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BEAUME'S HYDROMETERS.

119

When the instrument is intended for liquids lighter than water, it is called an alcoholimeter. In this case the point to which it sinks in water is near the bottom of the stem, and is marked

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10; the zero of the scale is the point to which it sinks in a solution of 10 parts of salt to 90 of water, the density of which is about 1·085, the divisions in this case being numbered upward from zero.

In order to adapt the formulæ of last section to the case of graduations numbered upwards, it is merely necessary to reverse the signs of ni, n2, and N; that is we must put

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and as we have now n=10, d1=1, n=0, d2=1·085

the formulæ give1

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Beaume's Alcoholi

meters.

100

190

87A. Twaddell's Hydrometer.-In this instrument the divisions are placed not as in Beaumé's, at equal distances, but at distances corresponding to equal differences of density. In fact the specific gravity of a liquid is found by multiplying the reading by 5, cutting off three decimal places, and prefixing unity. Thus the degree 1 indicates specific gravity 1.005, 2 indicates 1010, &c.

80

60

50

40

30

20

10

88. Gay Lussac's Centesimal Alcoholimeter.-When a hydrometer is to be used for a special purpose it may be convenient to adopt a mode of graduation different in principle from any that we have described above, and adapted to give a direct indication of the proportion in which two ingredients are mixed in the fluid to be examined. It may indicate, for example, the quantity of salt in sea-water, or the quantity of alcohol in a spirit consisting of alcohol and water. Where there are three or more ingredients of dif- Alcoholiferent specific gravities the method fails. Gay-Lussac's alcoholimeter is graduated to indicate, at the temperature of 15° Cent.,

Fig. 87. Centesimal

meter.

1 On comparing the two formulæ for D in this section with the tables in the Appendix to Miller's Chemical Physics, I find that as regards the salimeter they agree to two places of decimals and very nearly to three. As regards the alcoholimeter, the table in Miller implies that c is about 136, which would make the density corresponding to the zero of the scale about 1.074.

the percentage of pure alcohol in a specimen of spirit. At the top of the stem is 100, the point to which the instrument sinks in pure alcohol, and at the bottom is 0, to which it sinks in water. The position of the intermediate degrees must be determined empirically, by placing the instrument in mixtures of alcohol and water in known proportions, at the temperature of 15°. The law of density, as depending on the proportion of alcohol present, is complicated by the fact that, when alcohol is mixed with water, a diminution of volume (accompanied by rise of temperature) takes place.

88A. Specific Gravity of Mixtures.-When two or more substances are mixed without either shrinkage or expansion (that is, when the volume of the mixture is equal to the sum of the volumes of the components), the density of the mixture can easily be expressed in terms of the quantities and densities of the components.

V3

First, let the volumes v1, v2, v3 . . . of the components be given, together with their densities d1, d2, dз. . .

Then their masses (or weights) are vid1, vžd2, v3d3

The mass of the mixture is the sum of these masses, and its volume is the sum of the volumes V1, V2, V3

; hence its density is

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Secondly, let the weights or masses m1, M2, M3

of the compo

nents be given, together with their densities d1, d2, dз . . .

Then their volumes are M1, M2, M3

d1 do dz

The volume of the mixture is the sum of these volumes, and its mass is m1+m+m ̧+ . . . ; hence its density is

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88 B. Graphical Method of Graduation.-When the points on the stem which correspond to some five or six known densities, nearly equidifferent, have been determined, the intermediate graduations can be inserted with tolerable accuracy by the graphical method of interpolation, a method which has many applications in physics besides that which we are now considering. Suppose A and B (Fig. 85) to represent the extreme points, and I, K, L, R intermediate points, all of which correspond to known densities. Erect ordinates (that is to say, perpendiculars) at these points, proportional to the respective densities, or (which will serve our purpose equally well)

GRAPHICAL METHOD OF GRADUATION.

121

erect ordinates II', KK', LL', RR', BC proportional to the excesses of the densities at I, K, L, R, B above the density at A. Any scale of

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parts. Having erected the ordinates, we must draw through their extremities the curve AI'K'L'R'C, making it as free from sudden. turns as possible, as it is upon the regularity of this curve that the accuracy of the interpolation depends. Then to find the point on the stem AB at which any other density is to be marked-say 1·60, we must draw through the 60th division, on the line of equal parts, a horizontal line to meet the curve, and, through the point thus found on the curve, draw an ordinate. This ordinate will meet the base line AB in the required point, which is accordingly marked 16 in the figure. The curve also affords the means of solving the converse problem, that is, of finding the density corresponding to any given point on the stem. At the given point in AB, which represents the stem, we must draw an ordinate, and through the point where this meets the curve we must draw a horizontal line to meet the scale of equal parts. The point thus determined on the scale of equal parts indicates the density required, or rather the excess of this density above the density of A.

CHAPTER XI.

VESSELS IN COMMUNICATION.-CAPILLARITY.

89. Equilibrium in Vessels in Communication.-When a liquid is contained in vessels communicating with each other, and is in equilibrium, it stands at the same height in the different parts of the system, so that the free surfaces all lie in the same horizontal plane. This is an immediate consequence of the fact that layers of equal

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pressure in a liquid are always horizontal (§ 64); for if we take any such layer at the bottom of the system, we must proceed upwards through the same vertical height in all parts of the system in order to reach the free surface which corresponds to the pressure. Thus, in the system represented by Fig. 89, the liquid is seen to stand at the same height in the principal vessel and in the variously shaped tubes communicating with it. If one of these tubes is cut off at a height less than that of the liquid in the principal vessel, and if it

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be made to terminate in a narrow mouth, the liquid will be seen to spout up nearly to the level of that in the principal vessel.

This tendency of liquids to find their own level is very important, and of continual application. Thus, a reservoir of water may have different pipes issuing from it and spreading out in all possible directions with any number of turns and windings; provided that the ends of these pipes lie below the level of the reservoir, the water will flow through the pipes and run out at their extremities. The velocity of exit, however, will depend on the form and arrangement of the pipes, as well as on the difference of level. This velocity must of course be taken into account in calculating the quantity of water that will flow in a given time; and in forming plans for the proper distribution of public supplies of water. It also determines the height to which a jet of water can be discharged from an opening at the end of the pipe.

90. Water-level. The well-known instrument called the waterlevel depends upon the property just mentioned. It consists of a

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metal tube bb, bent at right angles at its extremities. These carry two glass tubes aa, very narrow at the top, and of the same diameter. The tube rests on a tripod stand, at the top of which is a joint that enables the observer to turn the apparatus and set it in any direction. The tube is placed in a position nearly horizontal, and water, generally coloured a little, is poured in until it stands at about threefourths of the height of each of the glass tubes.

By the principle of equilibrium in vessels communicating with each other, the surfaces of the liquid in the two branches are in the same horizontal plane, so that if the line of the observer's sight just grazes the two surfaces, it will be horizontal.

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