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the temperature at the bottoin, and t that at the top; where

fore,

48237 { 1 + 0·0012 (t + t') = 48237 + 58 (t + t').

Or, to render the numbers more easily recollected, without much affecting the accuracy of the result, if H be the height required,

H =

= { 48000 +60 (t+t)}{

1/B
(B-b,1
B+3 B+

+

3

+ &c.}...(A). This is correct only on the supposition that B and b are reduced by calculation, or approximate tables, to the same temperature. But it is known from experience, that the height varies about 3 feet for every degree of difference of the attached thermometers at top and bottom. Now, if denote the temperature of the attached thermometer at the bottom, and

that

at the top, then ·3 (r—7′) must be the correction which is to be subtracted, when is greater than, otherwise added. Hence, finally,

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great heights, and need seldom be attended to, as the error for heights of about

5,000 feet, it will be nearly 40' of the whole.

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This term may therefore be safely rejected for any height usually measured barometrically; whence formula (B) becomes

B-b
B+b

H = {48000 + 60 (t + t') } } +6

3 (TT)........

....(C).

To assist the memory, if 48000+60 (t+t') be denoted by c,

B-b

B+b

by d, and -3 (—) by e, formula (C) becomes

Hcd-e............(c).

The whole of the numerical co-efficients are multiples of the number 3, the last of them; the second is twenty times the last, or 20 x 3 = 60; and the first is eight hundred times the second, or 800 × 60 = 48000, which is 16000 times the last, or

16000 × 3 = 48000; which will afford some facility in recovering them when indistinctly remembered *.

We shall now proceed to the application of this formula.

EXAMPLE I.

To determine the height of Allermuir, one of the Pentland Hills, the following mean of a number of observations, with excellent barometers, were taken after the manner of those employed to obtain the height of Benlomond, given in a former Number of this Journal.

The observations were made on the Calton Hill, of known height, 355 feet, with the Observatory barometer, by Mr Thomas Henderson, and on the summit of Allermuir, by myself, on the 26th of July 1828.

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It may be observed, that the number 60 is ten times the number of working days in a week, that the number 48000 is eight hundred times 60, and -3 is the twentieth part of 60, so that the whole of the co-efficients are derived from one number (6), the number of work days in a week, and by that means can hardly be forgotten.

By introducing a more refined calculation with logarithms, the height would have been about 10 or 12 feet more, though it is obtained at considerable risk of error, from errors either in the ordinary tables, or oversights in the steps of the calculation, which to unpractised persons frequently occur; whereas the foregoing requires only a very simple arithmetical computation, where no figure of real utility is suppressed, which is frequently done, to make the operation look simpler than it really is.

EXAMPLE II.

On the 12th of September 1829, the following observations were made with Mr Adie's sympiesometer.

At Edinburgh, 270 feet above the sea.

S'

272 fathoms, and t = 57° 7 Fahrenheit;

At the top of Allermuir, on the same day,

S = 490 fathoms, and t = 50°-3 Fahrenheit;
Hence SS' 490-272 218 fathoms ;

And tt 57°⋅7 + 50°•3 = 108°, which, from the
engraved scale on the instrument, gives the factor
m = 1.053;

Whence 218 x 1.053 229.5 fathoms, or 1377 feet.

If to this, 270 feet be added, for the height of the lower station, at Edinburgh, above the sea, we shall have 1647 feet for the height of Allermuir, above mean-tide at Leith. This exceeds the former by about 43 feet, which must be partly ascribed to the unfavourable state of the weather at the time, and partly to a small error in the foregoing formula. On the whole, I consider the mean of these results, or 1625 feet, to be nearly the true height, as I have found it from other observations.

EXAMPLE III.

In the month of August 1830, with a mountain-barometer of the best construction, the following observations were made at

Fort-William, and on the top of Ben Nevis, to determine its

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A small quantity in so considerable a height, considering the

simplicity of the last method.

As the weather was unfavourable, and the barometer changeable, with only one observer, who took the observations, first, at the bottom in the morning, then at the top about mid-day, and again at the bottom in the evening,—of which the mean of the first and last was reckoned the true height of the mercurial column at the bottom, there might have been some error arising from this cause. As the barometer continued to rise somewhat gradually, the error from this source must likely be small. At all events, whatever error attends the use of the one instrument likewise affects the other, since they accord so well when used together with equal care.

I have likewise calculated the height of Ben Nevis more rigorously by employing logarithms, and using the dew-points, besides other refinements, which would increase the height to about 4430 feet. As this exceeds all the heights I have ever met with attributed to it by at least 50 feet, I cannot say what confidence is to be placed in it, more especially, as I have been informed the Ordnance Surveyors make it only about 4360 feet, or 70 feet less. From the great distances of their stations, a very slight error in the angle of elevation, arising from the variable nature of terrestrial refraction, will produce a considerable error in their results in feet. Indeed, according to Mr B. Bevan's paper in the Philosophical Transactions for 1823, Part I., there are errors in the altitudes of some of the stations in England, of from 50 to 100 feet, in heights of between 700 and 900 feet! In this case I cannot say what confidence may be placed in that of Ben Nevis, though, in our measurement of Benlomond, the correspondence was as close as could be desired.

From the present measurements, too, it appears that the sympiesometer is an instrument which, when in good order, may be confidently trusted as giving results, when carefully used, very near the truth. The formula now investigated, appears to give results rather too small by about one-hundredth of the whole, and this is the reason why those by the sympiesometer seem to be, on comparison, too great. If one-hundredth of the height by the formula be added to itself, the final result would agree very closely with the logarithmic process.

54. SOUTH BRIDGE, EDINBurgh.

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