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atmosphere, temperature, wind, &c., but on an average is usually taken as 0.16 of the curvature cc', Fig. 152. We have therefore to deduct 0.16 cc' from the calculated height, or we may deduct 0.16 of the angle CAC' from the observed angle BAC.

BC = d

B.

Fig. 153. Trigonometrical Levelling.

In this case formula (4) becomes-
BC'd tan BAC + 0.00000002d2
Formula (5) becomes-

=

BC'd tan (BAC + 0.000069d). Formula (6) becomes

sin (BAC -0.00001316d) COS (BAC + 0.00015143d) Formula (7) becomes—

BC' = d

C

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B

(8)

(9)

+0.000000023936d2... (10)

sin (BAC +0.00006913d)
COS (BAC +0.00015143d)

(11)

Reciprocal Observations for Cancelling Refraction.— The effects of refraction may be eliminated by taking reciprocal observations at each station as shown at A and B, Fig. 153, where the angle of elevation a is measured at A and the angle of depression ẞ is measured at B.

In this case we have

BC' = d

sin(a + B) cos (a+B+0)

BC' = d

where o is the angle AOB as before. If the zenith distances & and d' are measured, we have― sin('-8) (13) cos (8' - 8+0) When o is very small compared with a and ẞ or 8 and 8' it may be neglected, and (12) and (13) become respectively— BC' = d tan (a +ß) . . (8' – 8) . .

BC' = d tan

A

α = A

B

B=B+

'h cos A

(ODA)

d sin I'

'h' cos
d sin 1"

B

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Fig. 154. Trigonometrical Levelling.

Reduction of the observed Angles from the Summits of Signals. When the stations cannot be seen from each other, signals are erected, as aa and вʊ, Fig. 154. In this case the observed angles are A and B, and these have to be reduced to a and B. Let the heights of the signals above the instrument be h and h', then

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(12)

(14) (15)

(16)

(17)

In (16) and (17) the expressions in brackets are in seconds

of arc.

If zenith distances are observed as A' and B', the corrected angles will be-

B

b

Fig. 155. Height of Spire.

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If the height of a conical spire, as AB, Fig. 155, cannot be measured, take the circumference at a and b, and deduce the radii

AB=

A

AB=

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Rh

R-r

If the slant height / be measured

R
R-r

(18)

(19)

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Fig. 156.-Heights by Observation to Sea Horizon.

r and R. If the vertical height h between a and b be measured, we have

B

{/+(R-r)}{1-(R − r)}

Measurement of Heights by Observation to the Sea Horizon. If in Fig. 156 the zenith distance & be measured, then expressing 8- 90° in seconds, and taking m as the coefficient of refraction (0.08 on an average), r as the earth's mean radius, we get

2

I

2

sin I'

ВС

c' = Jr (in 1")" (6 – 90°)"{ 1 + (1)*(8-90°)")

I -m

Im

All observed angles in trigonometrical levelling are first to be

N

reduced to allow for height of axis of instrument above ground or station; or the height of instrument must otherwise be allowed for.

Precise Spirit Levelling.-Precise spirit levelling may be defined as the carrying of a very long line of levels, usually from the sea coast to inland points, for the accurate determination of the elevation of the principal bench marks in a large general survey of a country such as the Ordnance Survey of the United Kingdom, the United States Coast and Geodetic Survey, &c. Lines of precise levels are carried across the country, upwards of 1,000 miles in length, and from the bench marks established on these lines the levelling of the whole country is executed.* Lines of precise levels are also required for the execution of special engineering works, the determination of water slopes, canals, &c. In order to carry such a long line of levels with accuracy, special precautions and methods of working must be observed.

Instruments: Level. The great accuracy effected in precise spirit levelling is as much owing to the methods employed and the precautions observed in making the observations as to the special instruments used.

When a special level is used, it is usually of a magnifying power of telescope of about 40 or 50 diameters, and with three levelling screws. A mirror is provided fixed above the bubble tube in which the bubble is seen by the observer, and its position is observed without the eye being moved from the eyepiece. There is a screw called the "elevating screw" under the support of the telescope, next to the eyepiece, by means of which the bubble is brought accurately to the centre of its run while taking the observation, so that its position as seen in the mirror is correct at the instant of reading the staff. This screw of course moves both the telescope and the bubble tube without altering their relative positions. The spirit level should be very sensitive, and one division

* It is not possible to carry elevations from one triangulation station to another with very great accuracy by means of vertical angles or "trigonometrical levelling" owing to the great variations and uncertainty of refraction. The elevations of the very high principal trigonometrical stations are, however, usually found by trigonometrical levelling. In any triangulation the elevations of at least the base lines above mean sea level must be found by a line of precise spirit levels from the coast in order that the triangulation distances may be reduced to mean sea level.

of the bubble tube should correspond to a vertical angle of about 3 seconds. The bubble tube should be chambered so that the length of the bubble may be adjusted to a nearly constant length under different temperatures. The diaphragm must have three horizontal cross hairs, one axial and two stadia hairs, as the distance of the staff from the instrument must be ascertained for each sight. For stadia hairs and stadia measurement of distances, see Chapter VI. The tripod legs should be covered with white cloth to shield them from the sun and render them less liable to variations of temperature.

""

Kern Level. The levelling instrument adopted by the International Geodetic Commission is made by Kern & Co., of Aarau, Switzerland, who make a specialty of levels for "precise work. It is of the wye type, similar to that shown in Fig. 107, and otherwise as just described. Usually a tangent or slow motion screw is provided for adjusting the cross hairs on to the staff. The motion given by this screw is merely horizontal or in azimuth. Even the ordinary spirit level, as made in the United States, generally has a tangent or slow motion screw. The bubble tube rests on the upper surface of the rings on the telescope which are supported in the wyes, and is carried in the hand between instrument stations.

A Kern level with a focal length of 14 in. has an object glass in. diameter, and a magnifying power of telescope of 50 diameters.

Levelling Staff.—The levelling staff is made in one piece, and is about 10 ft. long. It is divided to feet and 100th parts of a foot, the 1000ths of a foot or third place of decimals being read by estimation. If working on the metrical system, the staff is graduated to metres and centimetres, the millimetres or third place of decimals being read by estimation. The staff should be made of good well-seasoned dry pine, and should be about 4 in. wide on the face and have a vertical stiffening piece up the back of it, thus making its cross section T-shaped. A small spirit level is attached to the staff, to enable the staff-holder to hold the staff truly vertical. Two wooden handles are provided for holding the staff, one on each side of the graduated face. The foot of the staff should have an iron cylindrical piece about 1 in. diameter fixed on it, and the graduations of the staff are reckoned from the extremity of this. The zero of the graduations is, however, some

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