Page:Supplement to the fourth, fifth, and sixth editions of the Encyclopaedia Britannica - with preliminary dissertations on the history of the sciences - illustrated by engravings (IA gri 33125011196181).pdf/757

 $$\tfrac{1}{4}\pi z^2$$ is the surface of a transverse section of the wire-stalk.

$$v$$ is the volume of the bulb or body of the aræometer.

$$w$$ is the whole weight of the aræometer.

$$x$$ is the length of the stalk that is plunged in the water.

$$\tfrac{1}{4}x \pi z^2$$ is the volume of the immersed portion of the stalk.

When the aræometer floats in equilibrio, it displaces a volume of water equal to its own weight, therefore, $$w = g (v + \tfrac{1}{4}x \pi z^2)$$, and, $$g = \frac{w}{v + \tfrac{1}{4}x \pi z^2}$$, $$x = \frac{4(w - gv)}{2g \pi z^2}$$; $$w-gv$$ is the difference between the quantity of water displaced by the whole aræometer, and the quantity displaced by the bulb alone, $$w-gv$$, therefore, is the volume af water displaced by the immersed portion of the stalk, as the diameter of the stalk $$z$$ is very small, the cylinder of water $$w-gv$$, which has $$z$$ for its diameter, is likewise very small, and does not exceed a few grains in weight; therefore, a small variation in $$w$$ (the weight of the aræometer), or in $$g$$ (the density of the liquid), occasions a great variation in $$x$$ (the length of the immersed part of the stalk). The value of $$x$$ changes rapidly, when $$z$$ (the diameter of the stalk) is changed, because the value of $$x$$ is divided by $$z^2$$, which is the square of a very small quantity.

When the aræometer is immersed in a liquid of another specific gravity $$g^1$$, then the equation is $$x^1 = \frac{2(w-g^1 v)}{g^1 \pi z^2}$$; subtract the value of $$x^1$$ from that of $$x$$, and there results $$x-x^1 = \frac{2w(g-g^1)}{g g^1 \pi z^2}$$; this is the diminution in the length of the immersed part of the stalk, which takes place when the aræometer is transferred to a liquid of a greater density. By this formula, it is seen, that the sensibility of the aræometer, that is, the length of the portion of the stalk which emerges upon transferring the aræometer to a denser liquid, is augmented, in the first place, by increasing $$w$$ (the weight of water displaced by the aræometer), that is, by increasing the volume of the body of the aræometer; secondly, by diminishing $$z$$ (the diameter of the stalk), which is in the denominator of the value of $$x-x^1$$. Consequently, the faculty of the aræometer to show the different densities of liquids is, in general, expressed by the fraction $$\frac{w}{z^2}$$.

With regard to the vertical mobility of the aræometer, when put in motion by placing a small weight ($$s$$) in its exterior cup, substitute $$w+s$$ for $$w$$, then, $$x^1 = \frac{4(w+s-gv)}{2g \pi z^2}$$, take the difference between this and $$x = \frac{4(w-gv)}{2g \pi z^2}$$; this difference is $$x^1-x = \frac{4s}{g \pi z^2}$$. Which shows that the length of the portion of the stalk that a small weight causes to immerge, is proportional to $$\frac{s}{z^2}$$, or in the direct ratio of the small weight, and in the inverse ratio of the square of the diameter of the stalk.

When the small weight, the density of the liquid, and the length of that part of the stalk which is submerged on adding the small weight, are known, then this equation will give the diameter of the stalk in known quantities $$z = 2 \sqrt{\frac{s}{g \pi (x^1-x)}}$$.

When the weight of the whole aræometer is known in ounces, &c., and the specific gravity of one of two liquids (water for instance) is known, the difference of specific gravity between that liquid and another liquid may be had in known quantities.

$$g$$ is the specific gravity of water.

$$g^1$$ is the specific gravity of the second liquid, which is here supposed more dense.

$$w$$ is the weight of the volume of water displaced by the aræometer.

$$s$$ is a small additional weight placed on the exterior cup to keep the aræometer, when placed in the denser liquid, at the same point of immersion as when it floated in water.

$$w+s$$ is the whole weight of the apparatus when floating in the denser liquid. ws

The equation $$g^1 = \frac{w+s}{v + \tfrac{1}{4} x \pi z^2}$$ is obtained by substituting $$g^1$$ for $$g$$, and $$w+s$$ for $$w$$ in the equation, $$g = \frac{w}{v + \tfrac{1}{4} x \pi z^2}$$, which was given above. Divide by $$g = \frac{w}{v + \tfrac{1}{4} x \pi z^2}$$, and there results $$\frac{g^1}{g} = \frac{w+s}{w}$$, which gives the proportion of the density of the second Iiquid to the density of water. By subtraction there results $$\frac{g^1-g}{g} = \frac{s}{w}$$ and $$g^1-g = \frac{sg}{w}$$, that is, the difference between the density of the second liquid, and the density of water is found by multiplying the small weight by 1000 ounces, and dividing this product by the number of ounces, &c., which denote the weight of the aræometer uncharged.

Small bodies, whose specific gravities are known, serve to indicate the specific gravity of a liquid in which they just remain suspended. In this way, beads of glass, three or four tenths of an inch in diameter, are employed, each of which remains suspended in spirit of a certain specific gravity. The density of each of these beads, or rather bubbles, is regulated by the proportion between the quantity of glass and the cavity which the glass incloses. A piece of boes-wax, whose specific gravity, by the addition of lead, is such, that the body is just suspended in brine of a known density, is used as an aræometer In some salt works. The fresh egg of a common fowl is just sustained by brine of a certain specific gravity, and is employed as an aræometer.

The aræometer of Homberg, consists of a phial, with a slender neck and glass-stepper, so made, that it may be filled with the same volume of different liquids. It is employed in finding the specific gravity of liquids in the following way: 1st, The phial is filled with distilled water, and then weighed in a balance; 2dly, The phial is emptied, and again filled with the liquid, whose specific gravity is sought, and weighed in a balance, the proportion of the weight of the contents of the phial in the second process to the weight of its contents in the first, is the specific