Function Sentence Examples
But that, too, is a function of present technology.
In the sense he can function but the memory is probably gone for good.
Exportable technology can function around the world.
She didn't understand how humans could feel so much and still function when she was overwhelmed.
Tests revealed normal brain function, and there was no indication of any permanent damage.Advertisement
Her mind was beginning to function somewhere near normal.
And yes, everything and everyone in my life has a function and purpose.
The first is " the law that thought is a function of the brain."
It was then that my slowly reacting brain, flowing like cold molasses began to function, more than a gerbil driven wheel.
In many sees there were no vidames, their function being exercised by viscounts or chatelains.Advertisement
To this high conception of a preacher's function the prophet was faithful throughout his career.
Their function is not quite clear.
Depending on function, robots can come in all shapes and sizes, and I see no compelling reason to make them like humans.
Based 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.
He was performing the same function, but meeting him didn't darken Dean's mood appreciably.Advertisement
Unable to function from shock, he sat down on a rock overlooking the lake.
Free and peaceful societies function best when government is transparent and open.
Early 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.
The 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.
The polyps are all non-sexual individuals whose function is purely nutritive.Advertisement
Simspon concluded that for a given wind velocity dissipation is practically a linear function of ionization.
If 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.
Its function is to examine criminal cases and to decide whether they shall be referred for trial to the lower courts or the cours dassises.
Accepting 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.
As it is with mechanical improvements, so is it to a still greater degree with changes in the function of timbre in art.Advertisement
One 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.
Bach'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.
An instance of this characteristic function, specially remarkable because the composer has taken exceptional measures for it, is Beethoven's overture to Fidelio.
When Ethel did the inviting, the function was nearly always out of town.
Laplace 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.
Finally 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.
Perhaps the earliest of such specializations is connected with the reproductive function.
Their chief function is to ordain, and to act as "intercessors."
Mention 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.
The principal function of these courts is the hearing of appeals both civil and criminal from the courts of first instance; only in.
All 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.
The 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.
Besides 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.
As 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.
The 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.
The 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.
Another enzyme, rennet, which in the animal body is proteolytic, is frequently met with in plants, but its function has not been ascertained.
Every plant is constrained to carry Out its functions of germination, growth, nutrition, reproduction, &c., between certain limits of temperature, and somewhere between the extremes of these limits each function finds ao optimum temperature at which the working of the living machinery is at its best, and, other things being equal, any great departure from this may induce pathological conditions; and many disasters are due to the failure to provide such suitable temperaturese.g.
The special function of the so-called "Socratic dialectic" was to show the inadequacy of popular beliefs.
As your husband, what will he consider his most important function?
Nevertheless, the function of reproduction is occasionally exercised by larvae.
Chladni'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.
The function of this pair of muscles is the closing of the valves.
This latter function, however, is one unknown to the English common law.
Besides 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.
Their 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.
The 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.
The 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.
It seems likely that the coelomic pore-canals were originally excretory organs, but in the existing Enteropneusta the pore-canals (especially the collar canals) have, as we have seen, acquired new functions or become vestigial, and the function of excretion is now mainly accomplished by a structure peculiar to the Enteropneusta called the glomerulus, a vascular complex placed on either side of the anterior portion of the stomochord, projecting into the proboscis-coelom.
Interchanges 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.
This result again indicates that the parts of the molecules are effectively separate from each other, the function of the electric forces being merely directive.
Clausius 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.
Its exact function is not fully understood.
Two other pairs have been termed divaricators by Hancock, or cardinal muscles (" muscles diducteurs " of Gratiolet), and have for function the opening of the valves.
The 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.
Those 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.
R 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."
Hence 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.
The function Zap 1 a 2 P2 ...an n being as above denoted by a partition of the weight, viz.
Theorem.- The function symbolized by (n), viz.
The general monomial symmetric function is a P1 a P2 a P3.
The weight of the function is the sum of the numbers in the bracket, and the degree the highest of those numbers.
The sum of the monomial functions of a given weight is called the homogeneous-product-sum or complete symmetric function of that weight; it is denoted by h.; it is connected with the elementary functions by the formula 1 7r1l7r2!7r3!
Application to Symmetric Function Multiplication.-An example will explain this.
It 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,...
In the theory of forms we seek functions of the coefficients and variables of the original quantic which, save as to a power of the modulus of transformation, are equal to the like functions of the coefficients and variables of the transformed quantic. We may have such a function which does not involve the variables, viz.
From 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.
We 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.
Evidently 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.
It is apparent that a vitrified enamel may be made to perform, in part at any rate, the function of a porcelain glaze.
Spencer 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.
On the other hand, he advances too easily from the maxim that function is prior to, and makes, structure to the conclusion that the results of use and disuse are therefore immediately incarnated in structural adaptations capable of hereditary transmission.
The result was the formation of a cabinet belonging, in Fox's own words, partly to the king and partly to the country - that is to say, partly of Whigs who wished to restrain the king, and partly of the king's friends, represented by Lord Shelburne, whose real function was to baffle the Whigs.
The Word of God alone is there to do it."Nevertheless Luther assigned to the state, which he assumes to be Christian, the function of maintaining the Gospel and the Word of God in public life.
One or more reserve teeth, in various stages of development, lie between the folds of the gum and are ready to take the place of the one in function whenever it is lost by accident, or shed.
In the last ten years of his life Horace resumed his lyrical function for a time, under pressure of the imperial command, and produced some of the most exquisite and mature products of his art.
This capacity is then a function of the geometrical dimensions of the conductor, and can be mathematically determined in certain cases.
The mathematical importance of this function called the potential is that it is a scalar quantity, and the potential at any point due to any number of point charges ql, q2, q3, &c., distributed in any manner, is the sum of them separately, or qi/xl+q2/x2+q3/x3+&c. =F (q/x) =V (17), where xi, x2, x 3, &c., are the distances of the respective point charges from the point in question at which the total potential is required.
This is mathematically expressed by the statement that dE is an exact differential of a function of the co-ordinates defining the state of the body, which can be integrated between limits without reference to the relation representing the path along which the variations are taken.
He therefore employed the corresponding expression for a cycle of infinitesimal range dt at the temperature t in which the work dW obtainable from a quantity of heat H would be represented by the equation dW =HF'(t)dt, where F'(t) is the derived function of F(t), or dF(t)/dt, and represents the work obtainable per unit of heat per degree fall of temperature at a temperature t.
It simply asserts that the efficiency function F'(t), which is known as Carnot's function, is the same for all substances at the same temperature.
Holtzmann (1845) by the same assumption deduced the value J/T for the function F'(t), but obtained erroneous results by combining this assumption with the caloric theory.
Kelvin had previously proposed to define an absolute scale of temperature independent of the properties of any particular substance in terms of Carnot's function by making F'(t) constant.
He now proposed to define absolute temperature as proportional to the reciprocal of Carnot's function, so as to agree as closely as possible with the scale of the gas thermometer.
Since the two expressions (9) are the partial differential-coefficients of a single function E of the independent variables v and 0, we shall obtain the same result, namely d 2 E/d0dv, if we differentiate the first with respect to v and the second with respect to 0.
A similar expression for the variation of the specific heat S at constant pressure is obtained from the second expression in (8), by taking p and 0 as independent variables; but it follows more directly from a consideration of the variation of the function (E+pv).
Observing that F is a function of the co-ordinates expressing the state of the substance, we obtain for the variation of S with pressure at constant temperature, dS/dp (0 const) '=' 2 F/dedp =-0d 2 v/d0 2 (p const) (12) If the heat supplied to a substance which is expanding reversibly and doing external work, pdv, is equal to the external work done, the intrinsic energy, E, remains constant.
The specific heats may be any function of the temperature consistently with the characteristic equation provided that their difference is constant.
If we assume that s is a linear function of 0, s= so(I +aO), the adiabatic equation takes the form, s 0 log e OW +aso(0 - Oo) +R loge(v/vo) =o
It is found by experiment that the change of pv with pressure at moderate pressures is nearly proportional to the change of p, in other words that the coefficient d(pv)/dp is to a first approximation a function of the temperature only.
But this procedure in itself is not sufficient, because, although it would be highly probable that a gas obeying Boyle's law at all temperatures was practically an ideal gas, it is evident that Boyle's law would be satisfied by any substance having the characteristic equation pv = f (0), where f (0) is any arbitrary function of 0, and that the scale of temperatures given by such a substance would not necessarily coincide with the absolute scale.
This gives by equation (9) the condition Odp/d0 =p, which is satisfied by any substance possessing the characteristic equation p/0=f(v), where f(v) is any arbitrary function of v.
The characteristic equation of the fluid must then be of the form v/0=f(p), where f(p) is any arbitrary function of p. If the fluid is a gas also obeying Boyle's law, pv = f (0), then it must be an ideal gas.
Putting d0/dp=A/0 2 in equation (15), and integrating on the assumption that the small variations of S could be neglected over the range of the experiment, they found a solution of the type, v/0 =f(p) - SA /30 3, in which f(p) is an arbitrary function of p. Assuming that the gas should approximate indefinitely to the ideal state pv = R0 at high temperatures, they put f(p)=Rip, which gives a characteristic equation of the form v= Re/p - SA /30 2 .
The advantage of this type of equation is that c is a function of the temperature only.
Other favourite types' of equation for approximate work are (I) p=RO/v±f(v), which makes p a linear function of 0 at constant volume, as in van der Waal's equation; (2) v=RO/p+f(p), which makes v a linear function of 0 at constant pressure.
At higher densities it is probable that more complex aggregates would be formed, so that as the effect of the collisions became more important c would cease to be a function of the temperature only; experiment, indeed, shows this to be the case.
The function J = E-94,, has been called the " free energy " of the substance by Helmholtz, and 90 the " bound energy."
This function is also called the " thermodynamic potential at constant volume " from the analogy with the condition of minimum potential energy as the criterion of stable equilibrium in statics.
If J', J" represent the values of the function for unit mass of the substance of specific volumes v' and v" in the two states at temperature 0 and pressure and if a mass m is in the state v', and 1-m in the.
The function G is represented by the negative area D"DM under the isothermal, bounded by the isopiestic DM and the axis of pressure.
The increment of 00 is always greater than that of the total heat F=E+pv, except in the special case of an equilibrium change at constant temperature and pressure, in which case both are equal to the heat absorbed in the change, and the function G remains constant.
This is geometrically obvious from the form of the area representing the function on the indicator diagram, and also follows directly from the first law.
If G' and G" are the values of the function G for the two states in equilibrium at the same pressure and temperature, we must have G' =G".
Assuming the function G to be expressed in terms of p and 0, this condition represents the relation between p and 0 corresponding to equilibrium between the two states, which is the solution of the relation (v" - v')dp/dO=L/D, (5).
To find the border curve of equilibrium between the two states, giving the saturation pressure as a function of the temperature, we have merely to equate the values of G and G".
This simple result is generally true, and the corresponding expressions for G" and J" are valid, provided that c - b in formula (17) is a function of the temperature only.
Finally, the function of the archbishop as judge in a court of appeal, though it still subsists, is of little practical importance now that the clergy, in civil matters, are universally subject to the secular courts.
The "powers" (numina, not dei), which thus become the objects of worship, are spirits specialized in function and limited in sphere.
Migration in general may be described as a natural function of social development.
The development of a true insect society among the Hymenoptera is dependent on a differentiation among the females between individuals with well-developed ovaries (" queens ") whose special function is reproduction; and individuals with reduced or aborted ovaries (" workers ") whose duty is to build the nest, to gather food and to tend and feed the larvae.
In the hive-bee and among ants, on the other hand, there are constant structural distinctions between queen and worker, and the function of the queen bee in a hive is confined to egg-laying, the labour, of the community being entirely done by the workers.
The hinder abdominal segments in the male show a curious asymmetrical arrangement, the sixth segment bearing on its upper side a small stalked plate (strigil) of unknown function, furnished with rows of teeth.
The function of such an agency is performed in the United States of America by the express companies (see Express).
Thus, for refined purposes, it cannot be assumed with any certainty that the instantaneous scale-value of the heliometer is known, or that it is a function of the temperature.
Its highest function is the contemplation of the divine unity, discoverable under the manifold of objects.
The former contrivances consist essentially of levers or cams with toothed surfaces or gripping shoes mounted upon transverse axes attached to the sides of the cage, whose function is to take hold of the guides and support the cage in the event of its becoming detached from the rope.
By inference only, increasing complication of stomach with ruminating function superadded.
The corresponding solid figure, in its most general form, is such as would be constructed to represent the relation of a magnitude E to two magnitudes F and G of which it is a function; it would stand on a plane base, and be comprised within a cylindrical boundary whose cross-section might be of any shape.
The area of the trapezette, measured from the lower bounding ordinate up to the ordinate corresponding to any value of x, is some function of x.
The statement that the ordinate u of a trapezette is a function of the abscissa x, or that u=f(x), must be distinguished from u =f(x) as the equation to the top of the trapezette.
The simplest case is that in which u is constant or is a linear function of x, i.e.
The next case is that in which u is a quadratic function of x, i.e.
Generally, if the area of a trapezette for which u is an algebraical function of x of degree 2n is given correctly by an expression which is a linear function of values of u representing ordinates placed symmetrically about the mid-ordinate of the trapezette (with or without this mid-ordinate), the same expression will give the area of a trapezette for which u is an algebraical function of x of degree 2n + 1.
In the case, therefore, of any solid whose cross-section at distance x from one end is a quadratic function of x, the position of the crosssection through the centroid is to be found by determining the position of the centre of gravity of particles of masses proportional to So, S2, and 4S 1, placed at the extremities and the middle of a line drawn from one end of the solid to the other.
If u is an algebraical function of x of degree not exceeding p, and if the area of a trapezette, for which the ordinate v is of degree not exceeding p+q, may be expressed by a formula Aovo-1--yivi+..
This ordinate will be an algebraical function of x, and we can again apply a suitable formula.
The area of the section by a plane at distance x from the edge 0 is a function of x whose degree is the same as that of u.
The volume of the briquette for which u is a function of x and y is found by the operation of double integration, consisting of two successive operations, one being with regard to x, and the other with regard to y; and these operations may (in the cases with which we are concerned) be performed in either order.
If, as is usually the case, the ordinate throughout each strip of the trapezette can be expressed approximately as an algebraical function of the abscissa, the application of the integral calculus gives the area of the figure.
These results may be extended to the calculation of an expression of the form fxo u4(x)dx, where 0(x) is a definite function of x, and the conditions with regard to u are the same as in § 82.
It is also clearly impossible to express u as an algebraical function of x and y if some value of du/dx or duldy is to be infinite.
Even where u is an explicit function of x, so that f x udx may be expressed in terms of x, it is often more convenient, for construction of a table of values of such an integral, to use finite-difference formulae.
Its function is to deliberate on subjects of common concern to the entire denomination, and to publish such opinions and counsels as a majority may see fit to send forth to the churches.
There is a tendency, moreover, to accord to the conferences the function of determining the tests of ministerial standing in the Congregational denomination.
In some of the states the licensing of preachers, which was formerly left to the voluntary associations of ministers in the different localities, has been made a function of the state conferences.
The function of the angels is that of the supernatural servants of God, His agents and representatives; the Angel of Yahweh, as we have seen, is a manifestation of God.
In Scripture the function of the angel overshadows his personality; the stress is on their ministry; they appear in order to perform specific acts.
In some of them the hind toe, which has already ceased to have any function in the lapwing, is wholly wanting.
C. Burkitt's The Gospel History (1906) vigorously sketches the book's dominant characteristics and true function.
He established the Federal Department of Commerce and Labor, the secretary of which has a seat in the cabinet, and in which there exists a bureau of corporations possessing the specific function of inspecting and supervising interstate corporations - an entirely new feature in American government.
Its colour varies with the liturgical colour of the day, or of the function at which it is worn.
This function is named the "power" of the two spheres, and it is important in the investigation of systems of spheres.
As to those of the first description, it is from the British charters that they derive a capacity by which they are considered as a public body, or at all capable of any public function....
In the modern metal bridge every member has a definite function and is subjected to a calculated straining action.
The function of the flanges is to resist a horizontal tension and compression distributed practically uniformly on their cross sections.
Her letter to the emperor, pervaded with he religious and almost mystic sentiments which predominate in the queen's mind, particularly since the death of Prince Albert, seems to have made a deep impression on the sovereign who, amid the struggles of politics, had never completely repudiated the philanthropic theories of his youth, and who, on the battlefield of Solferino, covered with the dead and wounded, was seized with an unspeakable horror of war."Moreover, Disraeli's two premierships (1868, 1874-80) did a good deal to give new encouragement to a right idea of the constitutional function of the crown.
The laws of the Day of Atonement belong to the Priestly Code.4 There is no trace of this function before the exile; the earliest reference to any such special time of atonement being the proposal of Ezek.
In later Judaism, especially from about ioo B.C., great stress was laid on the Day of Atonement, and it is now the most important religious function of the Jews.
Mexican acquaintance with the signs related only to their secondary function as dies (so to speak) with which to stamp recurring intervals of time.
Religious or semi-religious ceremonies, however, play a great part in the life of the Siamese, and few weeks pass without some great function or procession.
But on the modern theory, which includes the play of electrical phenomena as a function of the aether, there are other considerations which show that this number io 2 is really an enormous overestimate; and it is not impossible that the co-efficient of ultimate inertia of the aether is greater than the co-efficient of inertia (of different kind) of any existing material substance.
Returning now to the aether, on our present point of view no such complications there arise; it must be regarded as a continuous uniform medium free from any complexities of atomic aggregation, whose function is confined to the transmission of the various types of physical effect between the portions of matter.
Trains of waves nearly but not quite homogeneous as regards wave-length will as usual be propagated as wave-groups travelling with the slightly different velocity d(VX-1)/dX-', the value of K occurring in V being a function of X determined by the law of optical dispersion of the medium.
In his train also are found Enyo, the goddess of war who delights in bloodshed and the destruction of cities; his sister, Eris, goddess of fighting and strife; and the Keres, goddesses of death, whose function it is especially to roam the battle-field, carrying off the dead to Hades.
It is consistent with this circle of ideas that initiation into the profound mysteries of the liturgy was regarded, together with the preservation of dogma, as the most exalted function of theology.
On the other hand the seer, diviner and prophet is a minister whose function it is to communicate God's will or word to man.
There was also a third kind of priest called the zammaru, whose function it was to sing hymns.
There remains, indeed, one other sacred function of great importance in the ancient world in which the Greek priests had a share.
Thus for the function = 1 I -x.
Introducing another letter z, and considering the function I+xz.
And similarly, considering the function I +z(x+x2+..)...+Nxnzk-..
Although Bessel was the first to systematically treat of these functions, it is to be noted that in 1732 Daniel Bernoulli obtained the function of zero order as a solution to the problem of the oscillations of a chain suspended at one end.
Mehler, who proved that a simple relation existed between the function of zero order and the zonal harmonic of order n.
The mandibles are normally five-jointed, with remnants of an outer branch on the second joint, the biting edge varying from strong development to evanescence, the terminal joints or " palp " giving the organ a leg-like appearance and function, which disappears in suctorial genera such as Paracytherois.
The variable first maxillae are seldom pediform, their function being concerned chiefly with nutrition, sensation and respiration.
In 1892 he distinguished the former as those in which the first antennae of the male have both members modified for holding the female, and the genital openings of the female have a ventral position, sometimes in close proximity, sometimes strongly lateral; the latter as those in which the first antennae of the male are similar to those of the female, the function of holding her being transferred to the male maxillipeds, while the genital openings of the female are dorsal, though at times strongly lateral.
He has observed that in young specimens of Siren lacertina (the larva is still unknown) the gills are rudimentary and functionless, and that it is only in large adult specimens that they are fully developed in structure and function; he therefore concludes that the sirens are the descendants of a terrestrial type of batrachians, which passed through a metamorphosis like the other members of their class, but that more recently they have adopted a permanently aquatic life, and have resumed their branchiae by reversion.
It is in this way that the function of the seer is closely connected (as in the case of Balaam) with sacrifices.
The rise of this function of the prophets is plainly parallel with the change which took place under the kings in the position of the priestly oracle; the Torah of the priests now dealt rather with permanent sacred ordinances than with the giving of new divine counsel for special occasions.
But in point of fact the function of the new prophecy was not to preserve but to destroy Israel, if Israel still meant the actual Hebrew nation, with its traditional national life.
The current theological formula for this two-sided position is that the prophets are at once preachers of the law and forerunners of the gospel; and, as it is generally assumed that they found the law already written, their originality and real importance is made to lie wholly in their evangelical function.
So long as the great problems of religion could be envisaged as problems of the relation of Yahweh to Israel as a nation the prophets continued to speak and to bring forth new truths; but the ultimate result was that it became apparent that the idea of moral government involved the destruction of Israel, and then the function of prophecy was gone because it was essentiall y national in its objects.
It had therefore to find out a new world and a new spiritual function, in order first to establish the existence of what it desiderated, and then to realize and describe what it had proved to exist.
Money, however, is in men's actual transactions the measure of value, as well as the vehicle of exchange; and the precious metals are best suited for this function, as varying little in their own value for periods of moderate length; for distant times, corn is a better standard of comparison.
The mathematical function log x or log x is one of the small group of transcendental functions, consisting only of the circular functions (direct and inverse) sin x, cos x, &c., arc sin x or sin-' x,&c., log x and e x which are universally treated in analysis as known functions.
The notation log x is generally employed in English and American works, but on the continent of Europe writers usually denote the function by lx or lg x.
The logarithmic function is most naturally introduced into analysis by the equation log x= x ?
Thus log x is the integral function of 1/x, and it can be shown that log x is a genuinely new transcendent, not expressible in finite terms by means of functions such as algebraical or circular functions.
A relation which is of historical interest connects the logarithmic function with the quadrature of the hyperbola, for, by considering the equation of the hyperbola in the form xy=const., it is evident that the area included between the arc of a hyperbola, its nearest asymptote, and two ordinates drawn parallel to the other asymptote from points on the first asymptote distant a and b from their point of intersection, is proportional to log bla.
The function log x as x increases from o towards w steadily increases from - co towards +co.
The exponential function possesses the properties (i.) exp (x+y) =exp x X exp y.
It is customary, therefore, to denote the exponential function by e x, and the result ex = I +x+x2/2 !
It follows from this property of the function that we cannot have for log x a series which shall be convergent for all values of x, as is the case with sin x and cos x, for such a series could only represent a uniform function, and in fact the equation log(I +x) =x -",, x2 +3x 3 -4x 4 + is true only when the analytical modulus of x is less than unity.
The exponential function, which may still be defined as the inverse of the logarithmic function, is, on the other hand, a uniform function of x, and its fundamental properties may be stated in the same form as for real values of x.
An alternative method of developing the theory of the exponential function is to start from the definition exp x = I +x+x2/2 !
The progress of navigation and the association of divinities of the sky with maritime affairs probably also assisted to bring about the change, although the memory of her earlier function as a goddess of childbirth survived till imperial times.
The ordinary pleasures of festivals were mingled with all this, such as dances in beast-masks, sham fights and children's games, but the type of a religious function was a sickening butchery followed by a cannibal feast.
Only what falls under the first and second of the divisions so indicated can be said to have discharged a function in developing philosophy; only so much constitutes Schelling's philosophy proper.
The function of Naturphilosophie is to exhibit the ideal as springing from the real, not to deduce the real from the ideal.
The absolute was left with no other function than that of removing all the differences on which thought turns.
Each of these limbs was twobranched, the external branch consisting of a slender fringed flagellum possibly respiratory in function, and the inner of a normal jointed ambulatory leg.
It is probable that the first of these forms is the primary one and the second in most cases a development from it due to (i.) the influence of other individual cults, (ii.) anthropomorphic tendencies, (iii.) the influence of chieftainship, hereditary and otherwise, (iv.) annual sacrifice of the sacred animal and mystical ideas connected therewith, (v.) syncretism, due either to unity of function or to a philosophic unification, (vi.) the desire to do honour to the species in the person of one of its members, and possibly other less easily traceable causes.
Its main function was that of passing the lex curiata which was necessary for the ratification both of the imperium of the higher magistracies of the people, and of the potestas of those of lower rank.
After the peace of Villafranca he was sent to organize the army of the Central Italian League (composed of the provisional governments of Tuscany, Modena, Parma and Romagna), and converted it in a few months into a well-drilled body of 45, 000 men, whose function was to be ready to intervene in the papal states on the outbreak of a revolution.
The Senates act as courts for the trial of state officers impeached by the house (in imitation of the British House of Lords and the Federal Senate), and have in some states Powers and the function of confirming or refusing appointments Funcons made by the governor.
He has also the almost mechanical function of representing the state for various formal purposes, such as demanding from other states the extradition of offenders, the issuing of writs for the election of members of the legislature and of members of the Federal House of Representatives, and the receiving of reports from various state officials or boards.
The vice-president of the United States is ex officio presiding officer of the Senate, and this is his only active function in the government.
The judicial function of the Senate is to sit as a high court for the trial of persons impeached by the House of Representatives, a vote of two-thirds of those present being needed for conviction.
This latter function is discharged by written messages addressed by the president to Congress, the message sent at the beginning of each session being usually the most important; but the suggestions made in these messages do not necessarily or directly induce legislation, although it is open to him to submit a bill or have one drafted by a minister presented to Congress through a member.
Nevertheless, the distribution of offices under the so-called spoils system remains the most important ordinary function of the president, and the influence he exerts over Congress and legislation is due mainly to his patronage.
In ordinary times the president may be almost compared to the managing clerk in a large business establishment, whose chief function is to select his subordinates, the policy of the concern being in the hands of the board of directors.
There is, of course, a committee for every state, and at the head of the whole stands .a national committee for the whole Union, whose special function it is to make arrangements for the conduct of party work at a presidential election.
The other and parallel branch of the party organization consists of the bodies whose function it is to nominate party candidates for elective posts, whether legislative or Pa,aty Noni.
The paired ctenidia are very greatly developed right and left of the elongated body, and form the most prominent organ of the group. Their function is chiefly not respiratory but nutritive, since it is by the currents produced by their ciliated surface that food-particles are brought to the feebly-developed mouth and buccal cavity.
Septibranchia Gills have lost their respiratory function, and are transformed into a muscular septum on each side between mantle and foot.
On the whole, then, the most probable conclusion is that the original ancestral form of the Mollusca was unsegmented, possessed one pair of true nephridia, and one pair of coelomic ducts whose function was to conduct the generative products to the exterior.
Near the base of the ctenidium is a patch of sensory epithelium innervated from the branchial nerve, forming a sense-organ called the osphradium, whose function is to test the water entering the branchial cavity.
Two days later he showed that he intended to exercise the right of the President to address Parliament direct - a right which had fallen into desuetude - by sending a message to the Chambers, in which he stated that it was his function as President "to be a guide and adviser for public opinion in times of crisis" and "to seek to make a rational choice between conflicting interests."
He does, however, lay much stress upon the naturally social character of man; and this points forward to that treatment of morality as a function of the social organism which characterizes modern ethical theory.
They were parish ministers and subject like their brethren to church courts; their added function was to plant churches, and place ministers, elders and deacons where required.
Elders are rulers; their function also is spiritual, though practical and disciplinary.
These investigations led him to the announcement of the fundamental law of action between elements of current, or currents in infinitely short lengths of linear conductors, upon one another at a distance; summed up in compact expression this law states that the action is proportional to the product of the current strengths of the two elements, and the lengths of the two elements, and inversely proportional to the square of the distance between the two elements, and also directly proportional to a function of the angles which the line joining the elements makes with the directions of the two elements respectively.
He discovered a function, which has been called the potential of one circuit on another, from which he deduced a theory of induction, completely in accordance with experiment.
After he had educated himself by the study of the phenomena of lines of magnetic force in his discoveries on electromagnetic induction, he applied the same conception to electrostatic phenomena, and thus created the notion of lines of electrostatic force and of the important function of the dielectric or non-conductor in sustaining them.
In the Isopoda the respiratory function has been taken over by the abdominal appendages, both rami or only the inner becoming thin or flattened.
Although the excretory function of these has been demonstrated by physiological methods, however, their morphological relations are not clear.
When feathered or provided with secondary barbs the setae will respond to movements or vibrations in the surrounding water, and have been supposed to have an auditory function.
Recent observations, however, make it very doubtful whether aquatic Crustacea can hear at all, in the proper sense of the term, and it has been shown that one function, at least, of the so-called otocysts is connected with the equilibration of the body.
In certain Copepoda and Ostracoda glands of the same type produce a phosphorescent substance, and others, in certain Amphipoda and Branchiura, are believed to have a poisonous function.
It should be noticed that the real function denoted by (ab) 2 is not the square of a real function denoted by (ab).
For 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.
If the forms be all linear and different, the function is an invariant, viz.
For these only will the symbolic product be replaceable by a linear function of products of real coefficients.
Similarly regarding 1 x 2 as additional parameters, we see that every covariant is expressible as a rational function of n fixed covariants.
To 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 .
The 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.
Two 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.
Every 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.
In 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).
Putting n equal to co, in a generating function obtained above, we find that the function, which enumerates the asyzvgetic seminvariants of degree 0, is 1 1-z2.1-z3.1-z4....1-z0 that is to say, of the weight w, we have one form corresponding to each non-unitary partition of w into the parts 2, 3, 4,...0.
The 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.
The 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.
The generating function is thus z2e-1 - 1 (1 -z 2) (1 -z 3) (1 -z 4)...
Q 1 The Unreduced Generating Function Which Enumerates The Covariants Of Degrees 0, 0' In The Coefficients And Order E In The Variables.
Their 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.
By 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 ?
Every 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.
In 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.
Moreover, 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.
If 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.
But they have lost their respiratory function.
Their present function in scorpions is not ascertained.
The 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.
Appendages 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.
Remaining pairs of appendages similar in form and function, each tipped with two or three claws.
Appendages 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.
Hitherto, 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.
The 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.
Most of the Scottish kings were crowned at Scone, the last function being held on the 1st of January 1651, when Charles II.
It has frequently been confused with determinism, which, however, differs from it categorically in assigning a certain function to the will.
These 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.
By 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.
Laplace published in 1779 the method of generating functions, the foundation of his theory of probabilities, and the first part of his Theorie analytique is devoted to the exposition of its principles, which in their simplest form consist in treating the successive values of any function as the coefficients in the expansion of another function with reference to a different variable.
The latter is therefore called the generating function of the former.
A 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.
The 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.
It is important to begin the study of graphics with concrete cases rather than with tracing values of an algebraic function.
In 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.
Then, 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.
From this point of view, the function which, by algebraical operations on i+o.x+o.x2+..., produces the series, is called its generating function.
In the same way, the generating function of the series I+2x+x 2 +0.x 3 +o.x 4 +...
Other cases arise when we consider the continuity of a function.
Indeed, 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.
We 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.
The second and third factors of (3) being each of the form sin 2u/u2, we have to examine the character of this function.
The (8), A function of the telescope is in fact to allow the use of a wider, and therefore more easily measurable, aperture.
The 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.
If 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 .
If 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.
In 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.
In 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.
The 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.
Naturally, as the temperature became lower, a new function was gradually acquired by his clothing, that of protecting the body of the wearer.
Such 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.
No writer in any literature, who has contented himself with so limited a function, has gained so great a reputation as Terence.
The 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.
Health is simply that condition of structure and function which, on examination of a sufficient number of examples, we find to be commonest.
Disease we may define, accordingly, as any departure from the normal standard of structure or function of a tissue or organ.
If, 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.
Once 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.
It 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.
Atrophy may follow primary arrest of function - disuse atrophy.
Increased 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.
According 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.
On 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.
Their principal function is to bring about the removal of foreign, dead or degenerating material.
The 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.
Although 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.
Their function being at an end they give way to these cells which carry on the process of absorption.
Oertel 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.
But 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.
Heidenhain, 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.
It 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.
Discovery 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.
Thus 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.
The discoveries of the separate paths of sensory and motor impulses in the spinal cord, and consequently of the laws of reflex action, by Charles Bell and Marshall Hall respectively, in their illumination of the phenomena of nervous function, may be compared with the discovery in the region of the vascular system of the circulation of the blood; for therein a key to large classes of normal and aberrant functions and a fertile principle of interpretation were obtained.
This 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.
It 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.
Diseases 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.
Many 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.
His "grandeur in social function" was unequalled and his interests were very wide.
The 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.
Function of receive antenna is extraction energy from electromagnetic field.
But 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."
His pages abound in symbols representing unknown functions, the form of the function being left to be ascertained by observation of facts, which he does not regard as a part of his task, or only some known properties of the undetermined function being used as bases for deduction.
This 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.
Ignoring 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.
If 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.
Uniplanar 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.
Motion 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.
Similarly, 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,, ?
For 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?
Irrotational 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.
When 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'.
Taking 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.
A 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.
The 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.
Conversely, 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.
Thus 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 =..
Liouville, Caspary, Jukovsky, Liapounoff, Kolosoff and others, chiefly Russian mathematicians; and the general solution requires the double-theta hyperelliptic function.
In 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.
Its chief function was to regulate the trade monopoly conveyed to the borough by the royal grant of gilda mercatoria.