The Various States of Matter

THE STATICS OF FLUIDS

The division of bodies into solid, liquid, and gas are very important in daily life. However, it has long since lost their absolute value from the scientific point of view.

So far as concerns the liquid and gaseous states particularly,

The already antiquated researches of Andrews confirmed the ideas of Cagniard de la Tour.

• It established the continuity of the liquid and gaseous states.

A group of physical studies has thus been constituted on what may be called the statics of fluids, in which we examine the relations existing between the pressure, the volume, and the temperature of bodies, and in which are comprised, under the term fluid, gases as well as liquids.

These researches deserve attention by their interest and the generality of the results to which they have led. They also give a remarkable example of the happy effects which may be obtained by the combined employment of the various methods of investigation used in exploring the domain of nature.

Thermodynamics has allowed us to obtain numerical relations between the various coefficients, and atomic hypotheses have led to the establishment of one capital relation, the characteristic equation of fluids.

On the other hand, experiment in which the progress made in the art of measurement has been utilized, has furnished the most valuable information on all the laws of compressibility and dilatation.

The classical work of Andrews was not very wide.

• He did not go much beyond pressures close to the normal and ordinary temperatures. Of late years several very interesting and peculiar cases have been examined by MM. Cailletet, Mathias, Batelli, Leduc, P. Chappuis, and other physicists.

Sir W. Ramsay and Mr S. Young have made known the isothermal diagrams[6] of a certain number of liquid bodies at the ordinary temperature. They have thus been able, while keeping to somewhat restricted limits of temperature and pressure, to touch upon the most important questions, since they found themselves in the region of the saturation curve and of the critical point.

But the most complete and systematic body of researches is due to M. Amagat, who undertook the study of a certain number of bodies, some liquid and some gaseous, extending the scope of his experiments so as to embrace the different phases of the phenomena and to compare together, not only the results relating to the same bodies, but also those concerning different bodies which happen to be in the same conditions of temperature and pressure, but in very different conditions as regards their critical points.

From the experimental point of view, M. Amagat has been able, with extreme skill, to conquer the most serious difficulties. He has managed to measure with precision pressures amounting to 3000 atmospheres, and also the very small volumes then occupied by the fluid mass under consideration. This last measurement, which necessitates numerous corrections, is the most delicate part of the operation. These researches have dealt with a certain number of different bodies.

Those relating to carbonic acid and ethylene take in the critical point. Others, on hydrogen and nitrogen, for instance, are very extended. Others, again, such as the study of the compressibility of water, have a special interest, on account of the peculiar properties of this substance. M. Amagat, by a very concise discussion of the experiments, has also been able to definitely establish the laws of compressibility and dilatation of fluids under constant pressure, and to determine the value of the various coefficients as well as their variations.

It should be possible to condense all these results into a single formula representing the volume, the temperature, and the pressure. Rankin and, subsequently, Recknagel, and then Hirn, formerly proposed formulas of that kind; but the most famous, the one which first appeared to contain in a satisfactory manner all the facts which experiments brought to light and led to the production of many others, was the celebrated equation of Van der Waals.

Professor Van der Waals arrived at this relation by relying upon considerations derived from the kinetic theory of gases. If we keep to the simple idea at the bottom of this theory, we at once demonstrate that the gas ought to obey the laws of Mariotte and of Gay-Lussac, so that the characteristic equation would be obtained by the statement that the product of the number which is the measure of the volume by that which is the measure of the pressure is equal to a constant coefficient multiplied by the degree of the absolute temperature. But to get at this result we neglect two important factors.

We do not take into account, in fact, the attraction which the molecules must exercise on each other. Now, this attraction, which is never absolutely non-existent, may become considerable when the molecules are drawn closer together; that is to say, when the compressed gaseous mass occupies a more and more restricted volume. On the other hand, we assimilate the molecules, as a first approximation, to material points without dimensions; in the evaluation of the path traversed by each molecule no notice is taken of the fact that, at the moment of the shock, their centres of gravity are still separated by a distance equal to twice the radius of the molecule.

M. Van der Waals has sought out the modifications which must be introduced into the simple characteristic equation to bring it nearer to reality. He extends to the case of gases the considerations by which Laplace, in his famous theory of capillarity, reduced the effect of the molecular attraction to a perpendicular pressure exercised on the surface of a liquid. This leads him to add to the external pressure, that due to the reciprocal attractions of the gaseous particles. On the other hand, when we attribute finite dimensions to these particles, we must give a higher value to the number of shocks produced in a given time, since the effect of these dimensions is to diminish the mean path they traverse in the time which elapses between two consecutive shocks.

The calculation thus pursued leads to our adding to the pressure in the simple equation a term which is designated the internal pressure, and which is the quotient of a constant by the square of the volume; also to our deducting from the volume a constant which is the quadruple of the total and invariable volume which the gaseous molecules would occupy did they touch one another.

The experiments fit in fairly well with the formula of Van der Waals, but considerable discrepancies occur when we extend its limits, particularly when the pressures throughout a rather wider interval are considered; so that other and rather more complex formulas, on which there is no advantage in dwelling, have been proposed, and, in certain cases, better represent the facts.

But the most remarkable result of M. Van der Waals’ calculations is the discovery of corresponding states. For a long time physicists spoke of bodies taken in a comparable state. Dalton, for example, pointed out that liquids have vapour-pressures equal to the temperatures equally distant from their boiling-point; but that if, in this particular property, liquids were comparable under these conditions of temperature, as regards other properties the parallelism was no longer to be verified. No general rule was found until M. Van der Waals first enunciated a primary law, viz., that if the pressure, the volume, and the temperature are estimated by taking as units the critical quantities, the constants special to each body disappear in the characteristic equation, which thus becomes the same for all fluids.

The words corresponding states thus take a perfectly precise signification. Corresponding states are those for which the numerical values of the pressure, volume, and temperature, expressed by taking as units the values corresponding to the critical point, are equal; and, in corresponding states any two fluids have exactly the same properties.

M. Natanson, and subsequently P. Curie and M. Meslin, have shown by various considerations that the same result may be arrived at by choosing units which correspond to any corresponding states; it has also been shown that the theorem of corresponding states in no way implies the exactitude of Van der Waals’ formula. In reality, this is simply due to the fact that the characteristic equation only contains three constants.

The philosophical importance and the practical interest of the discovery nevertheless remain considerable. As was to be expected, numbers of experimenters have sought whether these consequences are duly verified in reality. M. Amagat, particularly, has made use for this purpose of a most original and simple method. He remarks that, in all its generality, the law may be translated thus: If the isothermal diagrams of two substances be drawn to the same scale, taking as unit of volume and of pressure the values of the critical constants, the two diagrams should coincide; that is to say, their superposition should present the aspect of one diagram appertaining to a single substance. Further, if we possess the diagrams of two bodies drawn to any scales and referable to any units whatever, as the changes of units mean changes in the scale of the axes, we ought to make one of the diagrams similar to the other by lengthening or shortening it in the direction of one of the axes. M. Amagat then photographs two isothermal diagrams, leaving one fixed, but arranging the other so that it may be free to turn round each axis of the co-ordinates; and by projecting, by means of a magic lantern, the second on the first, he arrives in certain cases at an almost complete coincidence.

This mechanical means of proof thus dispenses with laborious calculations, but its sensibility is unequally distributed over the different regions of the diagram. M. Raveau has pointed out an equally simple way of verifying the law, by remarking that if the logarithms of the pressure and volume are taken as co-ordinates, the co-ordinates of two corresponding points differ by two constant quantities, and the corresponding curves are identical.

From these comparisons, and from other important researches, among which should be particularly mentioned those of Mr S. Young and M. Mathias, it results that the laws of corresponding states have not, unfortunately, the degree of generality which we at first attributed to them, but that they are satisfactory when applied to certain groups of bodies.[7]

If in the study of the statics of a simple fluid the experimental results are already complex, we ought to expect much greater difficulties when we come to deal with mixtures; still the problem has been approached, and many points are already cleared up.

Mixed fluids may first of all be regarded as composed of a large number of invariable particles. In this particularly simple case M. Van der Waals has established a characteristic equation of the mixtures which is founded on mechanical considerations. Various verifications of this formula have been effected, and it has, in particular, been the object of very important remarks by M. Daniel Berthelot.

It is interesting to note that thermodynamics seems powerless to determine this equation, for it does not trouble itself about the nature of the bodies obedient to its laws; but, on the other hand, it intervenes to determine the properties of coexisting phases. If we examine the conditions of equilibrium of a mixture which is not subjected to external forces, it will be demonstrated that the distribution must come back to a juxtaposition of homogeneous phases; in a given volume, matter ought so to arrange itself that the total sum of free energy has a minimum value. Thus, in order to elucidate all questions relating to the number and qualities of the phases into which the substance divides itself, we are led to regard the geometrical surface which for a given temperature represents the free energy.

I am unable to enter here into the detail of the questions connected with the theories of Gibbs, which have been the object of numerous theoretical studies, and also of a series, ever more and more abundant, of experimental researches. M. Duhem, in particular, has published, on the subject, memoirs of the highest importance, and a great number of experimenters, mostly scholars working in the physical laboratory of Leyden under the guidance of the Director, Mr Kamerlingh Onnes, have endeavoured to verify the anticipations of the theory.

We are a little less advanced as regards abnormal substances; that is to say, those composed of molecules, partly simple and partly complex, and either dissociated or associated. These cases must naturally be governed by very complex laws. Recent researches by MM. Van der Waals, Alexeif, Rothmund, Künen, Lehfeld, etc., throw, however, some light on the question.

The daily more numerous applications of the laws of corresponding states have rendered highly important the determination of the critical constants which permit these states to be defined. In the case of homogeneous bodies the critical elements have a simple, clear, and precise sense; the critical temperature is that of the single isothermal line which presents a point of inflexion at a horizontal tangent; the critical pressure and the critical volume are the two co-ordinates of this point of inflexion.

The three critical constants may be determined, as Mr S. Young and M. Amagat have shown, by a direct method based on the consideration of the saturated states. Results, perhaps more precise, may also be obtained if one keeps to two constants or even to a single one—temperature, for example—by employing various special methods. Many others, MM. Cailletet and Colardeau, M. Young, M.J. Chappuis, etc., have proceeded thus.

The case of mixtures is much more complicated. A binary mixture has a critical space instead of a critical point. This space is comprised between two extreme temperatures, the lower corresponding to what is called the folding point, the higher to that which we call the point of contact of the mixture. Between these two temperatures an isothermal compression yields a quantity of liquid which increases, then reaches a maximum, diminishes, and disappears. This is the phenomenon of retrograde condensation. We may say that the properties of the critical point of a homogeneous substance are, in a way, divided, when it is a question of a binary mixture, between the two points mentioned.

Calculation has enabled M. Van der Waals, by the application of his kinetic theories, and M. Duhem, by means of thermodynamics, to foresee most of the results which have since been verified by experiment. All these facts have been admirably set forth and systematically co-ordinated by M. Mathias, who, by his own researches, moreover, has made contributions of the highest value to the study of questions regarding the continuity of the liquid and gaseous states.

The further knowledge of critical elements has allowed the laws of corresponding states to be more closely examined in the case of homogeneous substances. It has shown that, as I have already said, bodies must be arranged in groups, and this fact clearly proves that the properties of a given fluid are not determined by its critical constants alone, and that it is necessary to add to them some other specific parameters; M. Mathias and M. D. Berthelot have indicated some which seem to play a considerable part.

It results also from this that the characteristic equation of a fluid cannot yet be considered perfectly known. Neither the equation of Van der Waals nor the more complicated formulas which have been proposed by various authors are in perfect conformity with reality. We may think that researches of this kind will only be successful if attention is concentrated, not only on the phenomena of compressibility and dilatation, but also on the calorimetric properties of bodies. Thermodynamics indeed establishes relations between those properties and other constants, but does not allow everything to be foreseen.

Several physicists have effected very interesting calorimetric measurements, either, like M. Perot, in order to verify Clapeyron’s formula regarding the heat of vaporization, or to ascertain the values of specific heats and their variations when the temperature or the pressure happens to change. M. Mathias has even succeeded in completely determining the specific heats of liquefied gases and of their saturated vapours, as well as the heat of internal and external vaporization.

§ 2. THE LIQUEFACTION OF GASES, AND THE PROPERTIES OF BODIES AT A LOW TEMPERATURE

The scientific advantages of all these researches have been great, and, as nearly always happens, the practical consequences derived from them have also been most important. It is owing to the more complete knowledge of the general properties of fluids that immense progress has been made these last few years in the methods of liquefying gases.

From a theoretical point of view the new processes of liquefaction can be classed in two categories. Linde’s machine and those resembling it utilize, as is known, expansion without any notable production of external work. This expansion, nevertheless, causes a fall in the temperature, because the gas in the experiment is not a perfect gas, and, by an ingenious process, the refrigerations produced are made cumulative.

Several physicists have proposed to employ a method whereby liquefaction should be obtained by expansion with recuperable external work. This method, proposed as long ago as 1860 by Siemens, would offer considerable advantages. Theoretically, the liquefaction would be more rapid, and obtained much more economically; but unfortunately in the experiment serious obstacles are met with, especially from the difficulty of obtaining a suitable lubricant under intense cold for those parts of the machine which have to be in movement if the apparatus is to work.

M. Claude has recently made great progress on this point by the use, during the running of the machine, of the ether of petrol, which is uncongealable, and a good lubricant for the moving parts. When once the desired region of cold is reached, air itself is used, which moistens the metals but does not completely avoid friction; so that the results would have remained only middling, had not this ingenious physicist devised a new improvement which has some analogy with superheating of steam in steam engines. He slightly varies the initial temperature of the compressed air on the verge of liquefaction so as to avoid a zone of deep perturbations in the properties of fluids, which would make the work of expansion very feeble and the cold produced consequently slight. This improvement, simple as it is in appearance, presents several other advantages which immediately treble the output.

The special object of M. Claude was to obtain oxygen in a practical manner by the actual distillation of liquid air. Since nitrogen boils at -194° and oxygen at -180.5° C., if liquid air be evaporated, the nitrogen escapes, especially at the commencement of the evaporation, while the oxygen concentrates in the residual liquid, which finally consists of pure oxygen, while at the same time the temperature rises to the boiling-point (-180.5° C.) of oxygen. But liquid air is costly, and if one were content to evaporate it for the purpose of collecting a part of the oxygen in the residuum, the process would have a very poor result from the commercial point of view. As early as 1892, Mr Parkinson thought of improving the output by recovering the cold produced by liquid air during its evaporation; but an incorrect idea, which seems to have resulted from certain experiments of Dewar—the idea that the phenomenon of the liquefaction of air would not be, owing to certain peculiarities, the exact converse of that of vaporization—led to the employment of very imperfect apparatus. M. Claude, however, by making use of a method which he calls the reversal [8] method, obtains a complete rectification in a remarkably simple manner and under extremely advantageous economic conditions. Apparatus, of surprisingly reduced dimensions but of great efficiency, is now in daily work, which easily enables more than a thousand cubic metres of oxygen to be obtained at the rate, per horse-power, of more than a cubic metre per hour.

It is in England, thanks to the skill of Sir James Dewar and his pupils—thanks also, it must be said, to the generosity of the Royal Institution, which has devoted considerable sums to these costly experiments—that the most numerous and systematic researches have been effected on the production of intense cold. I shall here note only the more important results, especially those relating to the properties of bodies at low temperatures.

Their electrical properties, in particular, undergo some interesting modifications. The order which metals assume in point of conductivity is no longer the same as at ordinary temperatures. Thus at -200° C. copper is a better conductor than silver. The resistance diminishes with the temperature, and, down to about -200°, this diminution is almost linear, and it would seem that the resistance tends towards zero when the temperature approaches the absolute zero. But, after -200°, the pattern of the curves changes, and it is easy to foresee that at absolute zero the resistivities of all metals would still have, contrary to what was formerly supposed, a notable value. Solidified electrolytes which, at temperatures far below their fusion point, still retain a very appreciable conductivity, become, on the contrary, perfect insulators at low temperatures. Their dielectric constants assume relatively high values. MM. Curie and Compan, who have studied this question from their own point of view, have noted, moreover, that the specific inductive capacity changes considerably with the temperature.

In the same way, magnetic properties have been studied. A very interesting result is that found in oxygen: the magnetic susceptibility of this body increases at the moment of liquefaction. Nevertheless, this increase, which is enormous (since the susceptibility becomes sixteen hundred times greater than it was at first), if we take it in connection with equal volumes, is much less considerable if taken in equal masses. It must be concluded from this fact that the magnetic properties apparently do not belong to the molecules themselves, but depend on their state of aggregation.

The mechanical properties of bodies also undergo important modifications. In general, their cohesion is greatly increased, and the dilatation produced by slight changes of temperature is considerable. Sir James Dewar has effected careful measurements of the dilatation of certain bodies at low temperatures: for example, of ice. Changes in colour occur, and vermilion and iodide of mercury pass into pale orange. Phosphorescence becomes more intense, and most bodies of complex structure—milk, eggs, feathers, cotton, and flowers—become phosphorescent. The same is the case with certain simple bodies, such as oxygen, which is transformed into ozone and emits a white light in the process.

Chemical affinity is almost put an end to; phosphorus and potassium remain inert in liquid oxygen. It should, however, be noted, and this remark has doubtless some interest for the theories of photographic action, that photographic substances retain, even at the temperature of liquid hydrogen, a very considerable part of their sensitiveness to light.

Sir James Dewar has made some important applications of low temperatures in chemical analysis; he also utilizes them to create a vacuum. His researches have, in fact, proved that the pressure of air congealed by liquid hydrogen cannot exceed the millionth of an atmosphere. We have, then, in this process, an original and rapid means of creating an excellent vacuum in apparatus of very different kinds—a means which, in certain cases, may be particularly convenient.[9]

Thanks to these studies, a considerable field has been opened up for biological research, but in this, which is not our subject, I shall notice one point only. It has been proved that vital germs—bacteria, for example—may be kept for seven days at -l90°C. without their vitality being modified. Phosphorescent organisms cease, it is true, to shine at the temperature of liquid air, but this fact is simply due to the oxidations and other chemical reactions which keep up the phosphorescence being then suspended, for phosphorescent activity reappears so soon as the temperature is again sufficiently raised. An important conclusion has been drawn from these experiments which affects cosmogonical theories: since the cold of space could not kill the germs of life, it is in no way absurd to suppose that, under proper conditions, a germ may be transmitted from one planet to another.

Among the discoveries made with the new processes, the one which most strikingly interested public attention is that of new gases in the atmosphere. We know how Sir William Ramsay and Dr. Travers first observed by means of the spectroscope the characteristics of the companions of argon in the least volatile part of the atmosphere. Sir James Dewar on the one hand, and Sir William Ramsay on the other, subsequently separated in addition to argon and helium, crypton, xenon, and neon. The process employed consists essentially in first solidifying the least volatile part of the air and then causing it to evaporate with extreme slowness. A tube with electrodes enables the spectrum of the gas in process of distillation to be observed. In this manner, the spectra of the various gases may be seen following one another in the inverse order of their volatility. All these gases are monoatomic, like mercury; that is to say, they are in the most simple state, they possess no internal molecular energy (unless it is that which heat is capable of supplying), and they even seem to have no chemical energy. Everything leads to the belief that they show the existence on the earth of an earlier state of things now vanished. It may be supposed, for instance, that helium and neon, of which the molecular mass is very slight, were formerly more abundant on our planet; but at an epoch when the temperature of the globe was higher, the very speed of their molecules may have reached a considerable value, exceeding, for instance, eleven kilometres per second, which suffices to explain why they should have left our atmosphere. Crypton and neon, which have a density four times greater than oxygen, may, on the contrary, have partly disappeared by solution at the bottom of the sea, where it is not absurd to suppose that considerable quantities would be found liquefied at great depths. [10]

It is probable, moreover, that the higher regions of the atmosphere are not composed of the same air as that around us. Sir James Dewar points out that Dalton’s law demands that every gas composing the atmosphere should have, at all heights and temperatures, the same pressure as if it were alone, the pressure decreasing the less quickly, all things being equal, as its density becomes less. It results from this that the temperature becoming gradually lower as we rise in the atmosphere, at a certain altitude there can no longer remain any traces of oxygen or nitrogen, which no doubt liquefy, and the atmosphere must be almost exclusively composed of the most volatile gases, including hydrogen, which M.A. Gautier has, like Lord Rayleigh and Sir William Ramsay, proved to exist in the air. The spectrum of the Aurora borealis, in which are found the lines of those parts of the atmosphere which cannot be liquefied in liquid hydrogen, together with the lines of argon, crypton, and xenon, is quite in conformity with this point of view. It is, however, singular that it should be the spectrum of crypton, that is to say, of the heaviest gas of the group, which appears most clearly in the upper regions of the atmosphere.

Among the gases most difficult to liquefy, hydrogen has been the object of particular research and of really quantitative experiments. Its properties in a liquid state are now very clearly known. Its boiling-point, measured with a helium thermometer which has been compared with thermometers of oxygen and hydrogen, is -252°; its critical temperature is -241° C.; its critical pressure, 15 atmospheres. It is four times lighter than water, it does not present any absorption spectrum, and its specific heat is the greatest known. It is not a conductor of electricity. Solidified at 15° absolute, it is far from reminding one by its aspect of a metal; it rather resembles a piece of perfectly pure ice, and Dr Travers attributes to it a crystalline structure. The last gas which has resisted liquefaction, helium, has recently been obtained in a liquid state; it appears to have its boiling-point in the neighbourhood of 6° absolute. [11]

§ 3. SOLIDS AND LIQUIDS

The interest of the results to which the researches on the continuity between the liquid and the gaseous states have led is so great, that numbers of scholars have naturally been induced to inquire whether something analogous might not be found in the case of liquids and solids. We might think that a similar continuity ought to be there met with, that the universal character of the properties of matter forbade all real discontinuity between two different states, and that, in truth, the solid was a prolongation of the liquid state.

To discover whether this supposition is correct, it concerns us to compare the properties of liquids and solids. If we find that all properties are common to the two states we have the right to believe, even if they presented themselves in different degrees, that, by a continuous series of intermediary bodies, the two classes might yet be connected. If, on the other hand, we discover that there exists in these two classes some quality of a different nature, we must necessarily conclude that there is a discontinuity which nothing can remove.

The distinction established, from the point of view of daily custom, between solids and liquids, proceeds especially from the difficulty that we meet with in the one case, and the facility in the other, when we wish to change their form temporarily or permanently by the action of mechanical force. This distinction only corresponds, however, in reality, to a difference in the value of certain coefficients. It is impossible to discover by this means any absolute characteristic which establishes a separation between the two classes. Modern researches prove this clearly. It is not without use, in order to well understand them, to state precisely the meaning of a few terms generally rather loosely employed.

If a conjunction of forces acting on a homogeneous material mass happens to deform it without compressing or dilating it, two very distinct kinds of reactions may appear which oppose themselves to the effort exercised. During the time of deformation, and during that time only, the first make their influence felt. They depend essentially on the greater or less rapidity of the deformation, they cease with the movement, and could not, in any case, bring the body back to its pristine state of equilibrium. The existence of these reactions leads us to the idea of viscosity or internal friction.

The second kind of reactions are of a different nature. They continue to act when the deformation remains stationary, and, if the external forces happen to disappear, they are capable of causing the body to return to its initial form, provided a certain limit has not been exceeded. These last constitute rigidity.

At first sight a solid body appears to have a finite rigidity and an infinite viscosity; a liquid, on the contrary, presents a certain viscosity, but no rigidity. But if we examine the matter more closely, beginning either with the solids or with the liquids, we see this distinction vanish.

Tresca showed long ago that internal friction is not infinite in a solid; certain bodies can, so to speak, at once flow and be moulded. M.W. Spring has given many examples of such phenomena. On the other hand, viscosity in liquids is never non-existent; for were it so for water, for example, in the celebrated experiment effected by Joule for the determination of the mechanical equivalent of the caloric, the liquid borne along by the floats would slide without friction on the surrounding liquid, and the work done by movement would be the same whether the floats did or did not plunge into the liquid mass.

In certain cases observed long ago with what are called pasty bodies, this viscosity attains a value almost comparable to that observed by M. Spring in some solids. Nor does rigidity allow us to establish a barrier between the two states. Notwithstanding the extreme mobility of their particles, liquids contain, in fact, vestiges of the property which we formerly wished to consider the special characteristic of solids.

Maxwell before succeeded in rendering the existence of this rigidity very probable by examining the optical properties of a deformed layer of liquid. But a Russian physicist, M. Schwedoff, has gone further, and has been able by direct experiments to show that a sheath of liquid set between two solid cylinders tends, when one of the cylinders is subjected to a slight rotation, to return to its original position, and gives a measurable torsion to a thread upholding the cylinder. From the knowledge of this torsion the rigidity can be deduced. In the case of a solution containing 1/2 per cent. of gelatine, it is found that this rigidity, enormous compared with that of water, is still, however, one trillion eight hundred and forty billion times less than that of steel.

This figure, exact within a few billions, proves that the rigidity is very slight, but exists; and that suffices for a characteristic distinction to be founded on this property. In a general way, M. Spring has also established that we meet in solids, in a degree more or less marked, with the properties of liquids. When they are placed in suitable conditions of pressure and time, they flow through orifices, transmit pressure in all directions, diffuse and dissolve one into the other, and react chemically on each other. They may be soldered together by compression; by the same means alloys may be produced; and further, which seems to clearly prove that matter in a solid state is not deprived of all molecular mobility, it is possible to realise suitable limited reactions and equilibria between solid salts, and these equilibria obey the fundamental laws of thermodynamics.

Thus the definition of a solid cannot be drawn from its mechanical properties. It cannot be said, after what we have just seen, that solid bodies retain their form, nor that they have a limited elasticity, for M. Spring has made known a case where the elasticity of solids is without any limit.

It was thought that in the case of a different phenomenon—that of crystallization—we might arrive at a clear distinction, because here we should he dealing with a specific quality; and that crystallized bodies would be the true solids, amorphous bodies being at that time regarded as liquids viscous in the extreme.

But the studies of a German physicist, Professor 0. Lehmann, seem to prove that even this means is not infallible. Professor Lehmann has succeeded, in fact, in obtaining with certain organic compounds—oleate of potassium, for instance—under certain conditions some peculiar states to which he has given the name of semi-fluid and liquid crystals. These singular phenomena can only be observed and studied by means of a microscope, and the Carlsruhe Professor had to devise an ingenious apparatus which enabled him to bring the preparation at the required temperature on to the very plate of the microscope.

It is thus made evident that these bodies act on polarized light in the manner of a crystal. Those that M. Lehmann terms semi-liquid still present traces of polyhedric delimitation, but with the peaks and angles rounded by surface-tension, while the others tend to a strictly spherical form. The optical examination of the first-named bodies is very difficult, because appearances may be produced which are due to the phenomena of refraction and imitate those of polarization. For the other kind, which are often as mobile as water, the fact that they polarize light is absolutely unquestionable.

Unfortunately, all these liquids are turbid, and it may be objected that they are not homogeneous. This want of homogeneity may, according to M. Quincke, be due to the existence of particles suspended in a liquid in contact with another liquid miscible with it and enveloping it as might a membrane, and the phenomena of polarization would thus be quite naturally explained. [12]

M. Tamman is of opinion that it is more a question of an emulsion, and, on this hypothesis, the action on light would actually be that which has been observed. Various experimenters have endeavoured of recent years to elucidate this question. It cannot be considered absolutely settled, but these very curious experiments, pursued with great patience and remarkable ingenuity, allow us to think that there really exist certain intermediary forms between crystals and liquids in which bodies still retain a peculiar structure, and consequently act on light, but nevertheless possess considerable plasticity.

Let us note that the question of the continuity of the liquid and solid states is not quite the same as the question of knowing whether there exist bodies intermediate in all respects between the solids and liquids. These two problems are often wrongly confused. The gap between the two classes of bodies may be filled by certain substances with intermediate properties, such as pasty bodies and bodies liquid but still crystallized, because they have not yet completely lost their peculiar structure. Yet the transition is not necessarily established in a continuous fashion when we are dealing with the passage of one and the same determinate substance from the liquid to the solid form. We conceive that this change may take place by insensible degrees in the case of an amorphous body. But it seems hardly possible to consider the case of a crystal, in which molecular movements must be essentially regular, as a natural sequence to the case of the liquid where we are, on the contrary, in presence of an extremely disordered state of movement.

M. Taminan has demonstrated that amorphous solids may very well, in fact, be regarded as superposed liquids endowed with very great viscosity. But it is no longer the same thing when the solid is once in the crystallized state. There is then a solution of continuity of the various properties of the substance, and the two phases may co-exist.

We might presume also, by analogy with what happens with liquids and gases, that if we followed the curve of transformation of the crystalline into the liquid phase, we might arrive at a kind of critical point at which the discontinuity of their properties would vanish.

Professor Poynting, and after him Professor Planck and Professor Ostwald, supposed this to be the case, but more recently M. Tamman has shown that such a point does not exist, and that the region of stability of the crystallized state is limited on all sides. All along the curve of transformation the two states may exist in equilibrium, but we may assert that it is impossible to realize a continuous series of intermediaries between these two states. There will always be a more or less marked discontinuity in some of the properties.

In the course of his researches M. Tamman has been led to certain very important observations, and has met with fresh allotropic modifications in nearly all substances, which singularly complicate the question. In the case of water, for instance, he finds that ordinary ice transforms itself, under a given pressure, at the temperature of -80° C. into another crystalline variety which is denser than water.

The statics of solids under high pressure is as yet, therefore, hardly drafted, but it seems to promise results which will not be identical with those obtained for the statics of fluids, though it will present at least an equal interest.

§ 4. THE DEFORMATIONS OF SOLIDS

If the mechanical properties of the bodies intermediate between solids and liquids have only lately been the object of systematic studies, admittedly solid substances have been studied for a long time. Yet, notwithstanding the abundance of researches published on elasticity by theorists and experimenters, numerous questions with regard to them still remain in suspense.

We only propose to briefly indicate here a few problems recently examined, without going into the details of questions which belong more to the domain of mechanics than to that of pure physics.

The deformations produced in solid bodies by increasing efforts arrange themselves in two distinct periods. If the efforts are weak, the deformations produced are also very weak and disappear when the effort ceases. They are then termed elastic. If the efforts exceed a certain value, a part only of these deformations disappear, and a part are permanent.

The purity of the note emitted by a sound has been often invoked as a proof of the perfect isochronism of the oscillation, and, consequently, as a demonstration a posteriori of the correctness of the early law of Hoocke governing elastic deformations. This law has, however, during some years been frequently disputed. Certain mechanicians or physicists freely admit it to be incorrect, especially as regards extremely weak deformations. According to a theory in some favour, especially in Germany, i.e. the theory of Bach, the law which connects the elastic deformations with the efforts would be an exponential one. Recent experiments by Professors Kohlrausch and Gruncisen, executed under varied and precise conditions on brass, cast iron, slate, and wrought iron, do not appear to confirm Bach’s law. Nothing, in point of fact, authorises the rejection of the law of Hoocke, which presents itself as the most natural and most simple approximation to reality.

The phenomena of permanent deformation are very complex, and it certainly seems that they cannot be explained by the older theories which insisted that the molecules only acted along the straight line which joined their centres. It becomes necessary, then, to construct more complete hypotheses, as the MM. Cosserat have done in some excellent memoirs, and we may then succeed in grouping together the facts resulting from new experiments. Among the experiments of which every theory must take account may be mentioned those by which Colonel Hartmann has placed in evidence the importance of the lines which are produced on the surface of metals when the limit of elasticity is exceeded.

It is to questions of the same order that the minute and patient researches of M. Bouasse have been directed. This physicist, as ingenious as he is profound, has pursued for several years experiments on the most delicate points relating to the theory of elasticity, and he has succeeded in defining with a precision not always attained even in the best esteemed works, the deformations to which a body must be subjected in order to obtain comparable experiments. With regard to the slight oscillations of torsion which he has specially studied, M. Bouasse arrives at the conclusion, in an acute discussion, that we hardly know anything more than was proclaimed a hundred years ago by Coulomb. We see, by this example, that admirable as is the progress accomplished in certain regions of physics, there still exist many over-neglected regions which remain in painful darkness. The skill shown by M. Bouasse authorises us to hope that, thanks to his researches, a strong light will some day illumine these unknown corners.

A particularly interesting chapter on elasticity is that relating to the study of crystals; and in the last few years it has been the object of remarkable researches on the part of M. Voigt. These researches have permitted a few controversial questions between theorists and experimenters to be solved: in particular, M. Voigt has verified the consequences of the calculations, taking care not to make, like Cauchy and Poisson, the hypothesis of central forces a mere function of distance, and has recognized a potential which depends on the relative orientation of the molecules. These considerations also apply to quasi-isotropic bodies which are, in fact, networks of crystals.

Certain occasional deformations which are produced and disappear slowly may be considered as intermediate between elastic and permanent deformations. Of these, the thermal deformation of glass which manifests itself by the displacement of the zero of a thermometer is an example. So also the modifications which the phenomena of magnetic hysteresis or the variations of resistivity have just demonstrated.

Many theorists have taken in hand these difficult questions. M. Brillouin endeavours to interpret these various phenomena by the molecular hypothesis. The attempt may seem bold, since these phenomena are, for the most part, essentially irreversible, and seem, consequently, not adaptable to mechanics. But M. Brillouin makes a point of showing that, under certain conditions, irreversible phenomena may be created between two material points, the actions of which depend solely on their distance; and he furnishes striking instances which appear to prove that a great number of irreversible physical and chemical phenomena may be ascribed to the existence of states of unstable equilibria.

M. Duhem has approached the problem from another side, and endeavours to bring it within the range of thermodynamics. Yet ordinary thermodynamics could not account for experimentally realizable states of equilibrium in the phenomena of viscosity and friction, since this science declares them to be impossible. M. Duhem, however, arrives at the idea that the establishment of the equations of thermodynamics presupposes, among other hypotheses, one which is entirely arbitrary, namely: that when the state of the system is given, external actions capable of maintaining it in that state are determined without ambiguity, by equations termed conditions of equilibrium of the system. If we reject this hypothesis, it will then be allowable to introduce into thermodynamics laws previously excluded, and it will be possible to construct, as M. Duhem has done, a much more comprehensive theory.

The ideas of M. Duhem have been illustrated by remarkable experimental work. M. Marchis, for example, guided by these ideas, has studied the permanent modifications produced in glass by an oscillation of temperature. These modifications, which may be called phenomena of the hysteresis of dilatation, may be followed in very appreciable fashion by means of a glass thermometer. The general results are quite in accord with the previsions of M. Duhem. M. Lenoble in researches on the traction of metallic wires, and M. Chevalier in experiments on the permanent variations of the electrical resistance of wires of an alloy of platinum and silver when submitted to periodical variations of temperature, have likewise afforded verifications of the theory propounded by M. Duhem.

In this theory, the representative system is considered dependent on the temperature of one or several other variables, such as, for example, a chemical variable. A similar idea has been developed in a very fine set of memoirs on nickel steel, by M. Ch. Ed. Guillaume. The eminent physicist, who, by his earlier researches, has greatly contributed to the light thrown on the analogous question of the displacement of the zero in thermometers, concludes, from fresh researches, that the residual phenomena are due to chemical variations, and that the return to the primary chemical state causes the variation to disappear. He applies his ideas not only to the phenomena presented by irreversible steels, but also to very different facts; for example, to phosphorescence, certain particularities of which may be interpreted in an analogous manner.

Nickel steels present the most curious properties, and I have already pointed out the paramount importance of one of them, hardly capable of perceptible dilatation, for its application to metrology and chronometry. [13] Others, also discovered by M. Guillaume in the course of studies conducted with rare success and remarkable ingenuity, may render great services, because it is possible to regulate, so to speak, at will their mechanical or magnetic properties.

The study of alloys in general is, moreover, one of those in which the introduction of the methods of physics has produced the greatest effects. By the microscopic examination of a polished surface or of one indented by a reagent, by the determination of the electromotive force of elements of which an alloy forms one of the poles, and by the measurement of the resistivities, the densities, and the differences of potential or contact, the most valuable indications as to their constitution are obtained. M. Le Chatelier, M. Charpy, M. Dumas, M. Osmond, in France; Sir W. Roberts Austen and Mr. Stansfield, in England, have given manifold examples of the fertility of these methods. The question, moreover, has had a new light thrown upon it by the application of the principles of thermodynamics and of the phase rule.

Alloys are generally known in the two states of solid and liquid. Fused alloys consist of one or several solutions of the component metals and of a certain number of definite combinations. Their composition may thus be very complex: but Gibbs’ rule gives us at once important information on the point, since it indicates that there cannot exist, in general, more than two distinct solutions in an alloy of two metals.

Solid alloys may be classed like liquid ones. Two metals or more dissolve one into the other, and form a solid solution quite analogous to the liquid solution. But the study of these solid solutions is rendered singularly difficult by the fact that the equilibrium so rapidly reached in the case of liquids in this case takes days and, in certain cases, perhaps even centuries to become established.