a b c lie in one plane, inclined to each other at angles of 120°, but the other two, d and e, are at right angles to this plane.

If b and c are united to carbon then the hydroxyl can occupy two different positions-viz. d or e.

In the first case the hydroxyl is near the radical R1, in the second it is nearer R. If the OH is at a, R and R1 are equidistant.

§ 59. The Absolute Dimensions of Molecules and Atoms.The molecules, the constitution of which has been discussed in the preceding paragraphs, are not infinitely small, although much smaller than any magnitude perceptible to our unaided or even to the aided senses. As to the magnitude of the molecules themselves, it is at present impossible to give any exact determinations; still the limits within which the dimensions must lie can be approximately determined. Such approximations may, as was shown by Sir William Thomson in 1871, be arrived at by the aid of various physical phenomena; his conclusions have been confirmed and extended by other investigators. From certain optical phenomena-for instance, from the dispersion accompanying the refraction of light-it may be concluded, with some degree of probability, that the molecules of transparent materials, such as glass, water, and the like, are greater than the ten-thousandth part of a wave-length of light, which latter amounts again to only a few tenthousandth parts of a millimetre. Similar conclusions may be drawn also from the diffusion of colouring matters on solution, and again from the contact electricity of metals, and the heat produced by the attraction of metallic plates oppositely electrified, from the minimum thickness which soap bubbles can attain without bursting, and especially from the properties of gases and of the liquids produced by their condensation.

The highly developed kinetic theory of gases shows, for instance, that certain relationships exist between the dimensions

of gaseous particles, their velocity, and the path which they traverse before they come in contact with one another. From these relationships approximations may be made as to the weight and the mass of the molecules, and at the same time also of the atoms.

All these investigations have proved with approximate agreement that the diameters of the molecules of different substances are smaller than the ten-millionth part of a millimetre, but at the same time not indefinitely smaller than this.

§ 60. Aggregation of the Molecules. Although the affinities of the atoms in a molecule are satisfied by union with each other, the total affinity, that is to say the molecule, still exerts an appreciable attraction for other molecules, for it is only by the aggregation of molecules that the particles of matter of which our senses are cognisant are produced. According to the foregoing speculation, enormous numbers of these molecules must be present even in the smallest visible and ponderable particle.

The mode of aggregation of these molecules must vary, and these differences will give rise to the different states of matter. In the solid condition the particles are held together in an unalterable position; in the liquid state they are so held that the particles move easily among one another in such a manner that no two particles remain neighbours for any length of time. Between these two conditions, forming as it were the passage between the extremes, we have the soft, plastic, and viscous states of matter in which the particles may move with greater or less difficulty, under the influence of the force of gravity or pressure, without destroying the continuity of the whole mass.

In the gaseous state the attraction of the particles for one another ceases, so that these separate particles move away ⚫into space unless they are prevented from doing so by impassable boundaries.

§ 61. The Effect of Heat.-In no one of these conditions can we assume that the particles are in a state of absolute rest; we must rather imagine that in each one the particles possess a certain motion, which is perceptible to us as heat,

and this movement becomes the more active the greater the amount of heat the bodies take up. The form of this motion is not fully understood; still in the solid state each particle can only move round a certain fixed position of equilibrium, this motion being either vibratory or rotatory.

In liquids the particles must be imagined as moving over one another, so that they leave no spaces between them, whereas in the gaseous or vaporous condition each particle is separated from the others, and moves rapidly in a straight line until it comes in contact with some hindrance by which it is diverted from its path. A consequence of an accelerated motion of the particles is to be found in the expansion of bodies by heat, because more space is required for these extended movements.

It is, however, a remarkable fact that in the passage from one state of aggregation to another bodies take up the heat which disappears as such, so that it is no longer recognised by the senses or by the thermometer. This so-called latent heat serves, doubtless in a great part, to produce those movements of the particles which are characteristic of the new condition; in part, perhaps, also to overcome the forces of attraction between the particles, assuming such forces to exist.

The expansion exhibited by the majority of substances in melting may also be attributed to the increase of these internal movements.

In addition to the motion of the molecules we must also assume that the atoms constituting these molecules are likewise in a state of motion, and this again would be altered by the application of heat. In monatomic molecules, consisting only of one atom, as, for instance, in the case of the molecule of gaseous mercury, which has been proved to be monatomic by Kundt and Warburg, such atomic movements will not

occur.

§ 62. Homogeneous Solid Bodies.-When similar molecules collect together to form a solid aggregate, a solid body is produced, which will have a structure determined entirely by the relative position of the particles. In the formless, or amorphous, condition the arrangement of the particles would be similar in each direction throughout the mass of the body,

whilst in the case of crystals in certain directions it would be found to be different from others, and these differences are perceptible not only in the plane surfaces forming the external boundaries of the crystals, but also in any fragment taken from any part of the interior of the crystal. These differences are shown in the cohesion, the hardness, the cleavage of the crystals in certain directions, the expansion by heat, the conduction of heat, the velocity and refraction of light, the colour of the same, and in some cases also in certain peculiar electrical phenomena produced by heating or cooling. Such differences can only find their explanation in a different arrangement of the molecules. We may assume that the molecules are brought nearer together in one direction than they are in another; but the reason for such an arrangement of the molecules must be sought for in the molecules themselves; so we must assume that the particles arrange themselves together, so that their axes are parallel to one another or are otherwise regularly arranged.

The systematic disposition of points in space has been geometrically investigated by Leonard Sohncke, and its relation to the different systems of crystals established. The greater the symmetry of the distribution of such points, the simpler is the crystal system; and in full accord with this it is found that substances of the simplest composition, as, for instance, the elements and the compounds composed of a few atoms, form, as a rule, crystals belonging to the regular and hexagonal systems; whereas molecules composed of many atoms for instance, the majority of organic compoundsyield aggregates which crystallise with little or no evidence of symmetry. In amorphous substances the particles must be imagined as arranged irregularly, for this is the only way in which the particles could be arranged so that in any finite mass all sections would be the same. In many of their properties, e.g. behaviour with polarised light, amorphous bodies resemble the substances crystallising in the regular system; but this is not the case for other properties, such as cohesion, hardness, and cleavage.

§ 63. Heterogeneous Solid Molecular Aggregates.-A solid body may also be produced by the grouping together of

different kinds of molecules. Many substances crystallise with water of crystallisation; still these compounds would appear rather to be homogeneous aggregates, for every molecule is united with a definite number of molecules of water, and the molecules so produced are regularly grouped into new and larger ones. A few only of the compounds containing water crystallise in the regular system: as, for instance, the alums, the twenty-four molecules of water being so arranged around the salt molecule as to produce an aggregate homogeneous in all directions. The so-called double salts are similarly constituted to the compounds containing water of crystallisation, and these must be reckoned amongst the homogeneous aggregates, and also all other combinations produced in accordance with the laws of stachiometry.

The mixed crystals of isomorphous bodies in which the constituents occur in varying and changing proportions must, on the other hand, be considered as heterogeneous aggregates. Thus, for example, the so-called vitriols-that is, the hydrated sulphates of magnesium, copper, zinc, iron, manganese, nickel, and cobalt-may crystallise together in any proportions. This is true also of other isomorphous substances. This crystallisation together takes place only when the compounds are of analogous constitution, and when the isomorphous constituents take up approximately the same space. If this condition is not exactly satisfied, then an angle of the crystal of one substance would be altered to a greater or less extent by the entrance into that crystal of another body. For instance, calcspar (CaCO3) crystallises in rhombohedra, the angle being 105° 5', whereas magnesite (MgCO3) crystallises in the same form, the angle of which is 107° 25'. When both crystallise together in the form of dolomite the angle is 106° 15'. This difference in angle of these crystals arises from the fact that the quantity of calcium carbonate represented by its formula occupies a greater space than the quantity of magnesium carbonate represented by its formula, and the increase of volume in consequence of this results in an extension of the crystal along its chief axis.

The expansion of the crystal by heat takes place chiefly in