it (which is the weight W of the body) being less than the pressure upwards (which is the weight of a quantity of fluid equal in magnitude to the fluid displaced) the body must on the whole be acted on by a pressure upwards, which is equal to the weight of the fluid displaced minus the weight of the solid. Let a body Q be taken, of greater S. G. than that of the fluid, and large enough to sink both itself and the body P whose S. G. is required when P is attached to it. Let Q be immersed in the fluid and balanced by weights in A. Next, leaving the weights in A, let P be attached to Q and both of them be immersed. The scale A will now preponderate, and to restore the equilibrium a portion (W') of the weights in it must be removed. We have now therefore, (W being the weight of the body P when placed in the scale B), = W' whole effective pressure on P when immersed; which acts upwards; = weight of an equal bulk of fluid minus weight of P (W;) weight of a bulk of fluid equal to that of P=W+ W'. Now, since by Prop. vII. W= MS, when the magnitudes of the bodies are the same, Woc S; .. W : W + W' :: S. G. of solid : S. G. of fluid :: S: S'; ...S = W+W'' Whence,-W and W' being known weights,-S can be found if S' be given. 19. PROP. XI. (1) To describe the common Hydrometer, and (2) to shew how to compare the Specific Gravities of two fluids by it. (1) The common Hydrometer consists of two spheres touching each other, and a long uniform slender stem, which if produced, would pass through the centers of both of them. The upper sphere is hollow, and the lower is filled with small shot or mercury, to serve as ballast for the instrument and to make it float steadily in a vertical position when put into a fluid. The stem is graduated by divisions of the same length. P (2) The instrument is made lighter than equal bulks of the fluids whose Specific Gravities it is used to compare. Suppose the bulk of the portion of the stem included between every two graduations to be one four thousandth part of the bulk of the whole instrument. When the Hydrometer floats on a fluid whose S. G. is S, let 20 divisions be above the surface; and when it floats on a fluid whose S. G. is S', let there be 30 divisions out. Now the weights displaced of the fluids are in both cases the same, namely the weight of the instrument; Prop. VI. If M and M' therefore be the magnitudes of the fluids displaced, we have, by Prop. vII, = MS weight of the Hydrometer = M'S'; ... S: S' :: M' : M and the ratio of the S. G.'s of the two fluids is determined. [A mark P is made at the point in the stem to which the instrument sinks in a fluid called "Proof Spirit", which is a mixture consisting of equal weights,-(not equal magnitudes),—of pure Alcohol and of Water. Alcohol being lighter than Water, if a mixture of these two fluids contain a greater weight of the former than it does of the latter, it will be lighter than Proof Spirit, and the Hydrometer therefore will displace a greater bulk of it than it does of Proof Spirit. Wherefore the surface of such a mixture will rise to a higher point in the stem than P, and in that case the mixture is said to be "above proof". On the other hand, if the weight of the Water contained in the mixture be greater than that of the pure Alcohol, the Hydrometer will not sink so low as to the point P, and the fluid is then said to be "below proof". When equal weights of pure Alcohol and of Water are mixed, the magnitudes of the fluids are nearly as 3 to 2;-so that out of every hundred cubic inches of Proof Spirit about sixty are pure Alcohol and forty are Water.] 65 CHAPTER III. ELASTIC FLUIDS. 20. PROP. XII. AIR has AIR has Weight. This is proved by these experiments. If a close vessel be weighed after the air has been exhausted from it, and afterwards when the air has been admitted into it, it is found that the weight is greater in the latter than in the former case. If a bladder be weighed in a vessel from which all the air has been exhausted by a method which will be described hereafter, first, when the bladder contains no air, and again after air has been forced into it, the weight required to balance the bladder is greater in the latter than in the former case. Whence we conclude that Air has weight.' 6 21. PROP. XIII. The elastic force of Air at a given temperature varies as the density. This is proved by experiment. Let ABCD be a tube, of which the bore is uniform (its section being an area A), and the legs AB and CD are vertical. The end A of the tube is open, and D is closed. A quantity of air is retained in the shorter leg by some mercury which rises to E and F in the two legs respectively. Through E the horizontal line Ee is drawn. = Then, The weight of the mercurial column Fe pressure downwards at e on a surface A; by Prop. IV. upwards at e...... upwards at E.................; ; Prop. I. Prop. 11. = pressure by the air in DE on the surface of mercury with which it is in contact; (because, the whole being at rest, the pressures upwards and downwards on the same horizontal plane must be equal.) H Similarly, if when more mercury is A poured in, its surfaces stand at G and H in the two legs, we have, drawing Gg horizontally through G, Weight of the mercurial column Hg F pressure by the air in DG on the surface g A of mercury with which it is in contact. = E Now if ED, Fe, GD, Hg be measured, B it is invariably found, however the quantities of air and of mercury used in the experiment be varied, that DE: DG :: Hg: Fe; .. content of DE: content of DG :: content of Hg : content of Fe. But since the same quantity of air is contained in DE and in DG, .. Density of the air in DE: density of the air in DG :: content of DG content of DE :: content of Fe content of Hg :: Weight of mercury in Fe weight of mercury in Hg, :: Pressure of air in DE on surface A pressure of air in DG on same surface; shewn above. as |