Example. To find the number of feet passed through by a falling body in the ninth second of its descent.

Here we have 9×321-2891-16-273152, which is the number of feet passed through in the ninth second of the descent.

MOTION OF FLUIDS.

The velocity with which water will flow out of a hole at the side or in the bottom of a cistern, will be the same as that which a heavy body will acquire in falling from the level of the water surface to the level of the orifice, and may easily therefore be computed by a reference to the laws of falling bodies. The atmosphere exerts a pressure of about 14.7 lbs. per square inch, or 2116.4 lbs. per square foot, on all bodies on the earth's surface; and if the atmosphere be pumped out of the space beneath a piston, while suffered to press on its upper surface, the piston will be forced downward in its cylinder with a pressure of 14.7 lbs. on each square inch of the piston's area. In a common sucking pump the water is drawn up after the piston, in consequence of the production of a partial vacuum beneath the piston; and the water in the well being subjected to the pressure of the atmosphere while the pressure is removed from the water in the pump barrel, the water rises in the suction pipe, and would continue to do so if the pump were raised further and further up, until a column of water had been interposed between the pumpbarrel and the well sufficiently high to balance the weight of the atmosphere. The water will cease to rise any higher after this altitude has been attained.

When we know the weight of a cubic inch or cubic foot of water, it is easy to tell the number of cubic inches or cubic feet that must be piled upon one another to produce a weight of 14.7 lbs. on the square inch or 2116-4 lbs. on the square foot; and it will be found to be 408 cubic inches in the case of the cubic inches, or a column 1 inch square and 34 feet high, or 34 cubic feet in the case of the cubic feet. Mercury being about 13.6 times heavier than water, a column of mercury 1 inch square and 30 inches high will weigh about 15 lbs. A column

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of air high enough to weigh 15 lbs., will be 773-29 times higher than a column of water of the same weight-water being 773-29 times heavier than air at the ordinary barometric density of 29.9 inches of mercury. In other words, the height of a column of air 1 inch square and the same density as that on the earth's surface, that will weigh 15 lbs., will be 34 × 773·29 — 25521·86 feet, or taking the atmospheric pressure at 14-7 lbs., the height will be 26214 feet. The velocity therefore with which water will rush into a vacuum, will be equal to that which a heavy body will acquire in falling through a height of 34 feet. The velocity with which mercury will flow into a vacuum, will be equal to that which a heavy body will acquire by falling through a height of 21 feet; and the velocity with which air will flow into a vacuum, will be equal to that which a heavy body will acquire by falling through a height of 26214 feet. Now the velocity which a heavy body will acquire in falling through 34 feet will be equal to the square root of 34, which is 5'8 multiplied by the constant number 8·021; or it will be 46.5218 feet per second, which consequently will be the velocity with which water will flow into a vacuum. The velocity with which mercury will flow into a vacuum will be 12.83 feet per second, for the square root of 21 is 1·6 nearly, and 1.6 multiplied by 8.02112.8336. The velocity with which air weighing 0.080728 lbs. per cubic foot will flow into a vacuum will be 1298.5999 feet per second; for the square root of 26214 is 161.9 nearly, which multiplied by 8.021-1298.5999 feet per second. The density of the air here supposed is the density at the temperature of melting ice. At the ordinary atmospheric temperatures the density will be somewhat less; and if the density be taken so that the height of the homogeneous atmosphere, as it is called, or of that imaginary atmosphere which produces the pressure and which is supposed to be of uniform density throughout its depth-is 27,818 feet, then the velocity of the air rushing into a vacuum will be a little greater than what it has been here reckoned at, or it will be 1338 feet per second. These velocities it will be understood are the theoretical velocities, which can in no case be exceeded; but which are fallen

short of in practice to a greater or less extent, depending on the size and form of the orifice through which the air enters, and other analogous circumstances.

The velocity with which steam or any vapour or gas whatever will rush into a vacuum, can easily be determined when we know its pressure and density; for taking into account the density, or the weight of one cubic foot, we have merely to see how many of these cubic feet must be piled upon one another to produce the given pressure or weight upon the square foot of base; and the velocity will be in every case the same as that which a heavy body would acquire in falling through the height of the column required to produce the weight. Thus it is found that the density of steam of the atmospheric pressure is about 1700 times less dense than water. Mr. Watt reckoned that a cubic inch of water produced a cubic foot or 1728 cubic inches of steam, having the same pressure as the atmosphere; and if the pressure of the atmosphere be equal to the pressure produced by 34 feet of water, then, if we reckon steam as 1700 tines less dense than water, it would require 1700 columns of steam, each 34 feet high, placed on top of one another, to exert the same weight or pressure as one column of water 34 feet high. Now 1700 hundred times 34 is 57800, which therefore is the height a column of steam 1700 times less dense than water would require to have in order to balance the pressure of the atmosphere or of 34 feet of water. The velocity which a body would acquire in falling through a height of 57800 feet, is 1926.6 feet per second; for the square root of 57800 is 240-2 nearly, and 240.2 multiplied by 8·021=1926.6442 feet per second, which is consequently the velocity with which steam of this pressure would rush into a vacuum. The velocity with which steam of a greater pressure than that of the atmosphere will rush into a vacuum, will not be sensibly greater than that of steam of the atmospheric pressure. For as the density of the steam increases in nearly the same ratio as its pressure, the column will require to be as much lower, by virtue of the increased density, as it requires to be higher to give the increased pressure. In other words, the height of the theoretical column of steam required

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to produce the pressure, will be nearly the same at all pressures; since a low column of dense steam will produce the same pressure as a high column of rare, and the density and pressure advance in nearly the same ratio. It may hence be concluded that steam of all pressures will rush into a vacuum with a velocity of about 2,000 feet per second, if the vacuum be perfect and the How unimpeded.

If steam, instead of being suffered to escape into a vacuum, be made to issue into a vessel containing steam of a lower pressure, the velocity of efflux will be the same as that which a heavy body would acquire in falling from the top of the column of steam required to produce the greater pressure, to the top of a lower column of the same steam adequate to produce the lesser pressure. Thus if we have steam with a pressure of two atmospheres, flowing into steam with a pressure of one atmosphere, then, inasmuch as the density or weight of the steam increases very nearly in the same proportion as its pressure, a cubic inch of steam with a pressure of two atmospheres will be about twice as heavy as a cubic inch of steam with a pressure of one atmosphere. Such steam, therefore, instead of being 1700 times less dense than water, will be the half of this or only 850 times less dense than water. A column of this steam, therefore, 850 times 34 feet=28900 feet high, will exert a pressure of one atmosphere, or about 15 lbs. on each square inch; and a column of twice this height, or 57800 feet, will exert a pressure of two atmospheres or 30 lbs. on each square inch. The velocity with which the steam will rush from one vessel to the other, will be the same as that which a heavy body would acquire in falling from the height of the column of the denser steam required to produce the higher pressure to the top of the column of the same steam of such height as would produce the less pressure; and as in this case the heights of such columns will be 1700 × 34 feet, and 850 × 34 feet, or 57800 and 28900 feet, the difference of height will be 28900 feet; and the velocity of efflux from one vessel into the other will be equal to that which a heavy body would acquire by falling through a height of 28900 feet. Now the square root of 28900 is 170; and 170

multiplied by 8·021=1363·57 feet per second, which is the velocity with which steam with a pressure of two atmospheres would rush into steam with a pressure of one atmosphere. This consequently may be reckoned as the velocity with which steam of 15 lbs. pressure above the atmosphere would rush into the atmosphere. Such velocities at different pressures are exhibited in the following table :

VELOCITY OF EFFLUX OF HIGH-PRESSURE STEAM INTO THE

This table is computed by taking the difference of the two pressures for the effective pressure, which effective pressure is expressed in pounds per square inch, divided by the weight of a cubic foot of the denser fluid in pounds, and the square root of the quotient is multiplied by 96. The denser the fluids are the less, it is clear, will be the velocity of efflux which a given difference of pressure will create; for the heights of the columns, and also the difference of their heights, will be small in the proportion of the density of the denser fluid. The more dense the fluid is, the larger becomes the mass of matter which a given pressure has to move. With steam of 16 lbs. pressure flowing into steam or air of 15 lbs. pressure, the moving pressure is 1 lb., and the velocity of efflux is 482 feet per second. With steam of 101 lbs. pressure flowing into steam or air of 100 lbs. pressure,