CHAPTER II.BAROMETRIC MEASUREMENTSOF ALTITUDES.
The text books in physics present formulas for computing heights from barometric observations, based on physical laws which we will briefly give.
If the density of the air were constant throughout, the measurement of heights would be a problem of the simplest character; for as mercury weighs 10,500 times as much as air at the sea level, the mercurial column would fall one inch for every 10,500 inches of ascent above the sea. But air is compressible, and, in accordance with Boyle’s law, its density varies with the pressure to which it is subjected.
Now suppose the atmosphere divided into layers of uniform thickness, but so thin that the density may be considered uniform throughout.
Leth= the thickness of each layer.W = weight of a cubic foot of air at pressure H.W₁ = weight of a cubic foot of air at H.H₀ H₁, &c. = pressures measured in inches of mercury.
Then the pressure upon the unit of surface of any layer is greater than that upon the surface of next higher layer, by the weight of a volume of air whose base is the unit of surface and whose height is the thickness of the layer. If one foot be the unit of surface, then this quantity would behW. And to express it by height of mercury column, it is necessary to multiply by
which gives
But W : W₀ : : H : 30.
W₀ being the weight of a cubic foot air at the level of the sea (=.0807 at 32°F).
We have from the above
,
and the above expression for diminution may be written
.
If H₀ H₁ H₂ represent the pressures at the surfaces of the successive layers, we shall have
Multiplying these equations and suppressing common factors, we get
Ifhbe taken at one foot thennwould represent the number of feet vertically between two stations at which the barometric pressures are Hnand H₀ respectively.
By substituting for W₀ its value and taking logarithms we have
whence
For use in accurate observations, corrections are required for temperature, humidity and variation in the force of gravity.
La Place’s formula which includes terms derived from the consideration of these conditions is obtained as follows:
Suppose a portion of the atmosphere included between two stations at different altitudes to be divided into very thin laminæ.
Letzbe the distance of one of these from the surface of the globe anddzits thickness.
Let P be the pressure upon a unit of surface upon the lower side of this layer; and W the weight per cubic meter of the air at this pressure.
Then the pressure on the upper side will be less than P by an amount equal to the weight of a column of air whose base is a unit and height is equal todz.
Whence
If W₀ be the weight of a cubic meter of air at the temperature 0°Cand a barometric pressure of 0.ᵐ76, the weight of this same volume at pressure P and temperature ϴ would be
abeing the coefficient of dilatation of air which is here taken at .004 in consequence of the constant presence of watery vapor.
This expresses the weight at the surface of the globe. If transferred to the heightz, the weight would be diminished in the ratio of the squares of the distances from the center of the earth. We should then have
Substituting in equation 1, dividing by P and integrating between 0 andz, we get, by calling the pressure at the lower station P₀
the logarithm being Napierian.
From this we obtain
But the pressures P₀ and P are in direct ratio of the mercury columns which we will designate byh₀andh. These columns also vary in weight in accordance with the law of inverse squares of distance from the earth’s center, so that
Substituting in the value ofz, we have
But aszis so very small compared with R₁ we may replaceby.
Alsomay be neglected.
We shall then have
The weight W₀ refers to the heighth₁ the lower of the twostations. At the surface of the earth, this weight would be greater in the ratio of. But ashis always small compared with R this correction may be neglected.
But there is another of more importance which should be taken into account. On account of the spheroidal form of the globe weight varies with the latitude. If G represent the weight of a body at latitude 45°, then at any other latitudel, its weight, is found by multiplying G by
This factor is to be applied to W₀ in the formula. This is accomplished by multiplying the above value ofzby
In order to simplify the expression we will substitute forthe mean between the temperatures of the upper and lower stations, designated byt₀ andt. The factorthen becomes
and the value ofzmay be written
If M be used to represent the modulus of the Napierian logarithms we may write
in which the logarithms are of the common kind.
This is La Place’s formula.hin the expression is not the barometric height directly observed at the upper station, but this height reduced to the temperature of the lower station.
The value ofhas been determined by trial of the formula upon known altitudes. Ramond in his survey of the Pyrenees determined its value to be 18336.
The unknown termzin the second member is determined by successive approximations.
The first value being
This being substituted, we may have
Finally, these being substituted in the above value ofzwe get
The terms of this formula are generally reduced to tabular form for practical use.
Guyot’s formula which is derived from this, reducing meters to feet and the constants depending on temperature being changed to accord with Fahrenheit’s scale, is
The three terms after the first are the corrections. The first being that for the temperature at the two stations. The second is the correction for the force of gravity depending on the latitude.
The third contains, first the correction for action of gravity on the mercury column at the elevationz, and second a correction required for decrease in density of air owing to decrease in action of gravity at the greater elevation. The factorsbeing the approximate difference in altitude of the stations.
Plantamour’s formula, which has been much used, differs slightly from Guyot’s. The first coefficient is 60384.3. The denominator of temperature term is 982.26 and a separate correction is used for humidity of the air.
To use either of these formulas tables are necessary, of which thoseprepared by Lieut. Col. Williamson[1]are the most elaborate.
For the Aneroid in ordinary practice, formulas of much less complexity may be profitably used. The corrections depending upon the gravity of the mercury column would, in any case, be omitted. The other corrections may in very nice work be retained. But a correction depending on the effect of changes of temperature on the metallic work of the instrument should be carefully remembered. First-class Aneroids claim to becompensated, but a greater portion will need a correction which the purchaser can determine for himself, by subjecting the instrument to different temperatures while the pressure remains constant.
A modification of Guyot’s formula adapted to aneroid work was suggested in an excellent paper on the use of the aneroid, read before the American Society of Civil Engineers, in January, 1871.
It is
D is the difference of altitude in feet. H andhare the barometric readings in inches.
T andtare the temperatures of the air at the two stations.
Table IIis prepared for the use of this formula.
Other formulas will be given in another chapter.
[1]The use of the Barometer on Surveys and Reconnoisances. By R. S. Williamson. New York: D. Van Nostrand. London: Trubner & Co.
[1]The use of the Barometer on Surveys and Reconnoisances. By R. S. Williamson. New York: D. Van Nostrand. London: Trubner & Co.