# Equilibrium Sentence Examples

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

**equilibrium**.**Equilibrium**is the product of an axiomatic system.When 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.As 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**.The 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.Let 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.But 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.The 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.The simplest case to consider is that of

**equilibrium**between solid and liquid, or liquid and vapour.The condition of stable

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

**equilibrium**is that G should be a minimum, for which reason it has been called the " thermodynamic potential at constant 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.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".AdvertisementAssuming 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".Although 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.A 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.The 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.AdvertisementIn 1738 appeared his Hydrodynamica, in which the

**equilibrium**, the pressure, the reaction and varied velocities of fluids are considered both theoretically and practically.The 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.On 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.As 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.If, 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.AdvertisementAt 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.In 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.The 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.When 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.Suppose for instance that two molecules, when at rest in

**equilibrium**, are at a distance a apart.As 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.At 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.A 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.When 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.By this time, twenty years of solitary confinement had disturbed Ivan's mental

**equilibrium**, though he does not seem to have been actually insane.A severe property-tax and an increase of customs duties in 1879 only for a moment achieved financial

**equilibrium**.Let 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.A 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.Even then, however, the results fell far short of anticipation, and the armies settled down into

**equilibrium**again.The 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.This 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.England and Russia together were to maintain the

**equilibrium**of the world.Legendre'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.In a word, this diet disturbed the

**equilibrium**of the state by enfeebling and degrading the middle classes.Seriously 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.He considered that the

**equilibrium**of Europe would be irretrievably upset were the Russian boundaries to be pushed into the heart of Germany.What 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.The 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.'A discussion of the

**equilibrium**of Saturn's rings led him to conclude in 1855 that they must be of a fluid nature.The 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.When, 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**.But 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.The 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.In 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.The 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.Willard 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.When the limit is reached the solution is said to be saturated, and the system is in

**equilibrium**.If 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.It 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.**Equilibrium**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.A 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.Thus 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.It 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.If 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.Thus, 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.To 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.We then have in

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

**equilibrium**, and, as more heat enters, the temperature rises and the vapour-pressure rises with it.But, if we fix arbitrarily the temperature the pressure of

**equilibrium**can have one value only.The 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.If 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.The phase rule combined with the latent heat equation contains the whole theory of chemical and physical

**equilibrium**.**Equilibrium**between these phases is obtained at the freezing point of the saturated solution under the pressure of the vapour.In 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.At the cryohydric point 0 we have four phases in

**equilibrium**at a definite pressure, temperature and composition of the liquid phase.Again, 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.If the salt crystallizes with a certain amount of water as well as with none, we get a second point of

**equilibrium**between four phases.At 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.When this process is complete the temperature rises, and we pass along a new curve giving the

**equilibrium**between anhydrous crystals, solution and vapour.In 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.When a crystal of the solid phase is present the

**equilibrium**of a solution is given by the solubility curves we have studied.Between 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.Hence 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.If 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.The 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.The 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**.It 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.The 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.The 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.But 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.In 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.Therefore 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.When 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.Similar '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.The 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.Although 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**.They 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.Hence 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.The 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.The 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.The 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.From 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.If 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**.Stevinus also distinguished stable from unstable

**equilibrium**.He proved the law of the

**equilibrium**on an inclined plane.One 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.If 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.Here 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**.Debray (ibid., 1879, 88, p. 1341), who found that at about 150o C. a condition of

**equilibrium**is reached.With 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.This 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.In intimate connexion with the Gottorp affair stood the question of the political

**equilibrium**of the north.His 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.He 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.By 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.By 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.Castlereagh, 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.'Afterwards, 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.The 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.His 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.The Calorimeter Was Suspended By A Steel Wire, The Torsion Of Which Made The

**Equilibrium**Stable.It 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.At the freezing-point, the solution must have the same vapour-pressure as the solid solvent, with which it is in

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

**equilibrium**of vapour-pressure.In 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.The 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.The 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.If 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**.As a simple example of the geometrical method of treating statical problems we may consider the

**equilibrium**of a particle on a rough inclined plane.If the conditions of

**equilibrium**require an obliquity greater than this, sliding will take place.The conditions of

**equilibrium**are X=o, Y=o, Z=o.The 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.As exceptional cases the system may reduce to a couple, or it may be in

**equilibrium**.Again 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.The fact that three independent conditions must hold for

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

**equilibrium**.Hence the necessary and sufficient conditions of

**equilibrium**are that the force-polygon and the funicular should both be closed.It 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.Hence the forces x, y, x, y are in

**equilibrium**.The 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.Hence 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.For 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.For 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.A 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.The 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.Since 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.An 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.If 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.Conversely, 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**.An 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.The 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.The displacement of G is at right angles to JG; this shows that for

**equilibrium**JG must be vertical.Again, 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.For

**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.If 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.Consider, 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.We 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.It 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.We assume that in limiting

**equilibrium**we have F tsR, everywhere, where u is the coefficient of friction.In the case of a chain hanging freely under gravity it is usually convenient to formulate the conditions of

**equilibrium**of a finite portion PQ.For investigations relating to the

**equilibrium**of a string in three dimensions we must refer to the textbooks.Hence 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).The position x=o is one of

**equilibrium**, but it is unstable.This 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**.The body now passes once (at most) through its

**equilibrium**position, and the vibration is therefore styled aperiodic.According to dAlemberts formulation, the extraneous forces together with the effective forces reversed fulfil the statical conditions of

**equilibrium**.As 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.A 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.In 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.Hence 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.In 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,..Every 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.Support 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.The 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.In 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.For 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.It 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.Now 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.Using 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.As 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.The 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.**Equilibrium**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.The five great powers, held in

**equilibrium**by Lorenzo de' Medici, dreamed that the peninsula could be maintained in statu quo by diplomacy.While 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.Under the pressure of war and other emergencies it has been found impossible to maintain this desirable

**equilibrium**.Plana (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**.Of 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."But 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.In 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.The 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.In this state of

**equilibrium**the great powers of Anterior Asia remained during the first half of the 6th century.In 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**.The 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.The condition of

**equilibrium**is that this expression (which we may for the sake of distinctness call the potential energy) shall be a minimum.He 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.In 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.The free portions of the surface then assume new forms depending on the

**equilibrium**of surface-tension.In 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.Thomson (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.When 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),.The 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.For 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.But 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.If 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.If 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.When the liquid is in

**equilibrium**it forms a thin film, the outer edge of which is all of the same thickness.The 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.The

**equilibrium**of C is therefore stable.It is easy to show that if C had been placed in any other position than the middle, its

**equilibrium**would have been stable.Hence if the length of the cylindric film is less than its circumference, it is in stable

**equilibrium**.But 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.Hence the

**equilibrium**of a cylindric film whose length is greater than its circumference is unstable.The 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.Accordingly, the

**equilibrium**is stable if A be less than the circumference; but unstable if A be greater than the circumference of the cylinder.The 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.The 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.This 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.Now if the pressure is equal on both sides of a liquid film, and if its mean curvature is zero, it will be in

**equilibrium**.Hence 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.If, 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.The surface of separation is in this case horizontal and stable, so that the

**equilibrium**is established of itself.The

**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**.When 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.If this quantity is of the same sign as z, the displacement will be increased, and the

**equilibrium**will be unstable.If it is of the opposite sign from z, the

**equilibrium**will be stable.Under these conditions the

**equilibrium**is stable for all small displacements of the surface.In Penaud's artificial bird the

**equilibrium**is secured by the addition of a tail.The steam engine weighed about 7 lb per horse-power, but the

**equilibrium**of the apparatus 'was defective.Largely 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.This suggested some tendency to return to a state of

**equilibrium**as between urban and rural districts.This 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.The 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.The model has no power of recuperation; in a comparatively short time

**equilibrium**is restored and the resemblance with protoplasm disappears.There 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.During the civil war which broke out between the kings of Neustria and Austrasia, his policy was to try to maintain a state of

**equilibrium**.The principles of compensation and

**equilibrium**are found also in the physical universe, the product of matter and force, whose cause is God.Force, 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.Moral 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.In an additional five thousand, a similar

**equilibrium**will obtain in the physical sphere, which will then itself pass away.This is the law of convective

**equilibrium**.The 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.Manuel 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.He 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.The 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.The 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.The ordinary laws of chemical

**equilibrium**have been applied to the case of the dissociation of a substance into its ions.The investigation of the figure of

**equilibrium**of a rotating fluid mass engaged the persistent attention of Laplace.As 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**.If 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**.An 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).The 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.A more common method of procedure, however, is to infer the general relations of the thermodynamic potential from a consideration of the phenomena of

**equilibrium**.But 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.The 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**.Let a frame (without redundant members), and the external forces which keep it in

**equilibrium**, be represented by a diagram constituting one of these two plane figures, then the lines in the other plane figure or the reciprocal will represent in direction and magnitude the forces between the joints of the frame, and, consequently, the stress on each member, as will now be explained.The external forces on a frame or bridge in

**equilibrium**under those forces may, by a well-known proposition in statics, be represented by a closed polygon, each side of which is parallel to one force, and represents the force in magnitude as well as in direction.The true philosopher, therefore, is not the Pyrrhonist, trying to maintain an impossible

**equilibrium**or suspense of judgment, but the Academic, yielding gracefully to the impressions or maxims which he finds, as matter of fact, to have most sway over himself.'The phenomena of solution and of vapour pressure constitute cases of

**equilibrium**, and conform to the laws deduced by Gibbs, which thus yield a valuable method of investigating and classifying the equilibria of solutions.The phase rule combined with the latent heat equation enables us to trace the general phenomena of

**equilibrium**in solutions, and to elucidate and classify cases even of great complexity.Such a system is in the thermo dynamic

**equilibrium**.Thus, if the poise be at the zero end of the steelyard at the left-hand side of the machine, when the load comes upon the platform the screwed shaft carries the poise along the steelyard till

**equilibrium**is established, and the end of the steelyard drops.One of these was the notion of " punctuated

**equilibrium**".Some simple consequences The simplest attractor in the quotient system is an

**equilibrium**.Disclosure is the name of the game; emotional

**equilibrium**being reached when everybody 's contribution is reciprocated by all others.Sedimentation

**equilibrium**A lower rotor speed is used than is required for complete sedimentation of the sample.These are all approximately in thermal

**equilibrium**, the small deviations from which produce the weather.Adding a catalyst does n't produce any greater percentage of sulfur trioxide in the

**equilibrium**mixture.Exercise helps to regulate our homeostasis

**equilibrium**by ensuring that our hormones are functioning properly.Alcohol rehab is a process where an alcohol dependant person restores his/her physiological, psychological and mental

**equilibrium**by participation in a structured and professionally monitored program.In addition to being a pet, pond fish help to maintain the

**equilibrium**of the water system.This helps maintain the delicate

**equilibrium**between human and nature.Usually intravenous rehydration is very effective, allowing the child's body to return to its normal fluid

**equilibrium**.Diagnosis in these situations deserves the attention and care of a doctor with specialized skills in diseases of the ear, nose, throat,

**equilibrium**, and neurological system.They are a means of integrating recent and past learning and of establishing psychological

**equilibrium**.Other children may not be able to recover their

**equilibrium**without help from an adult.Babies are unable to keep themselves in a state of

**equilibrium**, as they lack the skills to control either the intensity or the duration of those emotions.An air ionizer compensates for this lack of

**equilibrium**and effectively eliminates most harmful airborne particles.Strive for emotional

**equilibrium**Cancer, and you'll go very far indeed!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.He rose despite his whirling

**equilibrium**.Gregory initiated the policy of establish ing an

**equilibrium**between the parties, which was carried ou by his successor Nicholas III.Meanwhile the podest still subsists; but he is no longer equal to the task d maintaining an

**equilibrium**of forces.The 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**.These 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**.What 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.A fixed weight is placed on one coil and the current is varied gradually until the balance is just in

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

**equilibrium**.Archimedes 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**.In 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.When 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**.It 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.He 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.These 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.Take 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.Then, since these three forces maintain

**equilibrium**, and R makes equal angles with P and Q, therefore P and Q must be equal.This 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.As 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.This 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.For 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.When 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.So 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.But 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.The 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.These 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).With 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?**Equilibrium**and Stability of a Ship or Floating Body.In 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.Peace 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.The 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.A 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).An example alone is needed finally to destroy the

**equilibrium**.The 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.To 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.Tirpitz 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.If a force Q acting at R maintains

**equilibrium**, QR/4 = (P - p)r =T.If 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.But, taking the history of the republic as a whole, that

**equilibrium**between the several organs of the government which the Constitution was intended to secure has been substantially maintained.But such a pressure represents the

**equilibrium**osmotic pressure discussed above.To the bottom of each cylinder is rigidly attached a heavy solid cylinder of lead, and these are the regulators of the position of

**equilibrium**of the cylinders when they rotate under the action of the load..The resistance to slipping of a flat belt on a pulley may be obtained by considering the

**equilibrium**of a small arc of the pulley surface subtending an angle dB at the centre, and having tensions T and T+dT at its extremities.Among the new truths detected by him was the valuable mechanical principle that if any number of bodies be so connected that, by their motion, their centre of gravity can neither ascend nor descend, then those bodies are in

**equilibrium**.There were no longer two rival monarchies in France; the feudal

**equilibrium**was destroyed, to the advantage of the duchy of France.The act of presentation (Vorstellen) then becomes partly transformed into an effort, and its product, the idea, becomes in the same proportion less and less intense till a position of

**equilibrium**is reached; and then at length the remainders coalesce.We have thus a statics and a mechanics of mind which investigate respectively the conditions of

**equilibrium**and of movement among presentations.