equilibrium

She squinted through her fingers and braced herself against one wall to counter the effects the drugs had on her equilibrium as she moved down the long hallway.

7064Equilibrium is the product of an axiomatic system.

4934When the two chemicals are mixed they have a short reaction time before they reach equilibrium.

3721Meanwhile the podest still subsists; but he is no longer equal to the task d maintaining an equilibrium of forces.

3535When the strong magnetizing field is gradually diminished to zero and then reversed, the needles pass from one stable position of rest to another through a condition of instability; and if the field is once more reversed, so that the cycle is completed, the needles again pass through a condition of instability before a position of stable equilibrium is regained.

2725Gregory initiated the policy of establish ing an equilibrium between the parties, which was carried ou by his successor Nicholas III.

2021According to the hypothesis of Waldeyer and Thiersch there is perfect equilibrium between the normal epithelium and its supporting structure, the connective tissue, but with advancing age this balance is upset owing to the connective tissue gradually losing its restraining power.

1920The cell, together with this balancing electromotive force, is thus a reversible system in true equilibrium, and the thermodynamical reasoning applicable to such systems can be used to examine its properties.

1818From a hilltop you can see a fish leap in almost any part; for not a pickerel or shiner picks an insect from this smooth surface but it manifestly disturbs the equilibrium of the whole lake.

1818Conservatives, acceptance of the railway redemption contracts, consolidation of the financial equilibrium, abolition 01 the !QrCe~

1616On the 23rd of March 1872, however, he succeeded in carrying his programme, which not only provided for the pressing needs of the moment, but laid the foundation of the much-needed equilibrium between expenditure and revenue.

1518- What is commonly understood by thermochemistry is based entirely on the first law of thermodynamics, but of recent years great progress has been made in the study of chemical equilibrium by the application of the second law.

1417was suggested by 0111vier; the paragraph asserting that France would not allow a foreign power to disturb to her own detriment the actual equilibrium of Europe was inserted by the emperor.

1215Of especial interest is the 0 curve BD; along this line liquid and rhombic sulphur are in equilibrium, which means that at above 131° and 400 atmospheres the rhombic (and not the monoclinic) variety would separate from liquid sulphur.

1113It is evident that the undissociated part of each acid must eventually be in equilibrium with the free hydrogen ions, and, if the concentrations are not such as to secure this condition, readjustment must occur.

1113Still, the necessary freedom was supposed to be secured by interchanges which the solvent can pass but the solution cannot, the solvent will enter till a certain equilibrium pressure is reached.

1114From B the curve of equilibrium (BD) between rhombic and liquid sulphur proceeds; and from C (along CE) the curve of equilibrium between liquid sulphur and sulphur vapour.

1012When the Crispi cabinet fell in March 1896 Sonnino had the satisfaction of seeing revenue increased by ~3, 400,000, expenditure diminished by 2,800,000, the gold premium reduced from 16 to 5%, consolidated stock at 95 instead of 72, and, notwithstanding the expenditure necessitated by the Abyssinian War, financial equilibrium practically restored.

912Political geography takes account of the partition of the earth amongst organized communities, dealing with the relation of races to regions, and of nations to countries, and considering the conditions of territorial equilibrium and instability.

912Conceived in the Hildebrandine spirit, his reforms led by a natural sequence to strained relations between Church and State; the equilibrium which he established was unstable, and depended too much upon his personal influence with the Conqueror.

913The above may be illustrated by considering the equilibrium between rhombic and monoclinic sulphur.

913The electric forces set up tend to prevent further separation, and finally a state of equilibrium is reached, when no 1 Zeits.

811It can be shown 3 that in a uniform field an elongated piece of any non-crystalline material is in stable equilibrium only when its length is parallel to the lines of force; for diamagnetic substances, however, the directing couple is exceedingly small, and it would hardly be possible to obtain a uniform field of sufficient strength to show the effect experimentally.

811The dynamical equilibrium between rhombic, liquid and monosymmetric sulphur has been worked out by H.

812In almost all climes the tortoise and the frog are among the precursors and heralds of this season, and birds fly with song and glancing plumage, and plants spring and bloom, and winds blow, to correct this slight oscillation of the poles and preserve the equilibrium of nature.

819When no current is passing through the coil and the magnetic field is of zero strength, the needles arrange themselves in positions of stable equilibrium under their mutual forces, pointing in.

79His marvellous physical and moral equilibrium gave him an evenness of temper which always renaered his society charming.

712When there is equilibrium, rp = yq 2 .

712Since the current passing through the balance when equilibrium is obtained with a given weight is proportional to the square root of the couple due to this weight, it follows that the current strength when equilibrium is obtained is proportional to the product of the square root of the weight used and the square root of the displacement distance of this weight from its zero position.

69In order that the solutions of these should be isohydric and the concentrations of the hydrogen ions the same, we must have a very large quantity of the feebly dissociated acetic acid, and a very small quantity of the strongly dissociated hydrochloric, and in such proportions alone will equilibrium be possible.

69In a fresh solution a-glucose only exists, but on standing it is slowly transformed into -y-glucose, equilibrium being reached when the a and y forms are present in the ratio o 368:0.632 (Tanret, Zeit.

69From 1876, when equilibrium between expenditure and revenue had first been attained, taxation yielded steady annual surpluses, which in 1881 reached the satisfactory level of 2,120,000.

57What form it would ultimately take depended still on the balance between the forces of conservatism and change, the suspicious temper of the autocracy being revealed, during the years of unstable equilibrium, by the alternate concession and withdrawal of privileges, e.g.

44Excepting the in creases of deficit in 1868 and 1870, the annual deficits tended thence forward to decrease, until in 1875 equilibrium between expendituri and revenue was attained, and was maintained until 1881.

47Further, the ocean and the atmosphere stand in equilibrium with each other; if there is excess of carbonic acid anywhere in the sea it is absorbed by the atmosphere and vice versa, and so also with the oxygen.

47Differences of temperature and atmospheric pressure must disturb this equilibrium, but the movements of both ocean and atmosphere lead to a high degree of uniformity in both envelopes as regards their gaseous constitutions.

47Almost equally important was the twenty years' truce of Zsitvatoriik, negotiated by Bocskay between the emperor and the sultan, which established for the first time a working equilibrium between the three parts of Hungary, with a distinct political preponderance in favour of Transylvania.

47- On the theory that crystal form and structure are the result of the equilibrium between the atoms and molecules composing the crystals, it is probable, a priori, that the same substance may possess different equilibrium configurations of sufficient stability, under favourable conditions, to form different crystal structures.

48Lay the compass upon the cardboard, and observe the rate at which its needle vibrates after being displaced from its position of equilibrium; this will vary greatly in different regions.

48The earth, or other planet, does not actually move round the sun; yet it is carried round the sun in the subtle matter of the great vortex, where it lies in equilibrium, - carried like the passenger in a boat, who may cross the sea and yet not rise from his berth.

36His next and most important publication was his famous paper "On the Equilibrium of Heterogeneous Substances" (in two parts, 1876 and 1878), which, it has been said, founded a new department of chemical science that is becoming comparable in importance to that created by Lavoisier.

36When a current is passed through the instrument it causes one end of the movable system to tilt downwards, and the other end upwards; the sliding weight is then moved along the tray by means of a silk cord until equilibrium is again established.

36Thomsen deduces the actual values of X, Y, Z to be 14.71, 13.27 and zero; the last value he considers to be in agreement with the labile equilibrium of acetylenic compounds.

36The line BC, representing the equilibrium between monoclinic and liquid sulphur, is thermodynamically calculable; the point B is found to correspond to 131° and 400 atmospheres.

36Similar investigations applied to the general case of chemical equilibrium lead to an expression of exactly the same form as that given by C. M.

36If 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.

36In his eighteenth year, while still a student in Edinburgh, he contributed two valuable papers to the Transactions of the same society - one of which, " On the Equilibrium of Elastic Solids," is remarkable, not only on account of its intrinsic power and the youth of its author, but also because in it he laid the foundation of one of the most singular discoveries of his later life, the temporary double refraction produced in viscous liquids by shearing stress.

37Some acetic acid is formed, and this process will go on till the solutions of the two acids are isohydric: that is, till the dissociated hydrogen ions are in equilibrium with both.

38In 1744 Alembert applied this principle to the theory of the equilibrium and the motion of fluids (Trcite de l'equilibre et du mouvement des fluides), and all the problems before solved by geometricians became in some measure its corollaries.

24Alexander insisted still more strongly on this claim, and in the convention which he concluded with the First Consul in October 1801 it was agreed that the maintenance of a just equilibrium between Austria and Prussia should be Napoleon.

24He rose despite his whirling equilibrium.

10A fixed weight is placed on one coil and the current is varied gradually until the balance is just in equilibrium.

10attention to the possible figures which would satisfy the conditions of equilibrium.

10The surface of the glass was hardened, but the inner layers remained in unstable equilibrium.

10For if the body is removed, and replaced by the fluid as at first, this fluid is in equilibrium under its own weight and the thrust of the surrounding fluid, which must be equal and opposite, and the surrounding fluid acts in the same manner when the body replaces the displaced fluid again; so that the resultant thrust of the fluid acts vertically upward through the centre of gravity of the fluid displaced, and is equal to the weight.

10With uniform temperature, taking h constant in the gas-equation, dp / dz= =p / k, p=poet/ k, (9) so that in ascending in the atmosphere of thermal equilibrium the pressure and density diminish at compound discount, and for pressures p 1 and 1, 2 at heights z 1 and z2 (z1-z2)11?

10= loge(P2891) =2.3 logio(p2/p1) (io) In the convective equilibrium of the atmosphere, the air is supposed to change in density and pressure without exchange of heat by conduction; and then PIN = (e/e0) n+1, d5 -(n-{--I) P -(n+I)R ' y - where is the ratio of the specific heat at constant pressure and constant volume.

10In the more general case of the convective equilibrium of a spherical atmosphere surrounding the earth, of radius a, (1-1?-=(n+ I) Po --a 2 dr, (12) gravity varying inversely as the square of the distance r from the centre; so that, k = po/po, denoting the height of the homogeneous atmosphere at the surface, 0 is given by (n+I)k(I -9/6 0) =a(I -a/r), (13) or if c denotes the distance where 0=o, 0 _a (14) 0 r c -a' When the compressibility of water is taken into account in a deep ocean, an experimental law must be employed, such as p - po=k(P - Po), or P/po=I+(p-p0)/A, A=kpo, (15) so that A is the pressure due to a head k of the liquid at density under atmospheric pressure po; and it is the gauge pressure required on this law to double the density.

10Equilibrium and Stability of a Ship or Floating Body.

10of the displaced fluid; for equilibrium these two forces must be equal and opposite in the same line.

10The conditions of equilibrium of a body, floating like a ship on the surface of a liquid, are therefore: (i.) the weight of the body must be less than the weight of the total volume of liquid it can displace; or else the body will sink to the bottom of the liquid; the difference of the weights is called the " reserve of buoyancy."

10(ii.) the weight of liquid which the body displaces in the position of equilibrium is equal to the weight W of the body; and (iii.) the C.G., B, of the liquid displaced and G of the body, must lie in the same vertical line GB.

10In addition to satisfying these conditions of equilibrium, a ship must fulfil the further condition of stability, so as to keep upright; if displaced slightly from this position, the forces called into play must be such as to restore the ship to the upright again.

10As a rule these equations are established immediately by determining the component acceleration of the fluid particle which is passing through (x, y, z) at the instant t of time considered, and saying that the reversed acceleration or kinetic reaction, combined with the impressed force per unit of mass and pressure-gradient, will according to d'Alembert's principle form a system in equilibrium.

10These equations were found by d'Alembert from two principles - that a rectangular canal, taken in a mass of fluid in equilibrium, is itself in equilibrium, and that a portion of the fluid, in passing from one place to another, preserves the same volume when the fluid is incompressible, or dilates itself according to a given law when the fluid is elastic. His ingenious method, published in 1752, in his Essai sur la resistance des fluides, was brought to perfection in his Opuscules mathematiques, and was adopted by Leonhard Euler.

11To use the apparatus, the long tube is placed in a vapour bath (c) of the requisite temperature, and after the air within the tube is in equilibrium, the delivery tube is placed beneath the surface of the water in a pneumatic trough, the rubber stopper pushed home, and observation made as to whether any more air is being expelled.

11The plummet is now placed in distilled water at 15°, and the beam brought to equilibrium by means of a rider, which we shall call I, hung on a hook; other riders are provided,;nth and ii b th respectively of I.

11Tirpitz himself maintains that his naval aspirations were directed not towards a war with Great Britain, but to the creation of a state of naval equilibrium or of German superiority, which would have enabled Germany to insist upon the unreserved cooperation of British policy in her world aims. It was probably true that Germany's policy was directed rather towards being so strong at sea as to make England unwilling to fight her unless absolutely necessary, than towards actually challenging British naval supremacy.

11Little remained to him of his light acquisitions; but he had convulsed Italy by this invasion, destroyed her equilibrium, exposed her military weakness and political disunion, and revealed her wealth to greedy and more powerful nations.

14When there is no magnetization, c c the yoke is in equilibrium; but as soon as the current °'°° is turned on the block C is drawn downwards as far as the screw R will allow, for, though the attractive forces F between B and C and between B' and C' are equal, the former has a greater moment.

14The following nine years mark the financial and commercial rehabilitation of Hungary, the establishment of a vast and original railway system which won the admiration of Europe, the liberation and expansion of her over-sea trade, the conversion of her national debt under the most favourable conditions and the consequent equilibrium of her finances.

14It had completed national unity, transferred the capital to Rome, overcome the chief obstacles to financial equilibrium, initiated military reform and laid the foundation of the relations between state and church.

16To explain this result, chemists suppose that both changes can occur simultaneously, and that equilibrium results when the rate at which AB and CD are transformed into AD and CB is the same as the rate at which the reverse change goes on.

16The long-promised abolition of the grist tax was not explicitly mentioned, opposition to the railway redemption contracts was transformed into approval, and the vaunted reduction of taxation replaced by lip-service to the Conservative deity of financial equilibrium.

00both army and navy, but advocated cordial relations with Berlin and Vienna as a guarantee against French domineering, and as a pledge that Italy would be vouchsafed time to effect her armaments without disturbing financial equilibrium.

00These changes may be brought about by external causes, such as the attacks of insects or of fungi, alterations in external conditions, &c., or by some unexplained internal disturbance of the morphological equilibrium.

00Chemical change which merely involves simple decomposition is thus seen to be influenced by the masses of the reacting substances and the presence of the products of decomposition; in other words the system of reacting substances and resultants form a mixture in which chemical action has apparently ceased, or the system is in equilibrium.

00At the present time we are quite uncertain what is the ultimate cause of new growths; in all probability there may be one or more aetiological factors at play disturbing that perfect condition of equilibrium of normal tissues.

00cooling the manufactured objects sufficiently slowly to allow the constituent particles to settle into a condition of equilibrium, is of vital importance.

00u3po ajXavuta), the science of the mechanics of water and fluids in general, including hydrostatics or the mathematical theory of fluids in equilibrium, and hydromechanics, the theory of fluids in motion.

00Archimedes maintained that each particle of a fluid mass, when in equilibrium, is equally pressed in every direction; and he inquired into the conditions according to which a solid body floating in a fluid should assume and preserve a position of equilibrium.

00In the hands of Blaise Pascal (1623-1662) hydrostatics assumed the dignity of a science, and in a treatise on the equilibrium of liquids (Sur l'equilibre des liqueurs), found among his manuscripts after his death and published in 1663, the laws of the equilibrium of liquids were demonstrated in the most simple manner, and amply confirmed by experiments.

00When generalizing the theory of pendulums of Jacob Bernoulli (1654-1705) he discovered a principle of dynamics so simple and general that it reduced the laws of the motions of bodies to that of their equilibrium.

00It was more fully developed in his Traite des fluides, published in 1744, in which he gave simple and elegant solutions of problems relating to the equilibrium and motion of fluids.

00He considered, at every instant, the actual motion of a stratum as composed of a motion which it had in the preceding instant and of a motion which it had lost; and the laws of equilibrium between the motions lost furnished him with equations representing the motion of the fluid.

00Take any two arbitrary directions in the plane of the paper, and draw a small isosceles triangle abc, whose sides are perpendicular to the two directions, and consider the equilibrium of a small triangular prism of fluid, of which the triangle is the cross section.

00Then, since these three forces maintain equilibrium, and R makes equal angles with P and Q, therefore P and Q must be equal.

00This is proved by taking any two points A and B at the same level, and considering the equilibrium of a thin prism of liquid AB, bounded by planes at A and B perpendicular to AB.

00As gravity and the fluid pressure on the sides of the prism act at right angles to AB, the equilibrium requires the equality of thrust on the ends A and B; and as the areas are equal, the pressure must be equal at A and B; and so the pressure is the same at all points in the same horizontal plane.

00This is proved by taking the two points A and B in the same vertical line, and considering the equilibrium of the prism by resolving vertically.

00When the body is floating freely like a ship, the equilibrium of this liquid thrust with the weight of the ship requires that the weight of water displaced is equal to the weight of the ship and the two centres of gravity are in the same vertical line.

00So also a balloon begins to rise when the weight of air displaced is greater than the weight of the balloon, and it is in equilibrium when the weights are equal.

00Analytical Equations of Equilibrium of a Fluid at rest under any System of Force.

00But by Green's transformation f flpdS = f f PPdxdydz, (2) thus leading to the differential relation at every point = dy dp The three equations of equilibrium obtained by taking moments round the axes are then found to be satisfied identically.

00The resultant force is therefore in the direction of the steepest pressure-gradient, and this is normal to the surface of equal pressure; for equilibrium to exist in a fluid the lines of force must therefore be capable of being cut orthogonally by a system of surfaces, which will be surfaces of equal pressure.

00These equations can be made to represent the state of convective equilibrium of the atmosphere, depending on the gas-equation p = pk =RA (6) where 0 denotes the absolute temperature; and then d9 d p R dz - dz (p) n+ 1' so that the temperature-gradient deldz is constant, as in convective equilibrium in (I I).

00Peace is considered not so much a state of equilibrium and friendly relations between parties, but rather as the rule of a third within a certain region - a house, an estate, a kingdom.

00The municipal finance has on the whole been sound, and notwithstanding the extra burdens assumed on the incorporation of the suburbs, the equilibrium of the communal budget was maintained up to the fall of the Liberal administration.

00A dissolved in B and B dissolved in A, since both of these solutions emit vapours of the same composition (this follows since the same vapour must be in equilibrium with both solutions, for if it were not so a cyclic system contradicting the second law of thermodynamics would be realizable).

00An example alone is needed finally to destroy the equilibrium.

00The surface of a charged conductor is an equipotential surface, because when the electric charge is in equilibrium there is no tendency for electricity to move from one part to the other.

00(i) Electrical Equilibrium and Potential.

00- If there be any number of charged conductors in a field, the electrification on them being in equilibrium or at rest, the surface of each conductor is an equipotential surface.

00Let a charge +Q be f t the ellipsoid a similar and slightly larger one, that distribution will be in equilibrium and will produce a constant potential throughout the interior.

00(5) in which dp/do is the rate of change of pressure with temperature when the two states are in equilibrium.

00- In order that a process may be strictly reversible, it is necessary that the state of the working substance should be one of equilibrium at uniform pressure and temperature throughout.

00The final state of the substance, when equilibrium has been restored, may be deduced from this condition, if the energy can be expressed in terms of the co-ordinates.

00But the line of constant energy on - the diagram does not represent the path of the transformation, unless it be supposed to be effected in a series of infinitesimal steps between each of which the substance is restored to an equilibrium state.

00The entropy tends to a maximum, and the state is one of stable equilibrium when the value of the entropy is the maximum value consistent with the conditions of the problem.

00Heterogeneous Equilibrium.

00- In a system, as distinguished from a homogeneous substance, consisting of two or more states or phases, a similar condition of equilibrium applies.

00The simplest case to consider is that of equilibrium between solid and liquid, or liquid and vapour.

00The condition of stable equilibrium of a system at constant temperature and volume is that the total J should be a minimum.

00This 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.

00state v", the value of J for unit mass of the mixture is mJ' + (1m) This must be a minimum in the state of equilibrium at constant temperature.

00Putting dJ /dm =o at constant volume, we obtain as the condition of equilibrium of the two states J' + p'v' = J" -}- p "v".

00This may be interpreted as the equation of the border curve giving the relation between p and 0, but is more easily obtained by considering the equilibrium at constant pressure instead of constant volume.

00The condition of stable equilibrium is that G should be a minimum, for which reason it has been called the " thermodynamic potential at constant pressure."

00The 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.

00If 0', E', v'; and 4)", E", v", refer to unit mass of the substance in the first and second states respectively in equilibrium at a temperature 0 and pressure p, the heat absorbed, L, per unit mass in a change from the first to the second state is, by definition of the entropy, equal to 0(4)"-4)'), and this by the first law is equal to the change of intrinsic energy, E" - E', plus the external work done, p(v" - v'), i.e.

00If 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".

00Assuming 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).

00To 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".

00Although the value of G in any case cannot be found without that of 0, and although the consideration of the properties of the thermodynamic potential cannot in any case lead to results which are not directly deducible from the two fundamental laws, it affords a convenient method of formal expression in abstract thermodynamics for the condition of equilibrium between different phases, or the criterion of the possibility of a transformation.

00A careful, calculating dynastic policy, which aimed at the establishment of an equilibrium by means of prudent compromises and defensive alliances, was, he rightly judged, the best guarantee for the future safety and glory of Poland.

00The field for recruiting its members, as well as its landed estates, became restricted by the Reformation in England and Germany, and the French knights gradually gained a preponderance which upset the international equilibrium of the Order.

00In 1738 appeared his Hydrodynamica, in which the equilibrium, the pressure, the reaction and varied velocities of fluids are considered both theoretically and practically.

00The answer to this was found experimentally by Arthur Schuster, who suspended the whole instrument in delicate equilibrium, and observed the effect of introducing the radiation.

00On the other hand, if the effects arose from balanced stresses set up inside the globe by the radiation, the effects on the vanes and on the case would be of the nature of action and reaction, so that the establishment of motion of the vanes in one direction would involve impulsion of the case in the opposite direction; but when the motion became steady there would no longer be any torque either on the vanes or on the case, and the latter would therefore come back to its previous position of equilibrium; finally, when the light was turned off, the decay of the motion of the vanes would involve impulsion of the case in the direction of their motion until the moment of the restoring torque arising from the suspension of the case had absorbed the angular momentum in the system.

00As far as the order to which he carried the approximations - which, however, were based on a simplifying hypothesis that the molecules influenced each other through mutual repulsions inversely as the fifth power of their distance apart--the result was that the equations of motion of the gas, considered as subject to viscous and thermal stresses, could be satisfied by a state of equilibrium under a modified internal pressure equal in all directions.

00If, therefore, the walls of the enclosure held the gas that is directly in contact with them, this equilibrium would be the actual state of affairs; and it would follow from the principle of Archimedes that, when extraneous forces such as gravity are not considered, the gas would exert no resultant force on any body immersed in it.

00At starting, when the full load is to be lifted, the balance chain uncoils, and continues to do so until the desired equilibrium between the working loads is attained, when it is coiled up again in the reverse direction, to be again given out on the return trip.

00If a force Q acting at R maintains equilibrium, QR/4 = (P - p)r =T.

00In like manner, after the French mathematicians had attempted, with more or less ingenuity, to construct a theory of elastic solids from the hypothesis that they consist of atoms in equilibrium under the action of their mutual forces, Stokes and others showed that all the results of this hypothesis, so far at least as they agreed with facts, might be deduced from the postulate that elastic bodies exist, and from the hypothesis that the smallest portions into which we can divide them are sensibly homogeneous.

00The fact that a solid body in its natural state is capable both of compression and of dilatation indicates that the molecules of the body must not be supposed to be fixed rigidly in position relative to one another; the further fact that a motion of either compression or of dilatation is opposed by forces which are brought into play in the interior of the solid suggests that the position of rest is one in which the molecules are in stable equilibrium under their mutual forces.

00When the molecules are oscillating about their equilibrium positions, there is no reason why their mean distance apart should be the same as when they are at rest.

00Suppose for instance that two molecules, when at rest in equilibrium, are at a distance a apart.

00As the temperature of a body increases the average energy of the molecules will increase, and therefore the range of their excursions from their positions of equilibrium will increase also.

00At a certain temperature a stage will be reached in which it is a frequent occurrence for a molecule to wander so far from its position of equilibrium, that it does not return but falls into a new position of equilibrium and oscillates about this.

00A molecule escaping from its original position in a body will usually fall into a new position in which it will be held in equilibrium by the forces from a new set of neighbouring molecules.

00When a stage is reached such that the number of molecules lost to the liquid by evaporation is exactly equal to that regained by condensation, we have a liquid in equilibrium with its own vapour.

00Now if the atoms are regarded as points or spherical bodies oscillating about positions of equilibrium, the value of n+3 is precisely six, for we can express the energy of the atom in the form (9 2 v a 2 v a2v E = z(mu 2 +mv 2 +mw 2 +x 2 ax2 + y2ay2-fz2az2), where V is the potential and x, y, z are the displacements of the atom referred to a certain set of orthogonal axes.

00By this time, twenty years of solitary confinement had disturbed Ivan's mental equilibrium, though he does not seem to have been actually insane.

00A severe property-tax and an increase of customs duties in 1879 only for a moment achieved financial equilibrium.

00Let an external force F act on the system, and for simplicity suppose its period is so great compared with that of the mechanism that we may take it as practically in equilibrium with the restoring force.

00A state of equilibrium was established, only momentarily disturbed by Kuropatkin's offensive on the Sha-ho in October, and by the Sandepu incident in the winter, until at last Oyama fought a battle on a grand scale and won it.

00Even then, however, the results fell far short of anticipation, and the armies settled down into equilibrium again.

00The lines in the diagram represent the directions of a series of forces which must all be in equilibrium; these lines may, for an object to be explained in the next paragraph, be conveniently named by the letters in the spaces which they separate instead of by the method usually employed in geometry.

00This polygon falls under the definition of a reciprocal figure given by Clerk Maxwell, if we consider the frame as a point in equilibrium under the external forces.

00If there are no redundant members in the frame there will be only two members abutting at the point of support, for these two members will be sufficient to balance the reaction, whatever its direction may be; we can therefore draw two triangles, each having as one side the reaction YX, and having the two other sides parallel to these two members; each of these triangles will represent a polygon of forces in equilibrium at the point of support.

0013 C figure of the three lines YX, XE, EY in the frame, and represents the three forces in equilibrium at the point YXE of the frame.

00England and Russia together were to maintain the equilibrium of the world.

00Legendre's second memoir was communicated to the Academie in 1784, and relates to the conditions of equilibrium of a mass of rotating fluid in the form of a figure of revolution which does not deviate much from a sphere.

00In a word, this diet disturbed the equilibrium of the state by enfeebling and degrading the middle classes.

00Seriously disturbed at the prospect of Russian aggrandizement, the idea occurred, almost simultaneously, to the courts of Berlin and Vienna that the best mode of preserving the equilibrium of Europe was for all three powers to readjust their territories at the expense of Poland.

00He considered that the equilibrium of Europe would be irretrievably upset were the Russian boundaries to be pushed into the heart of Germany.

00What it wanted most of all was peace, and by establishing something like a territorial equilibrium the congress did much to win that breathing space which was the cardinal need of all.

00The dynamical series of stages in nature, the forms in which the ideal structure of nature is realized, are matter, as the equilibrium of the fundamental expansive and contractive forces; light, with its subordinate processes - magnetism, electricity, and chemical action; organism, with its component phases of reproduction, irritability and sensibility.'

00A discussion of the equilibrium of Saturn's rings led him to conclude in 1855 that they must be of a fluid nature.

00The unequal intensities observed indicate a difference in the effectiveness of the channels through which energy is lost, and this need not be connected with the ultimate state of equilibrium when the body is kept at a uniform temperature.

00When, for instance, we observe the relation of the gas contained in a Plucker tube through which an electric discharge is passing, there can be little doubt that the partition of energy is very different from what it would be in thermal equilibrium.

00But the molecules affected by a spark discharge are not in any sense in equilibrium as regards their partition of energy and the word temperature " cannot therefore be applied to them in the ordinary sense.

00The difficulty that a number of spectroscopic lines seem to involve at least an equal number of electrons may be got over by imagining that the atom may present several positions of equilibrium to the electron, which it may occupy in turn.

00In order to exert force, or at all events that force of reciprocal pressure which we best understand, and on which, in impact, the third law of motion was founded, there are always at least two bodies, enduring, triply extended, mobile, each inert, mutually impenetrable or resistent, different yet similar; and in order to have produced any effect but equilibrium, some bodies must at some time have differed either in mass or in velocity, otherwise forces would only have neutralized one another.

00The war which ensued between the pope and the king of France ended in the complete defeat of the papacy, which was reduced to impotence (1303), and though the storm ceased during the nine months' pontificate of Benedict XI., the See of St Peter recovered neither its normal equilibrium Papacy nor its traditional character.

00Willard Gibbs, who considered the whole problem of physical and chemical equilibrium in papers published in 1877, though the application of his principles only began to make extensive progress about twenty years after the publication of his purely theoretical investigations.

00When the limit is reached the solution is said to be saturated, and the system is in equilibrium.

00If a crystal of the solid be added, the condition of supersaturation is destroyed, and the ordinary equilibrium of saturation is reached by precipitation of solid from solution.

00It is certain then that when dissolution occurs the available energy of the whole system is decreased by the process, while when equilibrium is reached and the solution is saturated the available energy is a minimum.

00Equilibrium requires that the available energy and therefore the area of contact should be a minimum, as is demonstrated in Plateau's beautiful experiment, where a large drop of oil is placed in a liquid of equal density and a perfect sphere is formed.

00A saturated solution is a system in equilibrium, and exhibits the thermodynamic relations which hold for all such systems. Just as two electrified bodies are in equilibrium when their electric potentials are equal, so two parts of a chemical and physical system are in equilibrium when there is equality between the chemical potentials of each component present in the two parts.

00Thus water and steam are in equilibrium with each other when the chemical potential of water substance is the same in the liquid as in the vapour.

00It is usual to call each part of the system of uniform composition throughout a phase; in the example given, water substance, the only component is present in two phases - a liquid phase and a vapour phase, and when the potentials of the component are the same in each phase equilibrium exists.

00If µ i and µ2 denote the potentials of any one component in two phases in contact, when there is equilibrium, we know that µ i =P2 If a third phase is in equilibrium with the other two we have also =123.

00Thus, when a system possesses two more phases than the number of its components, all the phases will be in equilibrium with each other at one definite composition, one definite temperature and one definite pressure, and in no other conditions.

00To take the simplest case of a one component system water substance has its three phases of solid ice, liquid water and gaseous vapour in equilibrium with each other at the freezing point of water under the pressure of its own vapour.

00We then have in equilibrium two phases only, and the temperature and pressure may change.

00We then have water and vapour in equilibrium, and, as more heat enters, the temperature rises and the vapour-pressure rises with it.

00But, if we fix arbitrarily the temperature the pressure of equilibrium can have one value only.

00The phenomena of equilibrium can be represented on diagrams. Thus, if we take our co-ordinates to represent pressure and temperature, the state of the systems p with ice, water and vapour in equilibrium is represented by the point 0 where the pressure is that of the vapour of water at the freezing point and the temperature is the freezing point under that pressure.

00If all the water be frozen, we have the vapour pressure curve of ice OB; while, if the pressure be raised, so that all the vapour vanishes, we get the curve OC of equilibrium between the pressure and the freezing point of water.

00The phase rule combined with the latent heat equation contains the whole theory of chemical and physical equilibrium.

00Equilibrium between these phases is obtained at the freezing point of the saturated solution under the pressure of the vapour.

00In representing on a diagram the phenomena of equilibrium in a two-component system we require a third axis along which p to plot the composition of a variable phase.

00At the cryohydric point 0 we have four phases in equilibrium at a definite pressure, temperature and composition of the liquid phase.

00Again, starting from 0, by the abstraction of heat we can remove all the liquid and travel along the curve OD of equilibrium between the two solids (salt and ice) and the vapour.

00If the salt crystallizes with a certain amount of water as well as with none, we get a second point of equilibrium between four phases.

00Taking the point 0 to denote the state of equilibrium between ice, hydrate; saturated solution and vapour, we pass along OA till a new solid phase, that of Na2S04, appears at 32.6°; from this point arise four curves, analogous to those diverging from the point O.

003 gives the equilibrium between sodium sulphate and water in this way.

00At that temperature crystals of the anhydrous Na 2 SO 4 appear, and a new fixed equilibrium exists between the four phases - hydrate, anhydrous salt, solution and vapour.

00When this process is complete the temperature rises, and we pass along a new curve giving the equilibrium between anhydrous crystals, solution and vapour.

00In this way two temperature points are obtained in the investigation - the higher giving a point on the equilibrium curve, the lower showing the non-variant point.

00A good example is the too° equilibrium of ferric chloride arid water, studied by B.

00At B we have the non-variant cryohydric point at which ice, the hydrate Fe2C16 12H20, the saturated solution and the vapour are in equilibrium at 55° C. As the proportion 26 of salt is increased, the melting point of the con glomerate rises, till, at the -40 maximum point C, we have the pure compound the hydrate with twelve molecules ¦¦ 0.b, E, ?

00When a crystal of the solid phase is present the equilibrium of a solution is given by the solubility curves we have studied.

00Between the saturation point and this lower temperature, the liquid holds in solution more of the solute than corresponds with equilibrium, and is said to be supersaturated.

00Hence the conditions necessary to secure equilibrium when the solid phase is present are not the same as those necessary to cause crystallization to start in a number of crystals at first excessively minute in size.

00If the temperature at which this dense spontaneous shower of crystals is found be determined for different concentrations of solution, we can plot a "supersolubility curve," which is found generally to run roughly parallel to the "solubility curve" of steady equilibrium between liquid and already existing solid.

0012) were inocu lated with crystals of A when its composition was that represented so by x, cooled very slowly and stirred, the conditions would be hose of equilibrium throughout.

00The conditions may then remain those of equilibrium along the curve f E, but before reaching f the solution may become supersaturated with B and deposit B crystals spontaneously.

00The possibility of these phenomena should be borne in mind when attempts are made to interpret the structure of crystalline bodies in terms of the theory of equilibrium.

00It was found, too, when water was placed on one side of such a membrane, and a sugar solution in a confined space on the other, that water entered the solution till a certain pressure was set up when equilibrium resulted.

00The importance of these experiments from the point of view of the theory of solution, lay in the fact that they suggested the conception of a perfect or ideal semi-permeable partition, and that of an equilibrium pressure representing the excess of hydrostatic pressure required to keep a solution in equilibrium with its pure solvent through such a partition.

00The loss in the solution bulbs gives the mass of solvent absorbed from the solution, and the loss in the solvent bulbs the additional mass required to raise the vapour pressure in the air-current to equilibrium with the pure solvent.

00But as we ascend in an atmosphere the pressure diminishes; hence the pressure of the vapour in the chamber is less the higher we go, and thus eventually we reach a state of equilibrium where the column of vapour is in equilibrium at the appropriate level both with solvent and solution.

00In practice the time required to reach these various conditions of equilibrium would be too great for experimental demonstration, but the theoretical consideration of vapour pressures is of fundamental importance.

00height of the column of solution would rise or fall and the equilibrium with the vapour be disturbed.

00be in equilibrium under the excess of hydrostatic pressure represented when the solution is very dilute by P=(p - p')p/0.

00But such a pressure represents the equilibrium osmotic pressure discussed above.

00Therefore the equilibrium osmotic pressure of a solution is connected with the vapour pressure, arid, in a very dilute solution, is expressed by the simple relation just given.

00When the solution and solvent are in equilibrium across the partition, the vapour pressure of the solution has been increased by the application of pressure till it is equal to that of the solvent.

00Similar 'considerations show that, since at its freezing point the vapour pressure of a solution must be in equilibrium with that of ice, the depression of freezing point produced by dissolving a substance in water can be calculated from a knowledge of the vapour pressure of ice and water below the freezing point of pure water.

00The solution then freezes, until the heat liberated is enough to raise the tern perature to the point of equilibrium given by the tendency of the solution taken in contact with ice to approach the true freezing point on one side and the temperature of the enclosure on the other.

00Although even good membranes of copper ferrocyanide are rarely perfectly semi-permeable, and in other membranes such as indiarubber, &c., which have been used, the defects from the theoretical values of the equilibrium pressure are very great, yet, in the light of the exact verification of theory given by the experiments described above, it is evident that such failures to reach the limiting value in no wise invalidate the theory of osmotic equilibrium.

00They merely show that, in the conditions of the particular experiments, the thermodynamic equilibrium value of the osmotic pressure cannot be reached - the thermodynamic or theoretical osmotic pressure (which must be independent of the nature of the membrane provided it is truly semi-permeable) is a different thing from the equilibrium pressure actually reached in a given experiment, which measures the balance of ingress and egress of solvent through an imperfect semi-permeable membrane.

0013 required for osmotic equilibrium through a semi-permeable wall below is now very great, since the osmotic pressure of strong solutions may reach many hundred atmospheres.

00Hence we must not assume that the density of the vapour in the surrounding atmosphere is constant, or that the solution, when equilibrium is reached, is of uniform concentration throughout.

00The osmotic pressure (defined as the difference in the hydrostatic pressures of the solution and solvent when their vapour pressures are equal and they are consequently in equilibrium through a perfect semi-permeable membrane) may also depend on the absolute values of the hydrostatic pressures, as may the vapour pressure of the liquids.

00The vapour at pressure p in equilibrium with the liquid is bounded by a solid piston C, which we can also move to change the pressure or volume.

00The relation between the equilibrium pressures P and P' for solution and solvent corresponding to the same value po of the vapour pressure is obtained by integrating the equation V'dP' = vdp between corresponding limits for solution and solvent.

00From this equation the osmotic pressure Po required to keep a solution in equilibrium as regards its vapour and through a semi-permeable membrane with its solvent, when that solvent is under its own vapour pressure, may be calculated from the results of observations on vapour pressure of solvent and solution at ordinary low hydrostatic pressures.

00If it be filled with a solution and the bottom immersed in the pure solvent, pressure equal to the osmotic pressure must be exerted on the piston to maintain equilibrium.

00Stevinus also distinguished stable from unstable equilibrium.

00He proved the law of the equilibrium on an inclined plane.

00One main object is the preservation of equilibrium in the growth of the several parts of the tree; and for this various minor details deserve attention.

00If the metal followed the laws of equilibrium, then whenever through change of temperature it entered a new region, it would forthwith adopt the constitution normal to that region.

00Here sulphur may indeed be removed to a very important degree in the form of manganese sulphide, which distributes itself between metal and slag in rough accord with the laws of equilibrium.

00This behaviour is explained by considering the non-ionized part of the diazonium hydroxide to exist in solution in a hydrated form, the equation of equilibrium being: C6H6.N.

00Debray (ibid., 1879, 88, p. 1341), who found that at about 150o C. a condition of equilibrium is reached.

00With this arrangement the lift - ram and the two balance pistons are always in equilibrium, or, in other words, the ever-changing displacement of the lift-ram is automatically in balance.

00° prismatic < - sulphur," indicating that there is equilibrium at the so-called "transition-point," 95.4°, and opposite change below and above.

00This comparison with fusion introduces a second notion, that of the "triple-point," this being in the melting-phenomenon the only temperature at which solid, liquid and vapour are in equilibrium, in other words, where three phases of one substance are co-existent.

00° cyanuric acid; the cyanic acid corresponds to sulphur vapour, being in equilibrium with either cyamelide or cyanuric acid at a maximum pressure, definite for each temperature.

00In reality such tautomeric compounds are apparently a mixture of two isomers in equilibrium, and indeed in some cases both forms have been isolated; then one speaks of desmotropy (Gr.

00Phys., 1862 (3), 65, p. 385 et seq.) have shown in the case of the formation of ethyl acetate from ethyl alcohol and acetic acid, a point of equilibrium is reached, beyond which the reacting system cannot pass, unless the system be disturbed in some way by the removal of one of the products of the reaction.

00In intimate connexion with the Gottorp affair stood the question of the political equilibrium of the north.

00His experiments and his treatise (written before 1651, published 1663) on the equilibrium of fluids entitle him to rank with Galileo and Stevinus as one of the founders of the science of hydrodynamics.

00He was economical, and gave up a third of his civil list in order to help forward the task of establishing an equilibrium in the annual budget, and he was always ready from his large private fortune to help forward all schemes for the social or industrial progress of the country.

00By the exercise of the most rigid economy in all branches this end was attained, though budgetary equilibrium was only secured by a variety of financial expedients, justified by the vital importance of saving Egypt from further international interference.

001887 there was practical equilibrium in the budget,, in -

00By mitigating the hardships of the corve, and improving the irrigation system, on which the prosperity of the country mainly depends, he had conferred enormous benefits on the fellahin, and had laid the foundation of permanent budgetary equilibrium for the future.

00Castlereagh, whose single-minded aim was the restoration of "a just equilibrium" in Europe, reproached the tsar to his face for a " conscience " which suffered him to imperil the concert of the powers by keeping his hold on Poland in violation of his treaty obligation.'

00Afterwards, owing to the increased attention given to stock-fattening and dairying, and to a rise in prices, farming reached a condition of equilibrium, and the most noticeable residuum of the period of depression was the large intrusion of the butcher and grazier class into the farmer class proper.

00to announce the restoration of equilibrium between expenditure and revenue for the first time since 1860.

00The former, from the Tertiary period even to the present day, has been -a region of compression; the latter, since the Carboniferous period at least, has been a region of equilibrium or of tension.

00His first care on arrival in India was to restore equilibrium to the finances, which were tottering under the burden imposed upon them by the Burmese War.

00If now a wheel W is forced up against q with a pressure equal to the weight of the moving part of the instrument, the whole weight of the moving part would rest upon W in unstable equilibrium; or if a pressure R, less than W, is employed, we have the end friction on the lower bearing removed to an extent = R sin 4),4), and the friction on the bearings of the upper pivot removed to the extent of F cos 4), - where 4 is the latitude of the place.

00The Calorimeter Was Suspended By A Steel Wire, The Torsion Of Which Made The Equilibrium Stable.

00It was shown by Homer Lane that a mass of gas held in equilibrium by the mutual gravitation of its parts actually grows hotter through radiating heat; the heat gained by the resulting contraction more than counterbalances that lost by radiation.

00At the freezing-point, the solution must have the same vapour-pressure as the solid solvent, with which it is in equilibrium.

00It is probable that osmotic pressure is not really of the same nature as gas-pressure, but depends on equilibrium of vapour-pressure.

00In this position the magnet is in equilibrium under the action of the torsion of the suspension and the couple exerted by the horizontal component, H, of the earth's field, this couple depending on the product of H into the magnetic moment, M, of the magnet.

00The magnet is in equilibrium under the influence of the couple VM due to the vertical component V, and the couple due to the fact that the centre of gravity is slightly on one side of the knife-edge.

00The subject is usually expounded under the two divisions of statics and kinetics, the former dealing with the conditions of rest or equilibrium and the latter with the phenomena of motion as affected by force.

00The laws of equilibrium are, it is true, necessaril~ included as a particular case under those of motion; but ther is no real inconvenience in formulating as the basis of static a few provisional postulates which are afterwards seen to b comprehended in a more general scheme.

00Stability of equilibrium.

00If three forces acting on a particle are represented as to magnitude and direction by the sides of a triangle taken in order, they are in equilibrium.

00if three forces acting on a particle be in equilibrium, and any triangle be constructed whose sides are respectively parallel to the forces, the magnitudes of the forces will be to ope another as the corresponding sides of the triangle.

00As a simple example of the geometrical method of treating statical problems we may consider the equilibrium of a particle on a rough inclined plane.

00If the conditions of equilibrium require an obliquity greater than this, sliding will take place.

00In the case of a body simply resting on an inclined plane, the reaction must of course be vertical, for equilibrium, and the slope a of the plane must therefore not exceed X.

00If a > X, a force P must be applied in order to maintain equilibrium; let 0 be the inclination of P to the plane, as shown in the left-hand diagram.

00(3) For equilibrium we must have R=o, which requires Xo, Y=o; in words, the sum of, the components of the system must be zero for each of two perpendicular directions in the plane.

00The conditions of equilibrium are X=o, Y=o, Z=o.

00The problem of determining the possible configurations of equilibrium of a system of particles subject to extraneous forces which are known functions of the positions of the particles, and to internal forces which are known functions of the distances of the pairs of particles between which they act, is in general determinate For if n be the number of particles, the 3n conditions of equilibrium (three for each particle) are equal in number to the 351 Cartesian (or other) co-ordinates of the particles, which are to be found.

00As exceptional cases the system may reduce to a couple, or it may be in equilibrium.

00Again it is necessary and sufficient for equilibrium that the sum of the projections of the forces on each of two perpendicular directions should vanish, and (moreover) that the sum of the moments about some one point should be zero.

00The fact that three independent conditions must hold for equilibrium is important.

00to the sides of a triangle at the middle points will be in equilibrium provided they are proportional to the respective sides, and act all inwards or all outwards.

00The position of equilibrium is determined by the considera tion that the reactions at A and B, which are by hypothesis normal t the planes, must meet at a point J on the vertical through G.

00The three conditions of equilibrium arc therefore ~(X) = o, ~(Y) = o, ~(xY yX) = o.

00If, however, the first and last sides of the funicular coincide, the two outstanding forces neutralize one another, and we have equilibrium.

00Hence the necessary and sufficient conditions of equilibrium are that the force-polygon and the funicular should both be closed.

00It is evident that a system of jointed bars having the shape of the funicular polygon would be in equilibrium under the action of the given forces, supposed applied to the joints; moreover any bar in which the stress is of the nature of a tension (as distinguished from a thrust) might be replaced by a string.

00Hence the forces x, y, x, y are in equilibrium.

00The complete figures obtained by drawing first the force-diagrams of a system of forces in equilibrium with two distinct poles 0, 0, and secondly the corresponding funiculars, have various interesting relations.

00The segments DE, nA then represent ~he upward pressures of the two supports on the beam, which pressures together with the given loads constitute a system of forces in equilibrium.

00Hence if a system of vertical forces be in equilibrium, so that the funicular polygon ii closed, the length which this polygon intercepts on the vertical through any point P gives the sum of the moments about P of all the forces on one side of this vertical.

00For instance, in the case 01 a beam in equilibrium under any given loads and the reactionf at the supports, we get a graphical representation of the distributior of bending moment over the beam.

00For the conditions of equilibrium of the forces on each pin furnish vi equations, viz, two for each point, which are linear in respect of the stresses and the extraneous forces.

00This system of equations must involve the three conditions of equilibrium of the extraneous forces which are already identically satisfied, by hypothesis; there remain therefore 2n ~ independent relations to determine the 2n3 unknown stresses.

00A frame of n joints and vi 3 bars may of course fail to be rigid owing to some parts being over-stiff whilst others are deformable; in such a case it will be found that the statical equations, apart from the thre identical relations imposed by the equilibrium of the extraneous forces, are not all independent but are equivalent to less thar 2,13 relations.

00The stresses produced by extraneous forces in a simple frame can be found by considering the equilibrium of the various joints in a proper succession; and if the graphical method be employed the various polygons of force can be combined into a single force-diagram.

00If an ideal section be drawn across the frame, the extraneous forces on either side must be in equilibrium with the forces in the bars cut across; and if the section can be drawn so \, as to cut only three bars, ~/\~ the forces in these can be found, since the problem.

00r As special cases, the system may reduce to a couple, or it may be in equilibrium.

00Since R2=X2+Y2+Z2, G2=L1+M2+N2, it is necessary and sufficient for equilibrium that the six quantities X, Y, Z, L, M, N, should all vanish.

00An exception to the general statement occurs when the six lines are such that they are possible lines of action of a system of six forces in equilibrium; they are then said to be in involution.

00If the extraneous forces are n equilibrium the total work which they would perform in any such displacement would be zero, since they reduce to a zero force and a zero couple.

00Conversely, we can show that if the, virtual work of the extraneous forces be zero for every infinitesimal displacement of the body as rigid, these forces must be in equilibrium.

00If this vanishes for all values of X, u, v, ~, s~, ~ we must have X, Y, Z, L, M, N = o, which are the conditions of equilibrium.

00The proper significance of the principle of virtual work, and of its converse, will appear more clearly when we come to kinetics (~ 16); for the present it may be regarded merely as a compact and (for many purposes) highly convenient summary of the laws of equilibrium.

00An important conclusion is that in any displacement of a system of bodies in equilibrium, such that the virtual work of all forces except gravity may be ignored, the depth of the centre of gravity is stationary.

00The question as to stability of equilibrium belongs essentially to kinetics; but we may state by anticipation that in cases where gravity is the only force which does work, the equilibrium of a body or system of bodies is stable only if the depth of the centre of gravity be a maximum.

00The displacement of G is at right angles to JG; this shows that for equilibrium JG must be vertical.

00Again, the locus of G is an arc of an ellipse whose centre is in the intersection of the planes; since this arc is convex upwards the equilibrium is unstable.

00For equilibrium, the altitude of the centre of gravity G must be stationary; hence G must lie in the same vertical line with the point of contact J of the two curves.

00If we imagine the bar in question t be removed, equilibrium will still persist if we introduce two equal and opposite forces S, of suitable magnitude, at the joints which it connected.

00Consider, for example, a frame whose sides form the six sides of a hexagon ABCDEF and the three diagonals AD, BE, CF; and suppose that it is required to find the stress in CF due to a given system of extraneous forces in equilibrium, acting on the joints.

00We may note that a frame of n joints which is just rigid must have 3116 bars; and that the stresses produced in such a frame by a given system of extraneous forces in equilibrium are statically determinate, subject to the exception of critical forms.

00It is assumed that the form can be sufficiently represented by a plane curve, that the stress (tension) at any point P of the curve, between the two portions which meet there, is in the direction of the tangent at P, and that the forces on any linear element s must satisfy the conditions of equilibrium laid down in I.

00It follows that the forces on any finite portion will satisfy the conditions of equilibrium which apply to the case of a rigid body (~ 4).

00We assume that in limiting equilibrium we have F tsR, everywhere, where u is the coefficient of friction.

00In the case of a chain hanging freely under gravity it is usually convenient to formulate the conditions of equilibrium of a finite portion PQ.

00For investigations relating to the equilibrium of a string in three dimensions we must refer to the textbooks.

00If M be the mass, and x the vertical displacement from the position of equilibrium, the equation of motion is of the form M~=-Kx, (7

00Hence if 1 be the length of the string, and x the horizontal displacement of the bob from the equilibrium position, the horizontal component of gravity is mgx/l, whence The motion is therefore simple-harmonic, of period r= 27ri/(l/g).

00The position x=o is one of equilibrium, but it is unstable.

00This applies to the inverted pendulum, with u =g/l, but the equation (9) is then only approximate, and the solution therefore only serves to represent the initial stages of a motion in the neighborhood of the position of unstable equilibrium.

00The body now passes once (at most) through its equilibrium position, and the vibration is therefore styled aperiodic.

00According to dAlemberts formulation, the extraneous forces together with the effective forces reversed fulfil the statical conditions of equilibrium.

00As a final example we may note the arrangement, often employed in physical measurements, where a body performs small oscillations about a vertical axis through its mass-centre G, under the influence of a couple whose moment varies as the angle of rotation from the equilibrium position.

00Stability of Equilibrium.

00q,,) be one of equilibrium, the equatiops (14) of 22 must be satisfied by th~ ~i,.

00A necessary and sufficient condition of equilibrium is therefore that the value of the potential energy should be stationary for infinitesimal variations of the co-ordinates.

00If, further, V be a minimum, the equilibrium is necessarily stable, as was shown by P. G.

00In the motion consequent on any slight disturbance the total energy T+V is constant, and since T is essentially positive it follows that V can never exceed its equilibrium value by more than a slight amount, depending on the energy of the disturbance, This implies, on the present hypothesis, that there is an upper limit to the deviation of each co-ordinate from its equilibrium value; moreover, this limit diminishes indefinitely with the energy of the original disturbance.

00Hence if the system be started from rest in a configuration for which V is less than in the equilibrium configuration considered, this quantity must sUll further decrease (since T cannot be negative), and it is evident that either the system will finally come to rest in some other equilibrium configuration, or V will in the long run diminish indefinitely.

00In discussing the small oscillations of a system about a configuration of stable equilibrium it is convenient so to choose the generalized co-ordinates qi, q2,..

00., (2) a constant term being irrelevant, and the terms of the first order being absent since the equilibrium value of V is stationary.

00We may note further that when ~ is small the displacement q has the equilibrium value Q/c, the same as would be produced by a steady force equal to the instantaneous value of the actual force, the inertia of the system being inoperative.

00Every particle of the system executes in general a simple vibration of the imposed period 27r/il, and all the particles pass simultaneously through their equilibrium positions.

00Support of .tructures.Every structure, as a whole, is maintained in equilibrium by the joint action of its own weIght, of the external load or pressure applied to it from without and tending to displace it, and of the resistance of the material which supports it.

00The principles of the support, as a whole, of a structure resting on the land, are so far identical with those which regulate the equilibrium and stability of the several parts of that structure that the oni principle which seems to require special mention here is one whic comprehends in one statement the power both of liquids and of loose earth to support structures.

00In order that the surfaces which abut at the Joint JK maybe pressed together, the resistance required by the conditions of equilibrium CR, must be a thrust and not a pull; and in that case the force by which the surfaces are pressed together is equal and opposite to the normal component CP of the resistance.

00For the condition of equilibrium of forces not parallel is that they shall be represented in direction and magnitude by the sides and diagonals of certain parallelograms, and of parallel forces that they shall divide certain straight lines in certain ratios; and the parallel projection of a parallelogram is a parallelogram, and that of a straight line divided in a given ratio is a straight line divided in the same ratio.

00It is usefu in practice, by enabling the engineer easily to deduce the condition of equilibrium and stability of structures of complex and unsym metrical figures from those of structures of simple and symnietrica figures.

00Now from the laws of statics it is known that, in order that a system of forces applied to a system of connected points may be in equilibrium, it is necessary that the sum formed by putting together the products of the forces by the respective distances through which their points of application are capable of moving simultaneously, each along the direction of the force applied to it, shall be zero, products being considered positive or negative according as, the direction of the forces and the possible motions of their points of application are the same or opposite.

00* S~atsc Equilibrium of MechanismsThe principle stated in the preceding section, namely, that the energy exerted is equal to forces acting in the respective directions of motion at two points of a mechanism, one being the point of application of the effort, and the other the point of application of the resistance, to be readily found.

00The general problem may then be thus stated: Given a mechanism of which r is the fixed link, and s and t any other two links, given also a force, f~ acting on the link s, to find the force f, acting in a given direction on the link t, which will keep the mechanism in static equilibrium.

00of that reaction and of the effort are then found by the principles of the equilibrium of three forces already stated in 7.

00Using this principle the method of finding the balance weights to be added to a given system of reciprocating weights in order to produce a system of forces on the frame continuously in equilibrium is exactly the same as that just explained for a system of revolving weights, because for the purpose of finding the balance weights each reciprocating weight may be supposed attached to the crank pin which operates it, thus forming an equivalent revolving system.

00As the mortality amongst boys, especially during the first year, is considerably above that of the other sex, numerical equilibrium between the two is established in early youth, and in most cases girls outnumber boys, except for a few years between twelve and sixteen.

00The forces underlying the movement may differ from time to time in their respective intensity, and, in highly exceptional cases, may approach equilibrium, their natural tendencies being interrupted by special causes, but the instances of general decline are confined to wild and comparatively small communities brought into contact with alien and more civilized races.

00Equilibrium was maintained by diplomacy, in which the humanists played a foremost part, casting a network of intrigue over the nation which helped in no small measure to stimulate intelligence and create a common medium of culture, but which accustomed statesmen to believe that everything could be achieved by wire-pulling.

00The five great powers, held in equilibrium by Lorenzo de' Medici, dreamed that the peninsula could be maintained in statu quo by diplomacy.

00While holding this office he devoted his energies chiefly to the acquisitions of Texas, in order to preserve the equilibrium between the South and the constantly growing North.

00Under the pressure of war and other emergencies it has been found impossible to maintain this desirable equilibrium.

00Plana (1781-1846), and later Robert Murphy (1806-1843), made them the subject of their investigations on the mode in which electricity distributes itself on conductors when in equilibrium.

00Of other papers in which he dealt with this and kindred branches of physics may be mentioned "Observations with a Rigid Spectroscope," "Heating of a Disc by Rapid Motion in Vacuo," "Thermal Equilibrium in an Enclosure Containing Matter in Visible Motion," and "Internal Radiation in Uniaxal Crystals."

00But before an equal space of time had further elapsed, it had itself become a pest; the most recent information, however, is to the effect that its numbers are now on the decline, and that the disturbed faunal equilibrium is being readjusted.

00In a word, the natural equilibrium of Swedish society was seriously threatened by the preponderance of the nobility; and the people at large looked to the new king to redress the balance.

00The Caps succeeded in transferring L250,000 from the pockets of the rich to the empty exchequer, reducing the national debt by L575,179, and establishing some sort of equilibrium between revenue and expenditure.

00In this state of equilibrium the great powers of Anterior Asia remained during the first half of the 6th century.

00In the outer valleys of the Himalaya the sides are generally steep, so steep as to be liable to landslip, whilst the streams are still cutting down the river beds and have not yet reached the stage of equilibrium.

00The first part establishes the laws of the elasticity of a finite portion of the solid subjected to a homogeneous strain, and deduces from these laws the equations of the equilibrium and motion of a body subjected to any forces and displacements.

00The condition of equilibrium is that this expression (which we may for the sake of distinctness call the potential energy) shall be a minimum.

00He proceeded to an investigation of the equilibrium of a fluid on the hypothesis of uniform density, and arrived at the conclusion that on this hypothesis none of the observed capillary phenomena would take place, and that, therefore, Laplace's theory, in which the density is supposed uniform, is not only insufficient but erroneous.

00In particular he maintained that the constant pressure K, which occurs in Laplace's theory, and which on that theory is very large, must be in point of fact very small, but the equation of equilibrium from which he concluded this is itself defective.

00The free portions of the surface then assume new forms depending on the equilibrium of surface-tension.

00In this way he produced a great many of the forms of equilibrium of a liquid under the action of surfacetension alone, and compared them with the results of mathematical investigation.

00Thomson (afterwards Lord Kelvin) investigated the effect of the curvature of the surface of a liquid on the thermal equilibrium between the liquid and the vapour in contact with it.

00The conditions under which the thermal and mechanical equilibrium of two fluids, two mixtures,.

00When a liquid is in thermal and dynamical equilibrium with its vapour, then if p' and x' are the values of p and x for the vapour, and po and Xo those for the liquid, x' - xo=JL - p(I/p' - I/pc),.

00The amount of this increase and diminution is too small to be directly measured, though it has a certain theoretical importance in the explanation of the equilibrium of the superficial layer of the liquid where it is inclined to the horizon.

00For if in the triangle abc the side ab is taken so as to represent on a given scale the tension of the surface of contact of the fluids a and b, and if the other sides be and ca are taken so as to represent on the same scale the tensions of the surfaces between b and c and between c and a respectively, then the condition of equilibrium at 0 for the corresponding tensions R, P and Q is that the angle ROP shall be the supplement of abc, POQ of bca, and, therefore, QOR of cab.

00But when the surface-tension of A exceeds the sum of the tensions of the surfaces of contact of B with air and with A, it is impossible to construct the triangle of forces, so that equilibrium becomes impossible.

00If a and b are the two fluids and c the solid then the equilibrium of the tensions at the point 0 depends only on that of thin components parallel to the surface, because the surface-tensions normal to the surface are balanced by the resistance of the solid.

00If the tension of the surface between the solid and one of the fluids exceeds the sum of the other two tensions, the point of contact will not be in equilibrium, but will be dragged towards the side on which the tension is greatest.

00When the liquid is in equilibrium it forms a thin film, the outer edge of which is all of the same thickness.

00The surface of the liquid will therefore no longer be in equilibrium, and a current will be formed at and near the surface from the alcohol to the surrounding water, and this current will go on as long as there is more alcohol at one part of the surface than at another.

00The equilibrium of C is therefore stable.

00It is easy to show that if C had been placed in any other position than the middle, its equilibrium would have been stable.

00Hence if the length of the cylindric film is less than its circumference, it is in stable equilibrium.

00But if the length of the cylindric film is greater than its circumference, and if we suppose the disk C to be placed midway between A and B, and to be moved towards A, the pressure on the side next A will diminish, and that on the side next B will increase, so that the resultant force will tend to increase the displacement, and the equilibrium of the disk C is therefore unstable.

00Hence the equilibrium of a cylindric film whose length is greater than its circumference is unstable.

00The capillary tension endeavours to contract the surface of the fluid; so that the stability, or instability, of the cylindrical form of equilibrium depends upon whether the surface (enclosing a given volume) be greater or less respectively after the displacement than before.

00Accordingly, the equilibrium is stable if A be less than the circumference; but unstable if A be greater than the circumference of the cylinder.

00The smaller the causes by which the original equilibrium is upset, the more will the cylindrical mass tend to divide itself regularly into portions whose length is equal to 4.5 times the diameter.

00The disturbances by which equilibrium is upset are impressed upon the fluid as it leaves the aperture, and the continuous portion of the jet represents the distance travelled during the time necessary to produce disintegration.

00This catenoid, however, is in stable equilibrium only when the portion considered is such that the tangents to the catenary at its extremities intersect before they reach the directrix.

00To prove this, let us consider the catenary as the form of equilibrium of a chain suspended between two fixed points A and B.

00Now if the pressure is equal on both sides of a liquid film, and if its mean curvature is zero, it will be in equilibrium.

00Hence the film in the form of the catenoid which is nearest the axis is in unstable equilibrium under the condition that it is exposed to equal pressures within and without.

00If, however, the circular ends of the catenoid are closed with solid disks, so that the volume of air contained between these disks and the film is determinate, the film will be in stable equilibrium however large a portion of the catenary it may consist of.

00The surface of separation is in this case horizontal and stable, so that the equilibrium is established of itself.

00The equilibrium of the fluids would now be unstable if it were not for the tension of the surface which separates them, and which, when the orifice of the vessel is not too large, continues to preserve the stability of the equilibrium.

00When the equilibrium at last becomes unstable, the destruction of equilibrium takes place by the lighter fluid ascending in one part of the orifice and the heavier descending in the other.

00If this quantity is of the same sign as z, the displacement will be increased, and the equilibrium will be unstable.

00If it is of the opposite sign from z, the equilibrium will be stable.

00Under these conditions the equilibrium is stable for all small displacements of the surface.

00In Penaud's artificial bird the equilibrium is secured by the addition of a tail.

00The steam engine weighed about 7 lb per horse-power, but the equilibrium of the apparatus 'was defective.

00Largely with the view of studying the problem of maintaining equilibrium, several experimenters, including Otto Lilienthal, Percy Filcher and Octave Chanute, cultivated gliding flight by means of aeroplanes capable of sustaining a man.

00This suggested some tendency to return to a state of equilibrium as between urban and rural districts.

00This excess of male births, which is usual, has been ascertained to find its equilibrium, through a higher rate of infant mortality among the males, about the tenth year of life, and is finally changed by perilous male occupations and other causes, including the stronger tendency of males to emigration.

00equilibrium at the very outset incited sympathy, while his wit and humour made him the centre of every circle within which he moved.

00The nicely balanced conditions of solution produce a state of unstable equilibrium, with the result that internal streaming movements and changes of shape and changes of position in the model simulate closely the corresponding manifestations in real protoplasm.

00The model has no power of recuperation; in a comparatively short time equilibrium is restored and the resemblance with protoplasm disappears.

00There is headache, which, with the continuance of the drug, becomes exceedingly severe, the vision and equilibrium are affected, and there is often some gastro-intestinal irritation.

00During the civil war which broke out between the kings of Neustria and Austrasia, his policy was to try to maintain a state of equilibrium.

00root um-, stand, or cause to stand), the branch of mechanics which discusses the conditions of rest or equilibrium of forces (see Mechanics).

00The principles of compensation and equilibrium are found also in the physical universe, the product of matter and force, whose cause is God.

00Force, naturally expansive and operating on the homogeneous atoms which constitute elemental matter, is subject to the law of equilibrium, or equivalence of action and reaction.

00Moral and political phenomena are the result of the opposing forces of progress and preservation, and their perfection lies in the fulfilment of the law of equilibrium or universal harmony.

00In an additional five thousand, a similar equilibrium will obtain in the physical sphere, which will then itself pass away.

00This is the law of convective equilibrium.

00Convective equilibrium, which depends upon it, gives far too steep a temperature gradient, for it yields a temperature of 6000° only 200 m.

00The space within is filled with radiations corresponding to this temperature, and these attain a certain equilibrium which permits the energy of radiation to be spoken of as a whole, as a scalar quantity, without express reference to the propagation or interference of the waves of which it is composed.

00Manuel Murillo Toro (1872-1874) and Santiago Perez (1874-1876) saw the country apparently acquiring constitutional equilibrium, and turning its energies to the development of its matchless resources.

00He investigated the problem by means of the general differential equations of static equilibrium for dams of triangular and rectangular form considered as isotropic elastic solids.

00(2) The beam must take up a definite position of equilibrium for a given small difference of weight in the scale-pans.

00it will sway or vibrate up and down, ultimately coming to rest in its position of equilibrium.

00the pull of A, the weight, D, and the counterbalance, E, are in equilibrium.

00The disk carries a weighted brass cylinder rigidly attached to it, which is pulled into an oblique position by the steel band until equilibrium is established.

00We may conclude that in polymorphs the substance occurs in different phases (or molecular aggregations), and the equilibrium between these phases follows definite laws, being dependent upon temperature and pressure, and amenable to thermodynamic treatment (cf.

00The line BC, representing the equilibrium between monoclinic and liquid sulphur, is thermodynamically calculable; the point B is found to correspond to 131Ã‚° and 400 atmospheres.

00Of especial interest is the 0 curve BD; along this line liquid and rhombic sulphur are in equilibrium, which means that at above 131Ã‚° and 400 atmospheres the rhombic (and not the monoclinic) variety would separate from liquid sulphur.

00The study of marine life has in recent years become more general, and has become associated with very precise investigations into the chemical composition of sea-water, changes in chemical equilibrium, the effect of variations in salinity and temperature, the processes set up by marine bacteria, and so on.

00This equilibrium pressure is called the osmotic pressure of the solution, and thermodynamic theory shows that, in an ideal case of perfect separation between solvent and solute, it should have the same value as the pressure which a number of molecules equal to the number of solute molecules in the solution would exert if they could exist as a gas in a space equal to the volume of the solution, provided that the space was large enough (i.e.

00The ordinary laws of chemical equilibrium have been applied to the case of the dissociation of a substance into its ions.

00When there is no magnetization, c c the yoke is in equilibrium; but as soon as the current Ã‚°'Ã‚°Ã‚° is turned on the block C is drawn downwards as far as the screw R will allow, for, though the attractive forces F between B and C and between B' and C' are equal, the former has a greater moment.

00The investigation of the figure of equilibrium of a rotating fluid mass engaged the persistent attention of Laplace.

00An important development, covering such diverse matters as the equilibrium of forces and the algebraic theory of complex numbers (Ã¯¿½ 66), has relation to cases where the numerical quantity has direction as well as magnitude.

00As a part of these discoveries has arisen another but kindred doctrine that of hormones (Starling), juices prepared, not for excretion, not even for partial excretion, but for the fulfilment of physiological equilibrium.

00Referred to three fixed coordinate axes, a fluid, in which the pressure is p, the density p, and X, Y, Z the components of impressed force per unit mass, requires for the equilibrium of the part filling a fixed surface S, on resolving parallel to Ox, f flpdS = f f fpXdxdydz, (I) where 1, m, n denote the direction cosines of the normal drawn outward of the surface S.

00- The principle of Archimedes in § 12 leads immediately to the conditions of equilibrium of a body supported freely in fluid, like a fish in water or a balloon in the air, or like a ship (fig.

00The body is in equilibrium under two forces: - (i.) its weight W acting vertically downward body, and (ii.) the buoyancy of the of the displaced fluid, and acting B, the C.G.

00If G was above M, the tangent drawn from G to the evolute of B, and normal to the curve of buoyancy, would give the vertical in a new position of equilibrium.

00An elongated shot is made to preserve its axial flight through the air by giving it the spin sufficient for stability, without which it would turn broadside to its advance; a top in the same way is made to stand upright on the point in the position of equilibrium, unstable statically but dynamically stable if the spin is sufficient; and the investigation proceeds in the same way for the two problems (see Gyroscope).

00The majority of thermodynamical problems may be treated without any reference to entropy, but it affords a convenient method of expression in abstract thermodynamics, especially in the consideration of irreversible processes and in reference to the conditions of equilibrium of heterogeneous systems.

00A more common method of procedure, however, is to infer the general relations of the thermodynamic potential from a consideration of the phenomena of equilibrium.

00But at this critical moment the leading companies of the Hessian infantry arrived, re-established the equilibrium (though not before four Prussian batteries had been temporarily overrun by the enemy), and a most obstinate fight ensued.

00The plummet is now placed in distilled water at 15Ã‚°, and the beam brought to equilibrium by means of a rider, which we shall call I, hung on a hook; other riders are provided,;nth and ii b th respectively of I.

00The impact will accordingly result in all the molecules being set into motion, and by the time that the masses have ceased impinging on one another the molecules of which they are composed will be performing oscillations about their positions of equilibrium.

00Let a frame (without redundant members), and the