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(found as in article 594) be about 6094-5 feet, then is z equal to 38′ 33′′-7=0; therefore, 2r=o—(e+e') = 38′ 33′′7-33′ 19′′, and r=2′ 37′′-35. Hence, o—(e+r) 19' 16"-85-33' 25"-35 == 14' 8"-5-848"-5;
also, Lh = L sin 1"+ Lv + Lz = 6·6855749 + 2.9286518 +5·3711022 = 2.9853289;
h= =- 966.8 feet.
and So that the height of Wisp Hill above Cross-Fell is 966.8 feet-that is, the latter station is higher than the former by 966.8 feet. In this example the refraction amounts to about of the intercepted arc.
The formulas are easily proved. Since angle HCC' + C'CO = 90°, and in the isosceles triangle OCC', 10+ C =90, hence HCC' = o; and from this it is evident that v = 10 ~ (e + r) = (e + r) — 4o in this case. Similarly, it can be proved that vo (e' +r) But v=v'; hence, er 10—10—e -r, and 2ro
(e + e'). Again, in the triangle CC'W, angle C' may be considered to be a right angle without sensible error in computing h, which is small in comparison with z, and CC' = 2; hence, z : h = 1 : sin v, and h = z sin v. But if is expressed in seconds, sin v : sin l′′ = v : 1", as v is small; hence, sin v = v sin l′′, and h = vz sin l".
601. If the height of the eye when observing the vertical angle of position of a station, taken at another station, is higher than the latter station, which is generally the case, the height of the eye being that of the instrument, the observed angle will thus be too great or too small by a few seconds, according as it is an angle of depression or elevation. Thus, if z is the distance of the stations in feet, h' the height of the eye in feet above one station, and " the angle in seconds, subtended by the height h' at the distance 7%9 that is, at the other station; then z: h'=1: sin v", and sin v"v" sin 1",
; hence, "=
sin v" =
Sometimes, instead of observing the elevation or depression of the top of the second station, it is that of the top of the signal that is observed, and of course a correction must also be made on this account exactly similar to that in the last paragraph.
602. The original base of a survey is reduced to its length at the level of the sea at that place, and an imaginary sphere, having the same radius as the earth at that place, is that on which all the triangles of the survey are conceived to be drawn. The distance between any two stations of the survey thus determined, can be reduced to the level of the sea at their mean latitude, as in article 562, by computing the radius of curvature of the earth at the place. It would be an advantage were geodetic surveys in every country conducted in reference to the same imaginary sphere, such as that of equal volume with the earth, alluded to in article 562.
EXERCISE. At the station of Black Comb in Cumberland, Scilly Bank appeared depressed 49′ 14′′; and at Scilly Bank, Black Comb was observed to be elevated 31′ 31′′, the distance between the stations being 121,028 feet; required the refraction and the height of Black Comb above Scilly Bank, the height of the instrument at both stations being 5 feet.
Ans. The correction of the angles of depression and elevation for the height of the instrument or "9"; the refraction r = l' 14" or of the intercepted arc; and the difference of height of the two stations or h = 1422 feet.
In the survey the heights of the stations are stated to be 500 and 1919; and hence their difference = 1419 feet.
603. When observations are taken at only one of two stations, the amount of refraction at the time of observation cannot be determined; and therefore the mean refraction must be taken, which is about of the angle measured by the distance between the stations at the earth's centre. This correction being assumed as the value of r, angle v of article 600, and then the difference of height, can be found as in that article. The effect of horizontal refraction is to increase the height of the station observed by a quantity in feet equal nearly to of the square of the distance in miles. For
instance, for a distance of 10 miles, the height would be increased by× 100 = 11 feet.
Leth the height of an object just visible at the distance d, were there no refraction,
h' the height of another object at the same distance just visible when there is a mean refraction. Then if h, h', are expressed in feet and d in miles (art. 238, Part I.), hď2, h′ = z (d — 1d)2= d2 nearly, and h―h' f d2.
The effect of mean refraction, therefore, would increase the height for a distance of 20 miles by about × 202, or 44 feet.
If the refraction at the time of observation were only instead of, then would h' (d — } d)2 = z · (§)2 d2 = f • f f ď2 = ƒ d2 nearly. And therefore the error on the height, when the refraction is in this extreme state, arising from adopting the mean refraction, would be = ( § — $) d2 =d; which, for a distance of 20 miles, would be = 44 feet. But this extreme case rarely occurs. Were the refraction, then would h' (d— 1d)2 = 3 ⋅ (1%)2 ď2 d2. And the error arising from adopting the mean refraction would be (5) dd nearly; which, for a distance of 20 miles, would give × 400 = 6·2 feet. 604. To avoid such errors, it is of importance that the zenith distances should be mutually taken at every two stations of each triangle, and also simultaneously, in order that the refraction for each pair of reciprocal observations may be the same as nearly as possible. The relative heights of the stations being known by the foregoing process, the absolute heights can easily be found by determining that of one station. A station is to be chosen for this purpose near to the sea; but instead of computing its height by observing the depression of the horizon, which is very uncertain on account of the unknown horizontal refraction, rendered still more irregular by the vapours exhaled at the surface of the sea, the more correct method of levelling or of mensuration of heights, explained in Part I., is to be in preference adopted.
EXPLANATION OF THE USE OF THE TABLES.
I. The principle on which the table of Apparent Depression of the Horizon is constructed, and also its use, are explained in article 442.
II. An example of the use of the table of Correction of Mean Refraction, is given in article 440. The pressure or height of the barometer is found at the foot of the page; and opposite to the apparent altitude in the column above the pressure, is found the correction, which is additive or subtractive according as the pressure is greater or less than 29-6, and it is marked accordingly by the sign + or Similarly, the temperature is found at the top line of the table, and under it and opposite to the apparent altitude is the correction, which is additive or subtractive according as the temperature is less or greater than 50°, and it is accordingly marked with the sign + or — (See the example, article 440.)
III. The principle on which the table of Mean Refraction is constructed, and its use, are explained in article 439. The table given here contains the mean refraction for the pressure 29.6 and temperature 50°. When these elements differ from these standard values, the corrections can be computed by the formula in article 440.
APPARENT DEPRESSION OF THE HORIZON.
H denotes the height of the eye in feet, and D the dip in minutes and seconds.