s ...) 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."

00- If we form the separation function (J2) j1!j2!13!...

00appertaining to the function (li'l32l3...), each separation having a specification m" ` ' 8 m ï¿½2s m ï¿½38 multiply b P (is 2s 3s .ï¿½ï¿½), P Y by ls!

00and take therein the coefficient of the function (mi t tm7t t m 31 t ...), we obtain the same result as if we formed the separation function in regard to the specification (mï¿½ It t'tm2 32tm"`l3t...), multiplied by Alt!!

00and took ï¿½1a ï¿½ therein the coefficient of the function (mis mï¿½ 2s Es m 3s 3s ...).

00so that f (al, a 3, a3,.ï¿½.an) =f, a rational integral function of the elementary functions, is converted into f(a1 +12, a2+ p a1,...

00The introduction of the quantity p converts the symmetric function 1 2 3 into (XiX2X3+...) -Hu Al (X 2 A 3 .-) +/l02(X1X3.ï¿½.) +/103(A1X2.ï¿½.) +....

00p operators D upon a monomial symmetric function is clear.

00in terms of dl, d 2, d 3, ..., products denoting successive operations, by the same law which expresses the ele mentary function a s in terms of the sums of powers s l, s 2, s3,...

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

00Application to Symmetric Function Multiplication.-An example will explain this.

00It is thus possible to study simultaneously all the theories which depend upon operations of the group. Symbolic Representation of Symmetric Functions.-Denote the s 8 s elementar symmetric function a s by al a 2 a3 ...at pleasure; then, Y y si,, si,...

00Any function of differences of S a, S b, S c, ...

00being subsequently put equal to a, a non-unitary symmetric function will be produced.

001+Eaix+Esiy+ /al a2x 2 +Malt2xy -Z01023,2+ï¿½ï¿½ï¿½ The most general symmetric function to be considered is E 41 041 8424-3033..ï¿½ .conveniently written in the symbolic form (pigi p2g2 p3go...)ï¿½ Observe that the summation is in regard to the expressions obtained by permuting then suffixes I, 2, 3, ...n.

00The weight of the function is bipartite and consists of the two numbers Ep and Eq; the symbolic expression of the symmetric function is a partition into biparts (multiparts) of the bipartite (multipartite) number Ep, Eq.

00The number of partitions of a biweight pq into exactly i biparts is given (after Euler) by the coefficient of a, z xPy Q in the expansion of the generating function 1 - ax.

00It will be ï¿½ shown later that every rational integral symmetric function is similarly expressible.

00The Function hpg.

00- If, in the identity 1 (1 +anx = 1+aiox+aoly+a20x 2 +allxy+a02y 2 +..., we multiply each side by (I -ï¿½-P.x+vy), the right-hand side becomes 1 +(aio+1.1 ') x +(a ol+ v) y +...+(a p4+/ 1a P-1,4+ va Pr4-1) xPyq - - ...; hence any rational integral function of the coefficients an, say f (al °, aol, ...) =f exp(ï¿½dlo+vdol)f d a P-1,4, dot = dapg The rule over exp will serve to denote that i udio+ vdo h is to be raised to the various powers symbolically as in Taylor's theorem.

001 exp(Adlo + vdol) = (1+/oD10+ v Doi +..ï¿½+ VQ +.ï¿½.)f; now, since the introduction of the new quantities 1.1., v results in the addition to the function (plglp2g2p3g3...) of the new terms A PI Pg1 (p 2q2 p 3g3ï¿½ï¿½ï¿½) +/ AP2Pg2 (p 1 g 1P343 ...)+/ Z3vg3 (p l g i p 2 g 2 ...)+ ï¿½, we find DP141(plqip2q2p3q3ï¿½ï¿½ï¿½) = (p 2 q 2 p 3 q 3ï¿½ï¿½ï¿½), and thence D P141 D P242 D P343 ï¿½ï¿½.

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.

00F(a ' a ' a, ...a) =r A F(ao, a1, a2,ï¿½ï¿½ï¿½an), 0 1 2 n the function F(ao, al, a2,...an) is then said to be an invariant of the quantic gud linear transformation.

00;51, 2) = r F(ao, al, a2,...; xi, x2), the function F(a 01 a 1, a 2, ...

00From these formulae we derive two important relations, dp4 = or the function F, on the right which multiplies r, is said to be a simultaneous invariant or covariant of the system of quantics.

00We write;L 22 = a 1 a 2 .b 1 n-2 b2s 3 n - 3 3 n-3 3 n-3 3 a 3 = a 1 a 2 .b 1 b 2 .c 1 c2, and so on whenever we require to represent a product of real coefficients symbolically; we then have a one-to-one correspondence between the products of real coefficients and their symbolic forms. If we have a function of degree s in the coefficients, we may select any s sets of umbrae for use, and having made a selection we may when only one quantic is under consideration at any time permute the sets of umbrae in any manner without altering the real significance of the symbolism.

00It should be noticed that the real function denoted by (ab) 2 is not the square of a real function denoted by (ab).

00For a single quantic of the first order (ab) is the symbol of a function of the coefficients which vanishes identically; thus (ab) =a1b2-a2bl= aw l -a1ao=0 and, indeed, from a remark made above we see that (ab) remains unchanged by interchange of a and b; but (ab), = -(ba), and these two facts necessitate (ab) = o.

00If the forms be all linear and different, the function is an invariant, viz.

00For these only will the symbolic product be replaceable by a linear function of products of real coefficients.

00The first and fourth of these indicate that (a 2) w is a homogeneous function of X i, X2, and of /u1, ï¿½ 2 separately, and the second and third arise from the fact that (X / 1) is caused to vanish by both Da ï¿½ and Dï¿½A.

00Similarly regarding 1 x 2 as additional parameters, we see that every covariant is expressible as a rational function of n fixed covariants.

00and that thence every symbolic product is equal to a rational function of covariants in the form of a fraction whose denominator is a power of f x.

00To exhibit any covariant as a function of uo, ul, a n = (aiy1+a2y2) n and transform it by the substitution fi y 1+f2 y where f l = aay 1, f2 = a2ay -1, x y - x y = X x thence f .

00y1 = x 15+f2n; fï¿½ y2 =x2-f?n, f .a b = ax+ (a f) n, l; n u 2 " 2 22 2 +` n) u3 n-3n3+...+U 2jnï¿½ 3 n Now a covariant of ax =f is obtained from the similar covariant of ab by writing therein x i, x 2, for yl, y2, and, since y?, Y2 have been linearly transformed to and n, it is merely necessary to form the covariants in respect of the form (u1E+u2n) n, and then division, by the proper power of f, gives the covariant in question as a function of f, u0 = I, u2, u3,...un.

00To prove that this system is complete we have to consider (f, o) 2, 04') 1, (f,Q) 1, (f,Q) 2, (f,Q) 3, 0,Q) 1, (o,Q)2, and each of these can be shown either to be zero or to be a rational integral function of f, 0 Q and R.

002 - 9 m 2 (1 3 m 2)) 2 we have a cubic equation for determining m 2 as a function of the absolute invariant.

00The discriminant is the resultant of ax and ax and of degree 8 in the coefficients; since it is a rational and integral function of the fundamental invariants it is expressible as a linear function of A 2 and B; it is independent of C, and is therefore unaltered when C vanishes; we may therefore take f in the canonical form 6R 4 f = BS5+5BS4p-4A2p5.

00Two of these show that the leading coefficient of any covariant is an isobaric and homogeneous function of the coefficients of the form; the remaining two may be regarded as operators which cause the vanishing of the covariant.

00Every symmetric function denoted by partitions, not involving the figure unity (say a non-unitary symmetric function), which remains unchanged by any increase of n, is also a seminvariant, and we may take if we please another fundamental system, viz.

00Observe that, if we subject any symmetric function the diminishing process, it becomes ao 1 - P2 (p2p3...)ï¿½ Next consider the solutions of 0=o o which are of degree 0 and weight w.

00In order to obtain the seminvari ants we would write down the (w; 0, n) terms each associated with a literal coefficient; if we now operate with 52 we obtain a linear function of (w - I; 8, n) products, for the vanishing of which the literal coefficients must satisfy (w-I; 0, n) linear equations; hence (w; 8, n)-(w-I; 0, n) of these coefficients may be assumed arbitrarily, and the number of linearly independent solutions of 52=o, of the given degree and weight, is precisely (w; 8, n) - (w - I; 0, n).

00- zn +9 1 -z2.1 -z3....1-z8; and since this expression is unaltered by the interchange of n and B we prove Hermite's Law of Reciprocity, which states that the asyzygetic forms of degree 0 for the /t ie are equinumerous with those of degree n for the The degree of the covariant in the variables is e=nO-2w; consequently we are only concerned with positive terms in the developments and (w, 0, n) - (w - r; 0, n) will be negative unless nO It is convenient to enumerate the seminvariants of degree 0 and order e=n0-2w by a generating function; so, in the first written generating function for seminvariants, write z2 for z and az n for a;.

00the complete function may be written ll A2(z) i 2A2 (z/ ' A 2 z 1az2 1.1-a2; and this is the reduced generating function which tells us, by its.

00Putting n equal to co, in a generating function obtained above, we find that the function, which enumerates the asyzvgetic seminvariants of degree 0, is 1 1-z2.1-z3.1-z4....1-z0 that is to say, of the weight w, we have one form corresponding to each non-unitary partition of w into the parts 2, 3, 4,...0.

00it was noted that Stroh considers Method of Stroh.-In the section on " Symmetric Function," (alai +a 2 a 2 +...

00We may write function of such symbolic + (1+ai)(1+alt)...(16e)=1+A2t 2 +A3E 3 +...+A0 °.

00a irl aï¿½ 2 a a3 ...Ev 1 02 2 ?3 3 ...; and, if we express Ea l v2 2 0-3 3 in terms of A2, A3 i ..., and arrange the whole as a linear function of products of A2, A3,..., each coefficient will be a seminvariant, and the aggregate of the coefficients will give us the complete asyzygetic system of the given degree and weight.

00The existence of such a relation, as 0-1+0-2+.,.+cr2=0, necessitates the vanishing of a certain function of the coefficients A2, A 3, ...A 9, and as a consequence one product of these coefficients can be eliminated from the expanding form and no seminvariant, which appears as a coefficient to such a product (which may be the whole or only a part of the complete product, with which the seminvariant is associated), will be capable of reduction.

00The generating function is I - z2' 52 For 0 =3, (alai +a2a2+a3a3) 10; the condition is clearly a1a2a3 = A3 = 0, and since every seminvariant, of proper degree 3, is associated, as coefficient, with a product containing A3, all such are perpetuants.

00The general form is (3'2 A and the generating function 3 3.

00The general form of perpetuant is (4 K 3 A 2"`) and the generating function 1-z2.1-z3.1-z4 In general when 0 is even and =20, the condition is a l a 2 ...U 24 II(v 1 +a 2)II(a l +a 2 +cr 3)...II(Q 1 +a 2 -}-...

00The generating function is thus z2e-1 - 1 (1 -z 2) (1 -z 3) (1 -z 4)...

00Taking The First Generating Function, And Writing Az P, Bz4, 2 For A, B And Z Respectively, We Obtain The Coefficient Of Aobe'Zpo 0' 2W That Is Of A E B E 'Z ï¿½, In 1 Z 2 1 Azp. 1 Azp 2....1 A2 P 2.1 Az P .

00Q 1 The Unreduced Generating Function Which Enumerates The Covariants Of Degrees 0, 0' In The Coefficients And Order E In The Variables.

00Their Number For Any Weight W Is The Number Of Ways Of Composing W 3 With The Parts I, 2, And Thus The Generating Function Is Verified.

00We will choose from the forms in such manner that the product of letters A is either a power of A i, or does not contain A i; this rule leaves us with A2B 1 B 2 and A 2 B,Bs; of these forms we will choose that one which in letters B is earliest in ascending dictionary order; this is A2B 1 B 21 and our earliest perpetuant is (22)a(21)b - (221)a(2)b, and thence the general form enumerated by the generating function Z7 is (1-z)(1 - z2)2 (2 A2+2) a (2ï¿½2 +1 1ï¿½1 +1) b - (2 A2+2 1)a (2 M2+1 1, ai)b ...

00By the rules adopted we take A?B 2 B 3, which gives (12)a(32)b - (1)a(321)b+ao(3212)b, the simplest perpetuant of weight 7; and thence the general form enumerated by the generating function 1 -z.1-z2.1 - z3 ?

00(2A2+ 41ï¿½i +2)a(3ï¿½3+12ï¿½2+ 1) b, due to the generating function 2 15 (1 -z)(1 -z 2) 2 (1 -z3) For the case 0=r, 0'=4., the condition is a 1 r 1 T 2 r 3 7 4 (a i + T 1)(a + T 2)(a 1 T (a i + T 4) II (T s T t) 0; the calculation gives Selecting the product A;B 4 B 3 B2, we find the simplest perpetuant A1B4(A 1 B2+A1B3+B4)(-B3-A1B2B3-ATB4) =0.

00(14) a (4322) b - (13) a (432 2 1) b + (12)a (432212) b - (1) a (432213)b +ao(432214)b, and thence the general form (1 A i + 4) a (4ï¿½4 + 1.3 ï¿½ 3 + 121 1, 2 + 2) b - ... ?ao(4ï¿½4+13ï¿½a+l2ï¿½2+21 Al+4)b, due to the generating function 2 15 1 -z.

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: - (ï¿½b) 2 2)2 = a b - a b; (xï¿½ = 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.

00Every other concomitant is a rational integral function of these four forms. The linear covariant, obviously the Jacobian of a x and x x is the line perpendicular to x and the vanishing of the quadrinvariant a x is the condition that a x passes through one of the circular points at infinity.

00In general any pencil of lines, connected with the line a x by descriptive or metrical properties, has for its equation a rational integral function of the four forms equated to zero.

00The perfection of the form of a thing is its entelechy (ivr€AMXECa) in virtue of which it attains its fullest realization of function (De anima, Í.

00Moreover, it is not constant, being an apparently arbitrary function of H or of B; in the same specimen its value may, under different conditions, vary from less than 2 to upwards of 5000.

00If a coil of insulated wire is suspended so that it is in stable equilibrium when its plane is parallel to the direction of a magnetic field, the transmission of a known electric current through the coil will cause it to be deflected through an angle which is a function of the field intensity.

00Ann., 1880, 11, 399) that in weak fields the relation of the magnetization I to the magnetizing force H is approximately expressed by an equation of the form I =aH +bH2, or K=I/H =a+bH, whence it appears that within the limits of Baur's experiments the magnetization curve is a parabola, and the susceptibility curve an inclined straight line, x being therefore a known function of H.

00Soc., 1903, 71, 30), experimenting with wires of iron, steel and nickel, showed that in weak fields the change of resistance was proportional to a function a12-+b14-{-cl', where a, b and c are constants for each specimen.

00But they have lost their respiratory function.

00Their present function in scorpions is not ascertained.

00The coxal glands do not establish any special connexion between Limulus and Scorpio, since thay also occur in the same somite in the lower Crustacea, but it is to be noted that the coxal glands of Limulus are in minute structure and probably in function more like those of Arachnids than those of Crustacea.

00Appendages of 1st pair.tri-segmented, chelate; of 2nd pair chelate, with their basal segments subserving mastication; of 3rd, 4th, 5th and 6th pairs similar in form and function, except that in recent and Carboniferous forms the basal segments of the 3rd and 4th are provided with sterno-coxal (maxillary) lobes, those of the 4th pair meeting in the middle line and underlying the mouth.

00coxal (maxillary) process, the apical segment tipped with a single movable or immovable claw; appendages of 3rd pair different from the remainder, tactile in function, with at least the apical segment many-jointed and clawless.

00Remaining pairs of appendages similar in form and function, each tipped with two or three claws.

00Appendages of the 3rd, 4th, 5th and 6th pairs similar in form and function, tipped with two claws, their basal segments in contact in the median ventral line.

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.

00Hitherto, in order to prove the equality of two expressions for the same function, it was necessary to transform the one into the other, i.e.

00The subject he had chosen for his Habilitationsschrift was the "Representation of a Function by Means of a Trigonometrical Series," a subject which Dirichlet had made his own by a now well-known series of researches.

00Most of the Scottish kings were crowned at Scone, the last function being held on the 1st of January 1651, when Charles II.

00It has frequently been confused with determinism, which, however, differs from it categorically in assigning a certain function to the will.

00These researches derive additional importance from having introduced two powerful engines of analysis for the treatment of physical problems, Laplace's coefficients and the potential function.

00By his discovery that the attracting force in any direction of a mass upon a particle could be obtained by the direct process of differentiating a single function, Laplace laid the foundations of the mathematical sciences of heat, electricity and magnetism.

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.

00The latter is therefore called the generating function of the former.

00A direct and an inverse calculus is thus created, the object of the former being to determine the coefficients from the generating function, of the latter to discover the generating function from the coefficients.

00The first formal proof of Lagrange's theorem for the development in a series of an implicit function was furnished by Laplace, who gave to it an extended generality.

00Algebraic Forms; Binomial; Combinatorial Analysis; Determin Ants; Equation; Continued Fraction; Function; Theory of groups; Logarithm; Number; Probability; Series.

00the theory of the function and the theory of limits.

00Later, the difficulty recurs in an acute form in reference to the continuous variation of a function.

00It is important to begin the study of graphics with concrete cases rather than with tracing values of an algebraic function.

00In the first class come equations in a single unknown; here the function which is equated to zero is the Y whose values for different values of X are traced, and the solution of the equation is the determination of the points where the ordinates of the graph are zero.

00do not involve x, and the indices of the powers of x are all positive integers, is called a rational integral function of x of degree n.

00(iv.) Generally, let N be any rational integral function of n of degree r.

00Then, since nr rl is also a rational integral function of n of degree r, we can find a coefficient c r, not containing n, and such as to make N-c r nr ri contain no power of n higher than n r - 1.

00(iv.) Thus the problems of determining the roots of an equation P = o and of finding the factors of P, when P is a rational integral function of x, are the same.

00From this point of view, the function which, by algebraical operations on i+o.x+o.x2+..., produces the series, is called its generating function.

00In the same way, the generating function of the series I+2x+x 2 +0.x 3 +o.x 4 +...

00Thus we arrive at the differential coefficient of f(x) as the limit of the ratio of f (x+8) - f (x) to 0 when 0 is made indefinitely small; and this gives an interpretation of nx n-1 as the derived function of xn (ï¿½ 45)ï¿½ This conception of a limit enables us to deal with algebraical expressions which assume such forms as -° o for particular values of the variable (ï¿½ 39 (iii.)).

00Other cases arise when we consider the continuity of a function.

00In accordance with the Declaration of Corfu, the decision regarding the actual form of the State was left to a constituent assembly: but as the machinery of Belgrade was naturally quite inadequate to the task of administering a country three times the size of the Serbia of 1914, the provincial Governments of Croatia (with the Ban at its head), Slovenia, Dalmatia and Bosnia continued to function, though the local diets were no longer summoned.

00Indeed, the historical and present importance of this aspect or branch of zoological science is so great that the name " zoology " has until recently been associated entirely with it, to the exclusion of the study of minute anatomical structure and function which have been distinguished as anatomy and physiology.

00We have seen that the problem before us is independent of the law of the secondary wave as regards obliquity; but the result of the integration necessarily involves the law of the intensity and phase of a secondary wave as a function of r, the distance from the origin.

00The second and third factors of (3) being each of the form sin 2u/u2, we have to examine the character of this function.

003), representing the value of the function sin 2 u/u 2 from u=o to u=271.

00The (8), A function of the telescope is in fact to allow the use of a wider, and therefore more easily measurable, aperture.

00In the case of the circular aperture the distribution of light is of course symmetrical with respect to the focal point p=o, q=o; and C is a function of p and q only through 11 (p 2 -}-q 2).

00This integral is the Bessel's function of order unity, defined by J,(z) n (z cos 0) sin 24 d4)..

00The function of a lens in forming an image is to compensate by its variable thickness the differences of phase which would otherwise exist between secondary waves arriving at the focal point from various parts of the aperture.

00If we put for shortness 7 for the quantity under the last circular function in (I), the expressions (i), (2) may be put under the forms u sin T, v sin (T - a) respectively; and, if I be the intensity, I will be measured by the sum of the squares of the coefficients of sin T and cos T in the expression u sin T +v sin (T - a), so that I =u 2 +v 2 +2uv cos a, which becomes on putting for u, v, and a their values, and putting f =Q .

00If we write p = 27rR/A (6), we must regard p as a function of f, and we may take with sufficient approximation under any ordinary circumstances where p' denotes the value of p at 0, and is a constant, which is positive when the retarding plate is held at the side on which the blue of the spectrum is seen.

00In the former case the function of the telescope is simply to increase the dispersion, and the formation of the bands is of course independent of the particular manner in which the dispersion arises.

00In the limiting case in which the medium is regarded as absolutely incompressible S vanishes; but, in order that equations (2) may preserve their generality, we must suppose a at the same time to become infinite, and replace a 2 3 by a new function of the co-ordinates.

00The third equation gives 2 d dt2 = b dx (3), of which the solution is = f (bt (4), where f is an arbitrary function.

00The name " South African Republic " was adopted as the title of the state, and the new constitution made provision for a volksraad to which members were to be elected by the people for a period of two years, and in which the legislative function was vested.

00Naturally, as the temperature became lower, a new function was gradually acquired by his clothing, that of protecting the body of the wearer.

00Such objects might be imitated in other materials and by successive copying lose their identity, or their first meaning might be otherwise forgotten, and they would ultimately exercise a purely decorative function.

00No writer in any literature, who has contented himself with so limited a function, has gained so great a reputation as Terence.

00The localization of function in the cerebral and in the cerebellar cortex has doubtless been the main cause of this progress, and has proceeded poi passu with an extended insight into the structure and connexions of the parts concerned.

00Health is simply that condition of structure and function which, on examination of a sufficient number of examples, we find to be commonest.

00Disease we may define, accordingly, as any departure from the normal standard of structure or function of a tissue or organ.

00If, for instance, we find that instead of the natural number of Malpighian bodies in the kidney there are only half that number, then we are entitled to say that this defect represents disease of structure; and if we find that the organ is excreting a new substance, such as albumen, we can affirm logically that its function is abnormal.

00Once grant the above definition of disease, and even the most trivial aberrations from the normal must be regarded as diseased conditions, quite irrespective of whether, when structural, they interfere with the function of the part or not.

00It is much more likely, as Verworn alleges, that the nerves which influence the characteristic function of any tissue regulate thereby the metabolism of the cells in question - in other words, that every nerve serves as a trophic nerve for the tissues it supplies.

00Atrophy may follow primary arrest of function - disuse atrophy.

00Increased work thrown on to a tissue may produce hypertrophy, but, if this excessive function be kept up, atrophy will follow; even the blacksmith's arm breaks down owing to the hypertrophic muscle fibres becoming markedly atrophied.

00According to our present knowledge of physiological and pathological processes, we must regard the cell as the ultimate biological unit - a unit of structure and a unit of function; this was first put forward by Schleiden in 1838, and by Schwann in 1839, but we owe to Virchow the full recognition of the fundamental importance of the living cell in all the processes of life, whether in health or disease.

00On reaching the vicinity they leave the blood stream and join in the warfare - many performing their function of phagocytosis (q.v.), others falling victims to the toxins.

00Their principal function is to bring about the removal of foreign, dead or degenerating material.

00The vessels are only temporary channels by which is brought forward the food supply that is needed by the advancing army if it is suc-, cessfully to carry on its function; they probably also drain off the deleterious fluid substances formed by the cellular disintegration that has taken place in the part.

00Although we have not reached a stage of certainty regarding their origin, function and destiny, recent investigations have brought forward evidence to elucidate the importance of the part played by the different cells in the various types of the inflammatory process.

00Their function being at an end they give way to these cells which carry on the process of absorption.

00Oertel finds an explanation of this want of complete celldifferentiation, loss of function, and acquired vegetative activity in the non-homogeneous character of the nuclear chromatin elements of the cell, and maintains that the different properties of the cell are carried and handed down by the different orders of chromatin loops.

00But their disintegration is more commonly brought about by " phagocytosis " on the part of the phagocytic cells in the different organs concerned with the function of haemolysis, i.e.

00Heidenhain, on the other hand, rejected entirely the filtration view of lymph-formation, believing that the passage of lymph across the capillary wall is a true secretion brought about by the secretory function of the endothelial plates.

00It is clear that the knowledge of function (physiology) did not by any means keep pace with the knowledge of structure, and this was probably the reason why the important sect of the empirics were able entirely to dispense with anatomical knowledge.

00Discovery in these various directions then led physicians to regard fever and inflammation not as separable entities, but as fluctuating symptomgroups, due to swervings of function from the normal balance under contingent forces.

00Thus the digestive function, in its largest sense, is now seen to consist, not only in preparation and supply, but in no small measure also of protective and antidotal conversions of the matters submitted to it; coincidently with agents of digestion proper are found in the circuit of normal digestion "anti-substances" which neutralize or convert peptones in their poisonous phases; an autochthonous ferment, such as rennet for instance, calling forth an anti-rennet, and so on.

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.

00This law has important ethical and political bearings; but in the province of disease this advance of what may be compared to the interlocking of points and signals has had wide influence not only in altering our conceptions of disease, but also in enlarging our views of all perturbations of function.

00It is needless to point out how such paths of disease, in their association with characteristic symptoms, have illuminated the clinical features of disease as well as the processes of normal function.

00Diseases of the latter kind are especially interesting, as in them we see that parts of the nervous structure, separated in space, may nevertheless be associated in function; for instance, wasting of a group of muscles associated in function may depend on a set of central degenerations concurring in parts whose connexion, in spite of dissociation in space, we thus perceive.

00Many other diseases formerly regarded as primarily diseases of the nervous system are not such; but, by means of agents either introduced into the body or modified there, establish themselves after the affinities of these in contiguous associated parts of the structure, as in vascular, membranous or connective elements, or again in distant and peripheral parts; the perturbations of nervous function being secondary and consequential.

00Moreover, the intricacies of structure and function within the heart itself have been more fully discriminated (W.

00His "grandeur in social function" was unequalled and his interests were very wide.

00The function of the academic department is to control the teaching branch, internal examinations, &c., and that of the external department to control external examinations, while the university extension system occupies a third department.

00Function of receive antenna is extraction energy from electromagnetic field.

00But the Liberalism of his early years was gone for ever, and he had become reconciled to Metternich's view that, in an age of decay, the sole function of a statesman was to "prop up mouldering institutions."

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.

00This varies in metals from 594 (lithium) to 22.48 (osmium), and in one and the same species is a function of temperature and of previous physical and mechanical treatment.

00Ignoring temperature effect, and taking the density as a function of the pressure, surfaces of equal pressure are also of equal density, and the fluid is stratified by surfaces orthogonal to the lines of force; n ap, dy, P d z, or X, Y, Z (4) are the partial differential coefficients of some function P, =fdplp, of x, y, z; so that X, Y, Z must be the partial differential coefficients of a potential -V, such that the force in any direction is the downward gradient of V; and then dP dV (5) ax + Tr=0, or P+V =constant, in which P may be called the hydrostatic head and V the head of potential.

00If w vanishes throughout the fluid at any instant, equation (I I) shows that it will always be zero, and the fluid motion is then called irrotational; and a function 4) exists, called the velocity function, such that udx+vdy-{-wdz =-d, (13) and then the velocity in any direction is the space-decrease or downward gradient of cp.

00- In the uniplanar motion of a homogeneous liquid the equation of continuity reduces to du dv dx' dy-O' u= -d,y/dy, v = d i t/dx, (2) surface containing so that we can put _ (6) (9) we have (I) (2) (5) (I) where 4 is a function of x, y, called the streamor current-function; interpreted physically, 4-4c, the difference of the value of 4, at a fixed point A and a variable point P is the flow, in ft.

00dy I +v2" dy = 0' d4' Y' =o, and 2 must be a function of 4'.

00If other vortices are present, any one may be supposed to move with the velocity due to the others, the resultant stream function being = gy m log r =log IIrm; (9) the path of a vortex is obtained by equating the value of 1P at the vortex to a constant, omitting the rm of the vortex itself.

00Uniplanar Motion of a Liquid due to the Passage of a Cylinder through it.-A stream-function 4, must be determined to satisfy the conditions v24 =o, throughout the liquid; (I) I =constant, over any fixed boundary; (2) d,t/ds = normal velocity reversed over a solid boundary, (3) so that, if the solid is moving with velocity U in the direction Ox, d4y1ds=-Udy/ds, or 0 +Uy =constant over the moving cylinder; and 4,+Uy=41' is the stream function of the relative motion of the liquid past the cylinder, and similarly 4,-Vx for the component velocity V along Oy; and generally 1,1'= +Uy -Vx (4) is the relative stream-function, constant over a solid boundary moving with components U and V of velocity.

00In the general case 4'=11.+Uy-Vx+2R(x 2 +y 2) is the relative stream function for velocity components, U, V, R.

00208.) If 01 denotes the velocity function of the liquid filling the cylinder r = b, and moving bodily with it with velocity Ul, 41 = -U1x, (12) and over the separating surface r =b 4, = I U (+- a2) a2 +b2 (13) 1 Ul b2 - a 2 ...

00Motion symmetrical about an Axis.-When the motion of a liquid is the same for any plane passing through Ox, and lies in the plane, a function ' can be found analogous to that employed in plane motion, such that the flux across the surface generated by the revolution of any curve AP from A to P is the same, and represented by 2s-4 -11'o); and, as before, if d is the increase in due to a displacement of P to P', then k the component of velocity normal to the surface swept out by PP' is such that 274=2.7ryk.PP'; and taking PP' parallel to Oy and Ox, u= -d/ydy, v=dl,t'/ydx, (I) and 1P is called after the inventor, " Stokes's stream or current function," as it is constant along a stream line (Trans.

00Soc., 1842; " Stokes's Current Function," R.

001 = yv24, (2) y 2 y y y suppose; and in steady motion, + y 2 dx v-t ' = o, dH +y 2dy0 2P = o, so that 2 "/ y = - y2, 7 2 1,G = dH/d is a function of 1,G, say f'(> '), and constant along a stream line; dH/dv = 2qi', H -f (1/.) = constant, throughout the liquid.

00Similarly, with the function (19) (2n+ I) 3 ch (2n+ I) ITrb/a' (2) Changing to polar coordinates, x =r cos 0, y = r sin 0, the equation (2) becomes, with cos 0 =µ, r'd + (I -µ 2)-d µ = 2 ?-r3 sin 0, (8) of which a solution, when = o, is = (Ar'+) _(Ari_1+) y2,, ?

00(II) Thus cos 0 is the Stokes' function of a point source at 0, and Papb of a line source AB.

00For instance, with n = I in equation (9), the relative stream function is obtained for a sphere of radius a, by making it, y' =1y+2Uy 2 = 2U(r 2 -a 3 /r) sin?

00Irrotational Motion in General.-Liquid originally at rest in a singly-connected space cannot be set in motion by a field of force due to a single-valued potential function; any motion set up in the liquid must be due to a movement of the boundary, and the motion will be irrotational; for any small spherical element of the liquid may be considered a smooth solid sphere for a moment, and the normal pressure of the surrounding liquid cannot impart to it any rotation.

00The kinetic energy of the liquid inside a surface S due to the velocity function 4' f i (s given by T=2p + (d) 2+ (t) dxdydz, pff f 75 4 dS (I) by Green's transformation, dv denoting an elementary step along the normal to the exterior of the surface; so that d4ldv = o over the surface makes T = o, and then (d4 2 d4) 2 'x) + (dy) + (= O, dd?

00In a multiply connected space, like a ring, with a multiply valued velocity function ¢, the liquid can circulate in the circuits independently of any motion of the surface; thus, for example, 4) =mB=m tan - l y/x (5) will give motion to the liquid, circulating in any ring-shaped figure of revolution round Oz.

00Again, since d4)/dv =d /ds, d4)/ds= - d4y/dv, (13) T = 1 p f(1 9 d = - 2 p f4' d (14) With the Stokes' function, y for motion symmetrical about an axis.

00With a velocity function 49, the flow -f d 4 = 4)142, (2) (9) (to) (6) (22) Z Uy (I -a4,ic /r4), so that the flow is independent of the curve for all curves mutually reconcilable; and the circulation round a closed curve is zero, if the curve can be reduced to a point without leaving a region for which 4 is single valued.

00= -dQ+1dg2, and integrating round a closed curve (udx+vdy+wdz) =0, and the circulation in any circuit composed of the same fluid particles is constant; and if the motion is differential irrotational and due to a velocity function, the circulation is zero round all reconcilable paths.

002, so that is an elliptic function of t, except when c =a, or 3a.

00Put S2 1 =12 cos 4, 12 2 = -12 sin 4, d4 d52 1 dS22 Y a2+c2 122 7Ti = 71 22 CL2- c2(121+5221)J, a2 +c2 do a2+c2 + 4c2 z dt a'-c2 (a2+,c2)2 M+2c2(a2-c2 N-{-a2+c2 2 Ý_a 2 +c 2 (' 4c2 .?"d za 2 -c 2 c2)2 2'J Z M+ -c2) which, as Z is a quadratic function of i 2, are non-elliptic so also for; G, where =co cos, G, 7 7 = - sin 4.

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.

00The terms of 0 may be determined one at a time, and this problem is purely kinematical; thus to determine 4)1, the component U alone is taken to exist, and then 1, m, n, denoting the direction cosines of the normal of the surface drawn into the exterior liquid, the function 01 must be determined to satisfy the conditions v 2 0 1 = o, throughout the liquid; (ii.) ' = -1, the gradient of 0 down the normal at the surface of the moving solid; 1 =0, over a fixed boundary, or at infinity; similarly for 02 and 03.

00A system of confocal ellipsoids is taken y2 (3) a 2 +X b 2 +X c2 + A= I, and a velocity function of the form = x1 P, (4) where 4' is a function of X only, so that 4) is constant over an ellipsoid; and we seek to determine the motion set up, and the form of >G which will satisfy the equation of continuity.

00When the liquid is bounded externally by the fixed ellipsoid A = A I, a slight extension will give the velocity function 4 of the liquid in the interspace as the ellipsoid A=o is passing with velocity U through the confocal position; 4 must now take the formx(1'+N), and will satisfy the conditions in the shape CM abcdX ¢ = Ux - Ux a b x 2+X)P Bo+CoB I - C 1 (A 1 abcdX, I a1b1cl - J o (a2+ A)P and any'confocal ellipsoid defined by A, internal or external to A=A 1, may be supposed to swim with the liquid for an instant, without distortion or rotation, with velocity along Ox BA+CA-B 1 -C1 W'.

00Taking Ox along OS, the Stokes' function at P for the source S is p cos PSx, and of the source H and line sink OH is p(a/f) cos PHx and - (p/a) (PO - PH); so that = p (cos PSx+f cos PHx PO a PH), (q) and Ili = -p, a constant, over the surface of the sphere, so that there is no flow across.

00When S and S' lie on the same radius, taken along Ox, the Stokes' function can; be written down; and when S and S' coalesce a doublet is produced, with a doublet image at H.

00For a doublet at S, of moment m, the Stokes' function is M f cos PSx = - m p s3; and for its image at H the Stokes' function is m f cos PHx =m f 3 PH" (6) so that for the comnation _ a3 I I 2 4)-myb12 (f 3 PH PS 3) =m f 3 (pa ll 3 P53)' 3 and this vanishes over the surface of the sphere.

00There is no Stokes' function when the axis of the doublet at S does not pass through 0; the image system will consist of an inclined doublet at H, making an equal angle with OS as the doublet S, and of a parallel negative line doublet, extending from H to 0, of moment varying as the distance from O.

00A distribution of sources and doublets over a moving surface will enable an expression to be obtained for the velocity function of a body moving in the presence of a fixed sphere, or inside it.

00The method of electrical images will enable the stream function, )' to be inferred from a distribution of doublets, finite in number when the surface is composed of two spheres intersecting at an angle 7r/m, where m is an integer (R.

00= constant, _ ff 00 NdA N BA-AA X - JA (a' +X) (b 2 +A)P - abc' a2 -b2 ' and at the surface A = o, I I N Bo-A 0 N I R - (a2+b2) abc a 2 -b 2 abc a2b2 I /b 2 N = R I /b2 - I /a2 abc I 1 I Bo - AO' a 2 b 2 - a2 b2 a 2 b2 = R (a 2 - b 2) /(a 22 + /b2) 2 - r (B o - Ao) U Bo+Co - B I - CI' Since - Ux is the velocity function for the liquid W' filling the ellipsoid A = o, and moving bodily with it, the effective inertia of the liquid in the interspace is Ao+B1+C1 Bo+Co - B1 - C, If the ellipsoid is of revolution, with b=c, - 2 XBo - - C BI' and the Stokes' current function 4, can be written down (I) is (5) (7) (6) The velocity function of the liquid inside the ellipsoid A=o due to the same angular velocity will be = Rxy (a2 - b2)/(a2 + b2), (7) and on the surface outside _ N Bo -Ao c1)0xy abc 2 62' so that the ratio of the exterior and interior value of at the surface is ?o= Bo-Ao (9) 4)1 (a 2 -6 2)/(a2 + b) - (Bo - Ao)' and this is the ratio of the effective angular inertia of the liquid, outside and inside the ellipsoid X = o.

00A torsion of the ellipsoidal surface will give rise to a velocity function of the form 4)--- where SZ can be expressed by the elliptic integrals in a similar manner, since dX/P3.

00But supposing them determined for the motion of a body through a liquid, the kinetic energy T of the system, liquid and body, is expressible as a quadratic function of the components U, V, W, P, Q, R.

00Conversely, if the kinetic energy T is expressed as a quadratic function of x, x x3, y1, y2, y3, the components of momentum, the partial differential coefficient with respect to a momentum component will give the component of velocity to correspond.

00Thus if T is expressed as a quadratic function of U, V, W, P, Q, R, the components of momentum corresponding are dT dT dT (I) = dU + x2=dV, x3 =dW, dT dT dT Yi dp' dQ' y3=dR; but when it is expressed as a quadratic function of xi, 'x2, x3, yi, Y2, Y3, U = d, V= dx, ' w= ax dT Q_ dT dT dy 1 dy2 dy The second system of expression was chosen by Clebsch and adopted by Halphen in his Fonctions elliptiques; and thence the dynamical equations follow X = dt x2 dy +x3 d Y = ..., Z ..., (3) = dt1 -y2?y - '2dx3+x3 ' M =..

00Liouville, Caspary, Jukovsky, Liapounoff, Kolosoff and others, chiefly Russian mathematicians; and the general solution requires the double-theta hyperelliptic function.

00In the motion which can be solved by the elliptic function, the most general expression of the kinetic energy was shown by A.

00Clebsch to take the form T= 2p(x12 +x22)+2p'x32 + q (xiyi +x2y2) +q'x3y3 +2r(y12+y22)+2r'y32 so that a fourth integral is given by dy 3 /dt = o, y = constant; dx3 (4 y) (q + y) _ (y y) dt - xl 'x2 xl Y Y x l 2 - 1, y2 () = (x12 +x22) (y12 + y22) = (X 1 2 + X 2) +y22)-(FG-x3y3)2 = (x 1 y32-G2)-(Gx3-Fy3) 2, in which 2 = F 2 -x3 2, x l y l +x2y2 = FG-x3y3, Y(y1 2 +y2 2) = T -p(x12 +x22) -p'x32 -2q(xiyi 'x2y2)- 2 q ' x = (p -p') x 2 + 2 (- q ') x 3 y 3+ m 1, (6) m1 = T 2 i y 3 2 (7) so that dt3) 2 =X3, (8) where X3 is a quartic function of x3, and thus t is given by an elliptic (8) (6) (I) integral of the first kind; and by inversion x 3 is in elliptic function of the time t.

00sin o= F dl, (20) C3 do F2 h _ F2 cos 2 o F 2 sin z o F dt y - V C G c +2 c1 coso+H]; (21) 1 z so that cos 0 and y is an elliptic function of the time.

00From the scriptural doctrine of the essentially spiritual nature of the kingdom of Christ, Glas in his public teaching drew the conclusions: (1) that there is no warrant in the New Testament for a national church; (2) that the magistrate as such has no function in the church; (3) that national covenants are without scriptural grounds; (4) that the true Reformation cannot be carried out by political and secular weapons but by the word and spirit of Christ only.

00In 1739 the General Assembly, without any application from him, removed the sentence of deposition which had been passed against him, and restored him to the character and function of a minister of the gospel of Christ, but not that of a minister of the Established Church of Scotland, declaring that he was not eligible for a charge until he should have renounced principles inconsistent with the constitution of the church.

00His function as a teacher is proved by the fact that the Apostle Paul boasts of having sat at the feet of Gamaliel (Acts.

00Its chief function was to regulate the trade monopoly conveyed to the borough by the royal grant of gilda mercatoria.

00In short, the function of guarding and supervising the trade monopoly split up into various fragments, the aggregate of the crafts superseding the old general gild merchant.

00Officers, commonly called wardens in England, were elected by the members, and their chief function was to supervise the quality of the wares produced, so as to secure good and honest workmanship. Therefore, ordinances were made regulating the hours of labour and the terms of admission to the gild, including apprenticeship. Other ordinances required members to make periodical payments to a common fund, and to participate in certain common religious observances, festivities and pageants.

00To this end he examined such immediate vital products as blood, bile and urine; he analysed the juices of flesh, establishing the composition of creatin and investigating its decomposition products, creatinin and sarcosin; he classified the various articles of food in accordance with the special function performed by each in the animal economy, and expounded the philosophy of cooking; and in opposition to many of the medical opinions of his time taught that the heat of the body is the result of the processes of combustion and oxidation performed within the organism.

00Of the carbon dioxide and ammonia no exhaustion can take place, but of the mineral constitutents the supply is limited because the soil cannot afford an indefinite amount of them; hence the chief care of the farmer, and the function of manures, is to restore to the soil those minerals which each crop is found, by the analysis of its ashes, to take up in its growth.

00By the grant of an immunity to a proprietor the royal officers, the count and his representatives, were forbidden to enter his lands to exercise any public function there.

00The most important public function whose transformation into a private possession was assisted by the growth of the immunity was the judicial.

00This acted at once and without any consciousness of difference of function, as judiciary, as legislature, in so far as there was any in the feudal period, and as council, and it exercised final supervision and control over revenue and administration.

00Almost all the institutions of modern states go back to the curia regis, branching off from it at different dates as the growing complexity of business forced differentiation of function and personnel.

00When the function of protection and local supervision could be resumed by the general government the feudal age ended.

00proper function was to manage the tribes and collect tribute.

00The vapour tension may approximate to a linear function of the composition, and the curve will then be practically a straight line.

00It can hardly be doubted that the function of these avicularia is the protection of the tentacles and compensation-sac. The suggestion that they are concerned in feeding does not rest on any definite evidence, and is probably erroneous.

00The whole apparatus is so exactly analogous in structure to the poison-gland and tooth of a venomous snake as to suggest a similar function, and there is now evidence that it employs this organ as an offensive weapon.

00Although this Great Being evidently exceeds the utmost strength of any, even of any collective, human force, its necessary constitution and its peculiar function endow it with the truest sympathy towards all its servants.

00The sub-aquatic habits of the present species probably render such a function impossible, hence the absence of the glands.

00His main function made him in his early life a preacher even more emphatically than a teacher.

00To his function as a preacher we owe some of his most characteristic and stimulating works, especially the discourses by which it may be said he won his way to wide and influential recognition - Endeavours after the Christian Life, 1st series, 1843; 2nd series, 1847; Hours of Thought, 1st series, 1876; 2nd series, 1879; the various hymn-books he issued at Dublin in 1831, at Liverpool in 1840, in London in 1873; and the Home Prayers in 1891.

00Many suffered under this law, but the ultimate effect was to invest the press with new popularity, and very soon the newspapers conceived a device which effectually protected their literary staff, for they employed dummy editors whose sole function was to go to prison in lieu of the true editor.

00Evidently the idea of the great Yokoya experts, the originators of the style, was to break away from the somewhat formal monotony of ordinary engraving, where each line performs exactly the same function, and to convert the chisel into an artists i It is first boiled in a lye obtained by lixiviating wood ashes; it is next polished with charcoal powder; then immersed in plum vinegar and salt; then washed with weak lye and placed in a, tub of water to remove all traces of alkali, the final step being to digest in a boiling solution of copper sulphate, verdigris and water.

00It is apparent that a vitrified enamel may be made to perform, in part at any rate, the function of a porcelain glaze.

00Spencer welcomed the Darwinian theory, and enriched it with the phrase" survival of the fittest "; but he did not give up the (Lamarckian) belief in the hereditary transmission of the modifications of organisms by the exercise of function.

00