The Principles of Archimedes*—It is well known that the mechanics of floating bodies, and the laws which govern their equilibrium, were established and enunciated by Archimedes, the Sicilian, in the third century before our era. The following propositions, demonstrated in the first book of his treatise on this subject, embody the fundamental principles of the hydrometer :—

(a) The surface of any fluid at rest is the surface of a sphere whose centre is the same as that of the earth.

(b) Of solids, those which, size for size, are of equal weight with a fluid will, if let down into the fluid, be immersed so that they do not project above the surface, but do not sink lower.

The instrument being closed, its true weight is constant.

Let it be assumed that our experiments are actually made in vacuo, at the sea-level in lat. 45°. In these conditions the standard gram exerts a vertical pressure of 1 gram (true).

We weigh the hydrometer and find its weight to be W grams. We now float it in distilled water contained in a suitable cylinder. In the construction of the hydrometer the internal load has been so adjusted that, when immersed in distilled water of the standard temperature T, which is to remain unaltered during the whole of the experiments, the surface of the water shall cut the stem in some line C, near its junction with the body of the instrument. Then the weight of the water displaced by the hydrometer is exactly W grams.

When determining the specific gravity of a saline solution by the hydrometric method, it is necessary first to find the weight which must be added to the hydrometer to immerse it to the 50-mm. mark when floating in the saline solution at the chosen standard temperature. This added weight is found by a series of observations in exactly the same manner as with the hydrometer in distilled water (see § 14 et seq.).

The details of three series of observations with hydrometer No. 17 in a solution of ⅛ gram-molecule of cæsium chloride in 1000 grams of water at 19·50° are given as an example in Table C. This table is arranged in the same manner as Table A1.

A laboratory is an inhabited room, and in Northern Europe the temperature of such apartments lies generally between 12° and 20°C. Consequently we find that a large amount of specific gravity work has been done at 15°C. by Gerlach, at 19·5° C. by Kremers, and at 17·5° C. by others ; while the calorimetric work by Julius Thomsen, extending over the last half-century, was all done at the temperature 18° C.

General Tables, giving the Facts of Observation ; namely, W, the weight, in grams, of the solution which contains m gram-molecules of the salt dissolved in 1000 grams of water; S, the specific gravity of this solution at the temperature, T, referred to that of distilled water at the same temperature as unity. From these data, Δ, the displacement of the solution, is obtained by the equation Δ = W/S, and it is expressed in grams of water having the temperature T. The symbol adopted for this unit is GT. The numbers printed in italics refer to specific gravities observed at the temperature printed in italics.

In the tables giving the results of the experiments made with solutions of a particular salt, the weights given are those which would have been used if the weighings had actually been made in a vacuum ; and the standard temperature, T, at which all the operations have been made, is given at the top with the name of the salt, both being constants.

Of the variables, we have under m the quantity of the salt, expressed as the number, whole or fractional, of gram-molecules, which is dissolved in 1000 grams of water, under W the weight in grams of the solution so produced, and under S the specific gravity of the solution referred to that of distilled water as unity, both having the standard temperature T.

When successive equal quantities of a salt are dissolved in a constant quantity of water, the successive increments of displacement of the solution, so produced, are generally unequal. They are usually the greater, the greater is the amount of salt which has already been dissolved.

In order usefully to discuss the change produced by any physical action, it is advisable to compare it with that which would be produced if it acted in accordance with some law which can be specified with precision. If the results observed agree with those calculated in terms of the law postulated, it is good evidence that the particular physical action takes place under the law. If no such agreement appears, then the observed results must be compared with those calculated in terms of the specification of some other law.

The earlier introductory notes and discussion of the values of the specific gravity and displacement explain the precise meaning of these terms. A comparison of the values of the increment of displacement (v) caused by the dissolution of m grammolecules of a salt in 1000 grams of water with the difference (dΔ) of these consecutive increments of displacement possesses considerable interest.

The increment of displacement produced in 1000 grams of water at 19·5° C, when 1/2 gram-molecule of potassium chloride is dissolved in it, is 14·001 GT; when a molecularly equivalent amount of potassium bromide is dissolved in the same quantity of water, the increase in the displacement is 17·547GT; when the salt in solution is potassium iodide, the number is 22·778 GT. Replacing, therefore, the chlorine by bromine increases the displacement by 3·546 GT; and if the bromine be now replaced by iodine, there is a further increase of 5·231 GT in the displacement; or, replacing the chlorine by iodine causes an increment of displacement of 8·777 GT.

Although chloride of sodium is not a member of either of the enneads which form the principal material of this research, its importance in nature justifies its inclusion in it.

The observations made at 15·0° C. and recorded in Table No. 26, Class A, were not sufficient. A complete series of observations of specific gravity in which the values of m formed a geometric series, descending from 1 to 1/512, was made at 19·50° C.

When the hydrometer is closed its mass cannot be diminished, and it can be increased only by external additions, which in practice must be immersed either in air or in the liquid. The use of submerged weights is attended by so much inconvenience that it has to be avoided; consequently, when the instrument is closed, its mass is increased only by adding weights at the top of the stem. The extent to which such additions can be made depends on the stability of the instrument when floating in the experimental liquid. The instrument (No. 0) which I used exclusively during the voyage of the Challenger weighed, in round numbers, 160 grams, and the greatest weight which had to be added to it was 4·071 grams, which produced no disturbing effect whatever.

For this purpose we will consider in detail the items of the experimental determination of the weight of liquid displaced by the instrument which is designated “Hydrometer A”; it has for some time been in constant use.

A scheme for recording the items of observation is given in § 86.

In the first vertical column the line corresponding to each item is designated by a letter—a, b, etc. In the second column are the symbols used for the principal items, and in the body of the scheme the items are described or explained.

There are three such groups of salts in the ennead, namely, KBr and RbCl; KI, RbBr, and CsCl; and Rbl and CsBr. Experiments have been made on strong solutions of the first group, KBr and RbCl. To these “natural isomers” we have added an (artificial) isomer consisting of a mixture which contains KCl and KI in equal molecular proportions, so that it may be represented by the formula This mixture contains 31·00 per cent. of KCl and 69·00 per cent. KI.

For the purpose of determining these constants, hydrometric observations were made on solutions of the three salts the concentrations of which ranged from 1/2 to 1/1024 gram-molecule per thousand grams of water, the experiments being made with the closed hydrometers Nos. 3 and 17. Experiments were also made on strong solutions of calcium chloride and magnesium chloride, using the open hydrometers A and B ; the concentrations of these solutions varied from 1 gram-molecule per thousand grams of water to the highest attainable degree of supersaturation. It was when the experiments on a supersaturated solution of calcium chloride were in progress that the observations were made which revealed the remarkable state of unrest in that solution which preceded its partition into crystals and mother-liquor with liberation of heat.

The primary purpose for which the open hydrometer was designed was to investigate the specific gravity and the displacement of solutions having concentrations in the neighbourhood of that of saturation. In § 90 we have seen the satisfactory result of experiments made for this purpose with solutions of chloride of calcium containing 6·3 grm.-mols. CaCl2 per 1000 grams of water. Experiments in the same direction were made with solutions of chloride of calcium of still higher concentration. A parent solution was made, which, on the basis of published data relative to the solubility of the salt, should be supersaturated at 19·5°C. With it, it was intended to produce the solution saturated at this temperature, and to study its specific gravity and that of solutions formed by diluting it with small quantities of water.

The work on the specific gravity of dilute solutions at 19·5° C. reported in the early part of this memoir was interrupted by the arrival of the great anticyclone or heatwave of the summer of 1904, during which observations at a temperature of 19·5° were quite impossible. Indeed, the temperature of the laboratory, whether by night or day, hardly ever fell below 23° C. or rose above 25° C. It persisted over Northern Europe for nearly six weeks, and produced tropical conditions, which were evidenced alike by the high temperature of the air and by its insignificant diurnal variation.