H

Dividing both terms of the fractional coefficient of by

H

the denominator, and neglecting the quantity T— 32, in comparison with 9990, we have,

The quantity z denotes, not only the height, but also the volume of the column of air aB, at 32°. When the temperature is changed from 32°, the pressures remaining the same, this volume will vary, according to the law of GAY LUSSAC.

If we suppose the temperature of the entire column to be a mean between the temperatures at B and a, which we may do without sensible error, the height of the column will become, Equation (153),

Hence, to adapt Equation (157) to the conditions proposed, we must multiply the value of z by the factor,

shown above, and multiplying the resulting value of z, by the factor 1+ .00102(t + t' — 64), we have,

p The factor is constant, and may be determined as ма follows: select two points, one of which is considerably higher than the other, and determine, by trigonometrical measurement, their difference of level. At the lower point, take the reading of the barometer, of its attached thermometer, and of a detached thermometer exposed to the air. Make similar observations at the upper station. These observations, together with the latitude of the place, will give all the quantities entering Equation (158), except the factor in question. Hence, this factor may be deduced. It is found to be 60345.51 ft. Hence, we have, finally, the barometric formula,

To use this formula for determining the difference of level between two stations, observe, simultaneously, if possible, the heights of the barometer and of the attached and detached thermometers, at the two stations. Substitute these results for the corresponding quantities in the formula; also substitute for the latitude of the place, and the resulting value of z, will be the difference of level required.

;

If the observations cannot be made simultaneously at the two stations, make a set of observations at the lower station; after a certain interval, make a set at the upper station then, after an equal interval, make another set at the lower station. Take a mean of the results of observation at the lower station, as a single set, and proceed as before.

For the more convenient application of the formula for the difference of level between two points, tables have been computed, by means of which the arithmetical operations are much facilitated.

Work due to the Expansion of a Gas or Vapor.

201. Let the gas or vapor be confined in a cylinder closed at its lower end, and having a piston working air-tight. When the gas occupies a portion of the cylinder whose height is h, denote the pressure on each square inch of the piston by p; when the gas expands, so that the altitude of the column becomes x, denote the pressure on a square inch by y.

G P

h

Ꭺ

L

Fig. 172.

Since the volumes of the gas, under these suppositions, are proportional to their altitudes, we shall have, from MARIOTTE'S laws,

whence,

py: x: h;

xy = ph

If we suppose p and h to be constant, and x and y to vary, the above equation will be that of an equilateral hyperbola referred to its asymptotes.

Draw AC perpendicular to AM, and on these lines, as asymptotes, construct the curve NLH, from the equation, xyph. Make AG h, and draw GH parallel to AC; it will represent the pressure p. Make AM = x, and draw MN parallel to AC; it will represent the pressure y. In like manner, the pressure at any elevation of the piston may be constructed.

Let KL be drawn infinitely near to GH, and parallel with it. The elementary area GKLH will not differ sensibly from a rectangle whose base is p, and altitude is GK. Hence, its area may be taken as the measure of the work whilst the piston is rising through the infinitely small space GK. In like manner, the area of any infinitely small element, bounded by lines parallel to AC, may be taken to represent the work whilst the piston is rising through the

height of the element. If we take the sum of all the elements between the ordinates GH and MN, this sum, or the area GMNH, will represent the total quantity of work of the force of expansion whilst the piston is rising from G to M. But the area included between an equilateral hyperbola and one of its asymptotes, and limited by lines parallel to the other asymptote, is equal to the product of the coordinates of any point, multiplied by the Naperian logarithm of the quotient obtained by dividing one of the limiting ordinates by the other; or, in this particular case, it is equal to ph × 1(2). Hence, if we designate the quantity of work performed by the expansive force whilst the piston is moving over GM, by q, we shall have,

This is the quantity of work exerted upon each square inch of the piston; if we denote the area of the piston, by A, and the total quantity of work, by Q, we shall have,

Q = Aph × 1(2) = Aph × 1(7) · (160.)

If we denote by c the number of cubic feet of gas, when the pressure is P, and suppose it to expand till the pressure is y, we shall have, Ah = c; or, if A be expressed in square

Finally, if we suppose the pressure at the highest point to

be p', we shall have,

Q

=

144cp × 1(2),

an equation which gives the quantity of work of c cubic feet of gas, whilst expanding from a pressure p, to a pressure p'.

Efflux of a Gas or Vapor.

202. Suppose the gas to escape from a small orifice, and denote its velocity by v. Denote the weight of a cubic foot of the gas, by w, and the number of cubic feet discharged in one second, by c, then will the mass escaping in

сго

one second, be equal to and its living force will be

сго

equal to v. But, from Art. 148, the living force is

double the accumulated quantity of work. If, therefore, we denote the accumulated work by Q, we shall have,

But the accumulated work is due to the expansion of the gas, and if we denote the pressure within the orifice, by p, and without, by p', we shall have, from Art. 201,

Substituting for g, its value, 32 ft., we have, after