Графични страници
PDF файл
ePub

ARTICLE VII.

An Appendix to the Abstract of M. Ramond's Instructions for Barometrical Measurements. By Baden Powell, MA. of Oriel College, Oxford.

(Concluded from p. 274.)

In order to render more complete the foregoing compendium, and as some readers may wish for an account of the principles on which the formula is constructed, it may not be improper here to add for their convenience a brief explanation of it, together with some remarks on other points connected with the subject.

I. Outline of the Demonstration of the Formula.

M. Biot, in the small tract prefixed to his "Tables Barometriques Portatives," has given at large the demonstration of a formula which differs from the present only in some very slight modifications. I shall, therefore, do no more than present a sketch of his elegant investigation, the principles of which perhaps, be made sufficiently intelligible, without following him through all the detail of his analytical processes. The reader who is desirous of fully appreciating their beauty is referred to the original.

may,

As I here propose only to give a mere outline of the investigation of the formula, it will be superfluous to go through the elementary proof of the general theorem, which establishes the relation between pressure and elevation. We may set out by assuming that the difference of elevation, z M being the modulus of the common system of logarithms, and Ca coefficient involving the various corrections.

=

M

C

log. (1),

(1.) Our object is to discover the coefficient C. This M. Biot proceeds to do in the following manner : *-If we represent by & the density of the air under the pressure h, that of mercury being unity, we have = C h, and

8

Ch, and C. We may obtain, there

= h.

fore, the value of C, if we have, by very exact experiments, the ratio of the densities of air and mercury, under a given pressure of the atmosphere.

This ratio is not the same in all countries; for in all countries the weight of bodies has not the same intensity, as we learn

[ocr errors]

from experiments on the pendulum, and the ratio varies with

h

the intensity of gravity. Indeed is the density of the air under

• Mesures Barometriques, p. 7.

[ocr errors]

a given pressure, for instance, 29.921 inches, but according as the intensity of gravity is greater or less, a column of mercury of the constant height of 29.921 inches will weigh more or less; consequently air subjected to this pressure will be more or less compressed. Now by experiments with the pendulum in different latitudes, we find that calling the force of gravity in lat. 45°, unity, in a latitude 4, it will be expressed by 1 0·0028371 . cos. 24. The density being proportional to the weight will vary in the same ratio; that is to say, calling it & in lat. 45°, and under the pressure h, it will become for any other latitude, and under a column of mercury of the same height, [1-0-0028371 cos. 24]. The coefficient C, which expresses the ratio of the density to the height of the barometric column, ought to vary in the same proportion, and consequently becomes C [I −0·0028371 . cos. 2], which being substituted in the value of x, gives x = log. () in this way it will be suffi

M

C.[1-0-0028371. cos. 2 ]

[ocr errors]

cient to find the coefficient

:

M

C. [1-0-0028371. cos. 247 by experiment for a given latitude; for thus, 4 being known, we shall know also ; and the formula becomes applicable to all possible

M

C

1

1-0.002837. cos. 2

latitudes. The formula may be rendered more convenient by causing the denominator to disappear, which is easily done; for the fraction being developed in a series by division, becomes 1 + 0.002837 cos. 2 + 0·00000804857, cos. 2 2 4 + ...... or simply 1 + 0.002837 cos. 2 by confining ourselves to the first term, which is alone of sensible magnitude. Thus we shall have ≈ = [1 + 0·002837 cos. 24] log.

M

(2.) Thus far we have supposed that the value of the coefficient

8

Cor is the same in all the strata of the column of air; but it

h

is not so in nature; and many causes tend to make this ratio vary. The principal cause is the inequality of the temperature of the strata; for the elasticity of air is augmented by heat, so that with a less density, it can support an equal column of mercury, which makes the ratio, or C, vary.

This ratio also varies according to the greater or less quantity of aqueous vapour which is found suspended in the different strata; for this vapour weighs less than dry air of equal elastic

This expression is deduced from one given by Laplace, Mec. Cel. b. 10.-(See M. Ramond's First Memoir, Part II. p. 16.)

force, so that its presence in the different strata renders them proportionally capable of sustaining with a less density an equal column of mercury.

Lastly, the decrease of gravity as we recede further from the centre of the earth is another cause of the change; for by this decrease a column of mercury whose length is h, weighs so much the less, as we recede from the centre: if it weigh less, it compresses less the strata of air into which it is carried: thus the ratio of their density to the length of the column of mercury, or is no longer the same for these strata as for those which are below. If all other circumstances are alike, the densities of the strata of air which these columns compress will be likewise proportional to them. The ratio or C, therefore, ought to vary from one stratum to another proportionally to the force g.

[ocr errors]

8

h'

(3.) The amount of each of these corrections may be calculated on the following principles :

First, the action of temperature. From the influence of this cause, a mass of air whose volume is 1 at zero (centig.) becomes at t degrees, 1+t. 000375, the barometrical pressure remaining the same. Under a constant pressure, the densities of this mass are reciprocally as the volumes, and, therefore, if the density at zero be 1, the density at t degrees will be constant pressure; the ratio, or C, must, therefore, vary pro

portionally to this quantity.

[ocr errors]

1+t. 0.00375

.

under a

Secondly, the influence of aqueous vapour. According to the experiments of De Saussure and Watt, the weight of this vapour is to that of air as 10 to 14, while their elastic forces and temperatures are the same; that is to say, while the air and the vapour being at the same temperature, sustain equal columns of mercury. The substitution, therefore, of this vapour in the strata of the air, renders them specifically lighter without diminishing their elastic force. To obtain the value of this effect, let h be the barometrical pressure which supports a certain stratum of air: let us call F the elastic force of the aqueous vapour contained in it; that is to say, the part of the barometrical pressure which the vapour sustains. The whole weight of the stratum may be considered as composed of two parts, viz. of a certain quantity of vapour whose elastic force is F, and of a certain quantity of atmospheric air perfectly dry, whose elastic force is h.-F. Let p be the whole weight of the stratum, if it were composed entirely of dry air under the pressure h. The weight of the same volume of dry air under the pressure h - F will be pF). The weight of the same volume under the

h

h-F

pressure F will be 2. Lastly, if this volume remaining always

under the pressure F,were composed entirely of aqueous vapour, its pF

10

weight would be of the former; that is to say, 10 PF. Now

14

h

we know by very decisive experiments that in a mixture of vapour and air, which has attained a state of stable equilibrium, these two fluids are uniformly diffused throughout the whole space which they occupy. Thus the weight of the mixture in the preceding proportions will be equal to the sum of the weights of the air and vapour which occupy the given space under the pressures h F and F; that is to say, that this weight will be

[ocr errors][ocr errors][merged small][merged small]
[ocr errors]
[merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small]

duction of the vapour, the weight of the same volume of dry air submitted to the same pressure h, would be represented by p. The densities being proportional to the weights, if & represent the density of the stratum in the dry state, the density in the

[merged small][ocr errors][merged small][merged small]

(-), or 8. [1-] the

h

pressure remaining the same. Thus we see that the introduction of aqueous vapour in the strata of air makes the ratio or C, vary proportionally to (1).

8

,

The tension F is always very small at those temperatures at which barometrical observations are commonly made. Its value in metres for the point of extreme saturation may be calculated from a formula, given by Laplace, from the experiments of Dalton; whence we find,

At 0° centigrade F = 0.005122 metre; (= 0·20165 inch.) At 30° centigrade F = 0.031690 metre: (= 1.24765 inch.) and within these limits, which are nearly those of barometrical observations, the increase of F may be sufficiently well represented by arithmetical progression, and will be

F=0·005122 m. +0·0008649 m. t (=0·20165 in. +0-03304 t) in. t being the temperature centigrade. Although this formula is not rigidly accurate, it is sufficiently so in practice on account of the little effect which it has on the observed heights.

But before it can be applied to the state of the atmosphere, it requires to be modified. It relates to the point of extreme saturation at which the atmosphere is scarcely ever found; and consequently the value of F will almost always be rather greater than the truth. No general determination can be given of the quantity of vapour suspended in the atmosphere. This quantity is extremely variable on different days; it varies even from one stratum to another in a manner very irregular, and often abrupt, as we see on mountains where strata very little charged with

vapour succeed others which are at the maximum of humidity. However, setting aside these extraordinary circumstances, every thing leads us to believe that we shall follow nature most closely if we avoid these extreme cases; and thus what seems most simple is to take for the expression for F in the atmosphere the half of the value which corresponds to the point of extreme humidity; that is to say,

F = 0·002561 metre + t . 0·00043245 metre

(0-10082 inch + t.0-01652) inch.

In substituting this value in the expression for the coefficient C, it must be multiplied by the variable factor, but on account of the minuteness of this correction, and also on account of the small difference in the values of h within the limits of ordinary measurements, it will suffice to put for h, the constant value 0.76 m. 29.921 in. which is the mean pressure at the level of the sea. This substitution will possess also the advantage of giving a less correction for the humidity in the higher strata of the column, which agrees with nature; for the humidity of these strata generally diminishes in proportion as we ascend, and sometimes the most elevated are extremely dry. Adopting this simplification, we have

1

2 F

= 1

7 h

2

7.0.76 m.

=

[0·002561 m. + t . 0·00043245 in.] = 10009628-0001626. t.

Without sensible error this expression may be put under the

following form, [1

may

C =

-·0001627.t] which gives

•0009628][1-
A[10009628] .g [1 ⚫0001627.t]

1 + t .00375

The factor, depending on t, which is found in the numerator, be combined with that which arises from the temperature. On account of the smallness of the coefficient 0001627, we may without sensible error substitute

1-0001627 .t.

[blocks in formation]

Thus we have in the denominator the product [1+0001627.t] [1 + t. 0·00375]. In performing the multiplication we may neglect the product of 0001627 x 00375: and thus it becomes [1+0039127.t]. The coefficient of t in this result differs so little from, 004, or 250' that we may, without fear of error, substitute for it this last value, which will simplify the calculation. We have, therefore,

[blocks in formation]

Thirdly, the variation of the force of gravity must affect both the coefficient or ratio of densities of air and mercury, and also the observed heights of the column of mercury at the two

« ПредишнаНапред »