bulk of the fluid, equilibrium will still exist. And if the fluid so added become solid, the equilibrium will continue, and the pressure upwards of the surrounding fluid will remain the same as before.

Now the pressure downwards of the portion of the fluid which becomes solid is its weight. And as the pressure upwards of the surrounding fluid supports this weight, that pressure must be exactly equal and opposite to it.

The pressure downwards, therefore, of the original solid, before it was immersed (i. e. its weight MS, Prop. VII.), must, by immersion of the solid, have been diminished by a pressure upwards, arising from the surrounding fluid, exactly equal to the weight of the fluid displaced,—which weight, by Prop. VII., is equal to MS'.

.. Weight lost by the body: the whole weight of the body :: MS': MS :: S' : S :: S. G. of the fluid: S. G. of the body.

75. It appears from the proof of the last Proposition, that the pressure of a fluid on a body wholly immersed in it acts vertically upwards, and is equal to the weight of the fluid which the body displaces.

If this pressure be less than the weight of the body—that is, if the S. G. of the fluid be less than that of the solid-the pressure downwards arising from the weight of the solid will be greater than the pressure of the surrounding fluid upwards, and the body will therefore sink to the bottom of the vessel.

But if the pressure of the fluid upwards be greater than the weight of the body immersed, that is, if the S. G. of the fluid be greater than that of the solid,-the pressure upwards will be greater than the pressure downwards, and the body will therefore rise, until the conditions of Proposition VIII. are fulfilled, and will then float.

76. PROP. X. To describe the Hydrostatic Balance; and to shew how to find the Specific Gravity of a body by means of it,-1st, when its Specific Gravity is greater than that of the fluid in which it is weighed, 2ndly, when it is less.

The Hydrostatic Balance is the Common Balance, with a hook attached to the under part of one of its scales, so that bodies may be weighed either by putting them into the scale, or by suspending them from it and letting them A be immersed in a fluid as here represented.

B

1st. Let the S. G. (Specific Gravity) of the body be greater than that of the fluid.

Since the S. G. of the solid is greater than that of the fluid, the body will sink in the fluid. Art. 75.

Let SS. G. of the solid,

=

W weight required to
placed in the scale.

S'S. G. of the fluid.

balance the body when

W'= weight required when the body is immersed; .. WW' weight lost by immersion in the fluid.

=

But W and W' are known weights, and S' is supposed to be given, therefore S may be determined.

2ndly. Let the S. G. of the body be less than the S. G. of the fluid.

When the body, in this case, is forced under the surface of the fluid, the pressure downwards on 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 equal to the difference between the weight of the fluid displaced and the weight of the solid.

To determine this last pressure, take a body Q,-called the Sinker, 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 first immersed by itself 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 let a weight W" be placed in the scale B*. This weight therefore is equal to the tendency upwards of P when immersed; that is,

=

W" weight of a bulk of the fluid equal to that of P minus the weight of P (W);

.. weight of a bulk of fluid of the same size as P

= W+W".

Now since, (by Prop. vII.), W=MS, when the magnitudes of the bodies are the same, WS;

.. weight of P: weight of an equal bulk of fluid

But W and W" are known weights, and S' is supposed to be given; therefore S may be found.

It will be observed that it is not requisite for the exact weight of the Sinker to be known.

77. PROP. XI. To describe the Common Hydrometer; and to shew how to compare the Specific Gravities of two fluids by means of it.

*The better way in practice is to restore the equilibrium by removing a weight W" from the scale A; but the method described in the text perhaps renders the demonstration easier to be understood.

The common Hydrometer consists of two hollow spheres attached to each other, and of a cylindrical slender stem, whose axis, if produced, would pass through the centres of both the spheres. The upper sphere is empty; and the lower is filled with lead or mercury, so as to make the instrument P float steadily in a vertical position when put into a fluid. The stem is graduated by divisions of equal length.

The Hydrometer is made lighter than an equal bulk of any of the fluids whose Specific Gravities it is employed 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 vertically in a fluid whose S. G. is S, suppose 20 divisions are above the surface; and when it floats in a fluid whose S. G. is S', let there be 30 divisions out.

Now, the weights of the bulks which are displaced of the two fluids are the same, each being equal to the weight of the instrument, by Prop. VI. If M and M', therefore, be the magnitudes of the fluids displaced, by Prop. VII., M-S=weight of the Hydrometer = M'ׂS' ;

and the ratio of the Specific Gravities of the two fluids is thus determined.

78. 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 an equal bulk of Proof Spirit, and the Hydrometer therefore will displace a greater bulk of it than it does of Proof Spirit, that is, it will sink deeper in

the mixture than in Proof Spirit. Wherefore the surface of such a mixture will rise to a higher point in the stem than P. In such a case the mixture is said to be "above proof".

But 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".

QUESTIONS ON CHAP. III.

(1) When M is said to be the magnitude of a body, what does it mean? Is it a number, or what is it?

(2) What is the difference between Gravity and Specific Gravity? (3) In TABLES of Specific Gravities what is commonly taken for the unit, or standard?

(4) Can the same substance have a different Specific Gravity under different circumstances? Is water such a substance? If so, how can it be used as a standard?

(5) If M be expressed in cubic feet, and S in terms of the Specific Gravity of Water, in terms of what must W be expressed, in order that the equality W=MS may be true?

(6) What is the datum necessary for rendering the formula, W = MS, practically useful; so that, for instance, knowing the Specific Gravity of gold (194) you could apply the formula to find the weight of a cubic inch of gold?

(7) In Prop. VIII., where is it assumed, that the floating body is of uniform density?

(8) In the definition of Specific Gravity of a substance, is it assumed, that the substance is of uniform density?

(9) In Prop. VIII., if the whole body be just immersed, and float there, what conclusion do you draw as to the Specific Gravities of the body and fluid?

(10) Would Prop. VIII. apply to an empty ship constructed wholly of iron, and floating in smooth water?

(11) What becomes of the result in Prop. vIII., if the Specific Gravity of the body be greater than that of the fluid ?

(12) In Prop. IX. what is meant by 'weight lost'? Is the weight actually lost? In what case will a body 'lose' its whole weight in a fluid?