were provided with several barometers, one of which was kept constantly at the Great Falls of St. John, in a building to which the air had free access: the man who had it in charge made the observations at 8 A. M., at noon, and at 4 P. M. daily, and the travelling party always endeavoured to make their observations as nearly as possible simultaneously with those at the Falls. The instruments employed for such purposes are, in general, similar to the ordinary barometer but more delicate; occasionally, however, a Siphon Barometer, of a kind invented by M. Gay Lussac, and now frequently made in this country, has been used. A formula for determining the relative heights of stations, from the heights of the mercurial columns supported by the atmosphere at the stations, may be thus investigated.

429. The particles of the atmosphere which surrounds the earth gravitate towards the surface of the latter, but, being of an elastic nature, they exert an expansive force in every direction; and, when the atmosphere is tranquil, that part of the force which, at any point, acts from below upwards is in equilibrio with the gravity or weight of the column of air which is vertically above that point. Now, in any small volume of air within which the density may be considered as uniform, that density is proportional to the external force which, acting against it, tends to compress it: and this force of compression on every part of the surface of the volume is equal to the weight of the column of air which is incumbent on that part; it follows therefore that, since the weight of a volume of air of uniform density is proportional to the density, the weights of small portions of air, equal in volume, in every part of the atmosphere (gravity being supposed constant), will be the same fractional parts of the weights of the atmospherical columns above them. This law being admitted, let AB be part of the surface of the earth, AZ the indefinite height of a slender cylindrical column of the atmosphere, and let this be divided into strata of equal thickness Aa, ab, bc, &c. Then, taking w to represent the weight of the column zf,

1

n

Ꮓ

C

w may represent the weight of the small column ef

1

n+1

and w+-w, or w, will express the weight of the

n

n

column ze. Again, by the law above mentioned, 1("+1w)

n

n

W

will represent the weight of the small column de, and

D D

(n+1w), or (n+1)2 w, will be the weight of zd.

W

Continuing in this manner it will be found that the weights of the columns zf, ze, zd, &c., and also of the columns ef, ed, de, &c., will form a geometrical progression whose common n+1 ratio is : that is, the weights or pressures at ƒ, e, d, &c.

n

will form a geometrical progression, while the depths fe, fd, fc, &c., form an arithmetical progression.

430. But, by the nature of logarithms, if a series of natural numbers be in a geometrical progression, any series of numbers in an arithmetical progression will be logarithms of those natural numbers; therefore, if there were a kind of logarithms adapted to the relation between the densities of the air and the depths of the strata, on finding the densities of the air at any two places as between a and b and between c and d, such logarithms of those densities would express the depths fa and fd, and the difference between the logarithms would be equal to ad, which is the height of d above a. Thus ad= log. dens. at a — log. dens. at d, or

There are no such logarithms, but, from the general properties of logarithms, the formula may be adapted to those of the ordinary kind. Thus, the weight of an atmospherical column, as Af, is equal to the sum of the increasing weights of the series of strata from ƒ down to A, and may be represented by the product of the greatest term (the weight of the air in aa) by some constant number M, which is therefore the modulus of the system of logarithms whose terms are fe, fd, fc, &c., to fA. Now M may represent the height of a homogeneous atmosphere whose uniform density is equal to that of the stratum Aa, and whose weight or pressure on A, is equal to that of the real atmosphere; and since such column, at a temperature expressed by 31° (Fahrenheit's

* From the series 1, 2 (art. 213. Elem. of Algebra), we have n мw= log. (1+w)" where nмw may represent ƒ A, and (1+w)" the weight of the stratum in Aa: let the first member be represented by x and (1+w)" by y. Then, by subtraction, in the series 2, the increment da of nм w is мw; and after developing (1 +w)n, (1 +w)2+1 by the binomial theorem, we have, by subtraction, the value of dy, the increment of y : from the values of these increments it will be found that ydx мdy. But the integral of ydr expresses the sum of the weights of all the strata; therefore the integral of мdy, that is, мy, or the product of the greatest term (the weight of Aa) by the modulus м of the system, is equal to that sum or to the weight of the column aƒ.

=

thermometer), would hold in equilibrio at the level of the sea a column of mercury equal in height to 30 inches, it follows (the heights of two columns of homogeneous fluids, equal in weight, being inversely proportional to their densities) that the height of the column of homogeneous atmosphere at that temperature would be 4343 fathoms, and this may be considered as the value of M. The modulus of the common logarithms is 0.4343; and since, in different systems, the logarithms of the same natural number are to one another in the same proportion as their moduli, we have dens. at a dens. at d

0.4343 : 4343 :: com. log.

dens. at a

: 10000 com.log.dens. atď

and the last term is equivalent to atmospheric log.

dens. at a

dens. at d

But the heights of the columns of mercury in a barometer, at any two stations, a and d, are to one another in the same ratio as the densities of the atmosphere at those stations, or as the weights of the columns of air above them; therefore the height of d above a may be expressed, in fathoms, by the barom. at a formula barom. at d'

10000 common log.

or by its equivalent, 10000 (log. barom. at a - log. barom. at d). Thus the height of the column of mercury in a barometer being observed at any two stations, as a and d, as nearly as possible at the same time, there may be obtained the relative heights of the two stations.

In using the formula it is evident that the heights of the mercurial columns at the two stations ought to be reduced to those which would have been observed if the temperature of the air and of the mercury were 31°. In order to make such reduction, since the mean expansion of a column of mercury is a part expressed by .000111 of the length of the column for an increment of temperature equal to one degree of Fahrenheit's thermometer, if this number be multiplied by the difference between the temperature at each station and 31° (that temperature being expressed by the thermometer attached to the instrument), the product will be the expansion of the column (in parts of its length) for that difference. Therefore, multiplying this product into the observed height of the column, in inches, at each station, we have the expansion in inches; and this last product being subtracted from the observed height, if the temperature at the station be greater than 31°, or added if less, there remains the corrected height of the column. The logarithms of these corrected heights, at the two stations, being used in the last formula

above, the resulting value of ad will be a first approximation to the required height. By experiments it has been found that the relative height thus obtained varies by of its value for each degree of the thermometer in the difference between 31° and the mean of the temperature of the air at the two stations: consequently, if d be the difference between 31° and the mean of two detached thermometers, one at each station, the correction on account of the temperature of the

ad

air will be expressed by the formula d, where ad is the 435

first approximate value of the height. This correction is to be added to that approximate height when the mean of the detached thermometers is greater than 31°, and subtracted when less.

The result is very near the truth when the height of one station above the other does not exceed 5000 or 6000 feet and when the difference of temperature does not exceed 15 or 20 degrees: in other cases, more accurate formulæ must be employed, and that which is given by Poisson in his " Traité de Mécanique" (second ed. No. 628.), when the measures are reduced to English yards, and the temperatures to those which would be indicated by Fahrenheit's thermometer, is

H = A { log. barom. at a log. k};

in which н is, in yards, the required above the other, as d above a.

height of one station

T

· T

9990

k = height of barom. at d, × (1+

t and t' are the temperatures of the air, by detached thermometers at the two stations.

T and T' are the temperatures of the mercury, by attached thermometers.

a is the common latitude of the stations, or a mean of the latitudes of both if the stations be distant from each other in latitude.

431. The siphon barometer is a glass tube formed nearly as in the annexed figure, and containing mercury; it is hermetically sealed at both extremities, and has at A a very fine perforation, which allows a communication with the external air without suffering the mercury to escape. The atmosphere pressing on the mercury at N, balances the weight of a column of that fluid whose upper extremity may be at M. There is

M

a sliding vernier at each extremity of the column; and, the zero of the scale of inches being below N, the difference between the readings at M and N on the scale is the required height of the column of mercury.

M

In other mountain barometers the tube is straight, and its lower extremity, which is open, enters into a cistern AB containing mercury: the bottom of the cistern is of leather, and by means of a screw at C, that mercury can be raised or lowered till its upper surface passes through an imaginary line, on which, as at N, is the zero of the scale of inches ; M being the upper extremity of the column of mercury in the tube, the height MN is read by means of a vernier at M. The external air presses on the flexible A bottom of the cistern, and this causing the surface N of the mercury at N to rise, or allowing it to fall, the corresponding variations in the elasticity of the air in the part AN of the cistern, produce the same effect on the height of the mercurial column MN, as would be produced by the external air if it acted directly on the surface at N.

The barometer invented by Sir H. Englefield has no screw for regulating the surface of the mercury in the cistern with respect to the zero of the scale of inches; and the atmosphere, entering through the pores of the box-wood of which the cistern is formed, presses directly at N on the surface of the mercury; there can, consequently, be only one state of the atmosphere in which the surface is coincident with that zero. The exact number of inches and decimals, on the scale, at which the extremity м of the column of mercury stands when the surface at N coincides with the zero is found by the artist, and engraven on the instrument; and, when the top of the column is at that height (or at the neutral point, as it is called) no correction is necessary on account of the level of the mercury in the cistern. In other cases such correction is determined in the following manner: The ratio between the interior area of a horizontal section through the cistern, and the area of a like section through the bore of the tube, is ascertained by the artist and engraven on the instrument : let this ratio be as 60 to 1: then the lengths of cylindrical columns, containing equal volumes, being inversely proportional to the areas of the transverse sections, the required correction will be of the difference between the height of the neutral point, and that at which the top of the column stands in the tube. This correction must be added to, or subtracted from, the height read on the scale according as the top of the column is above or below the neutral point.