adjusted. To do this, apportion the error proportionally to the lengths of the sides. For instance, the length of the side AB is 1,060 ft., and the total length of the sides being 5,729 ft., we have— 1060 5729

Correction of latitude of AB =

× 9.2 = 1.7 ft.

As the latitude of AB is positive, and the positive latitudes are in excess, this is to be deducted from the latitude of AB, and we get— Corrected latitude of AB = 1015.5 - 1.7 = 1013.8.

Similarly the correction for BC is

I 202

5729

× 9.2 = 1.9, and the corrected

value of the latitude of BC is 300.3 - 1.9 = 298.4. The correction

ΠΙΟ

for CD is × 9.2 = 1.8, and as the positive latitudes are in excess,

5729

this falls to be added to the latitude of cp, and the corrected value is 390.2+1.8392. Similarly the corrections for the other latitudes are calculated and applied in the same manner, the departures are treated in exactly the same way, and we get the corrected latitudes and departures as given in Table II.

Upon adding the latitudes and departures we see that the sum of the latitudes is o.1 and the sum of the departures is o. The error of o.I still remaining in the latitudes is owing to the corrections being calculated to one place of decimals only. The traverse is now to be plotted by ruling a meridian through the starting point A and proceeding in the same manner as described for the plotting of the traverse on Plate III., Figs. 79 and 80. The correction of the latitudes and departures evidently affects the bearings and lengths of the lines AB, BC, &c. These may be measured with the protractor and scaled after the work has been plotted from the corrected latitudes and departures, or, if more

accuracy is desired, the new lengths and bearings may be calculated from the corrected latitudes and departures.

Thus the new length of any of the lines is obviously

=

√(corrected latitude)2 + (corrected departure)2 and the tangent of the angle made with the meridian corrected departure

=

corrected latitude

Adjustment of Closing Error when some of the Measurements may be considered more accurate than others. The above method of correcting and adjusting the closing error is based on the assumption that the chaining and the measurement of the bearings are equally in error, also that the relative accuracy of the lengths and bearings of all the lines is the same. When from special circumstances the measured lengths and bearings of some of the lines may be considered to be more accurate than others, the following method may be adopted. Take one of the lines as a standard and assume that the error in this line is 1; from this basis estimate what the probable error in each of the other lines would be, taking into consideration the special circumstances of each, such as any particular obstacles to measuring, roughness or steepness of ground, number of observations made to determine bearing, if check measurement of length made, and so on; these probable errors are to be for a distance equal to the length of the standard line. Each of the lines being thus weighted with its probable error as 1.5, 2, 3, 5, &c., multiply the length of each line by its probable error and then we have

Correction of lat. or dep. of any line = multiplied length of given line sum of all multiplied lengths x whole error in lat. or dep.

Adjustment of Closing Error when the Error is considered to be due to the Chaining only.-When it is considered that the bearings are practically correct, and that the closing error is due to the chaining alone, the correction of each line is to be computed as follows::

Correction of lat. or dep. of any

line

=

given lat. or dep.

sum of lats. or deps. > whole error in lat. or dep.

The closing error may be assumed to be entirely due to the

chaining when the bearings have been carefully measured with the theodolite and the bearing of the first line, when redetermined at the close of the traverse, is found to agree very nearly with its true or assumed bearing at the beginning of the traverse.

Amount of Closing Error allowable in practice.*—For purposes of comparison the closing error is taken as the ratio of the length of the line joining the initial and final points of the traverse (as plotted or computed from the field notes) to the length of the whole perimeter of the traverse. In ordinary open country the closing error should not exceed 1 in 300. work the closing error should average about 1 in 5,000. For special purposes where greater accuracy is required, as in tunnel work, &c., the precautions in measuring the lengths and angles described in Chapters VII. and XII. must be adopted.

In town

Compass Traverse Surveys.-A traverse survey executed with the compass only, e.g., with a circumferentor or surveying compass (Fig. 60), or with Whitelaw's theodolite (Fig. 61), is executed in precisely the same manner as those already described. The essential difference is that the bearing of each line is measured from the magnetic meridian, and angular errors are therefore not cumulative, as they are in the preceding methods. A long traverse hastily executed with the circumferentor will therefore be in general more accurate than the same traverse if hastily executed with the theodolite, and indeed the speed and accuracy attainable in compass surveys is remarkable compared with theodolite surveys. At each set up of the compass the bearings of two lines may be taken, the line in front and the line behind, and when great speed is necessary it is usual to set up the compass over every second station only. To guard against local attraction, however, the compass should be set up at each station, and the back bearing as well as the forward bearing of each line observed. If the back bearing does not differ by 180° from the forward bearing there has either been a mistake in reading the forward bearing or else there is some local attraction. In the latter case the angles between the lines must be booked in place of the bearings, until the correspondence of the back and forward bearings indicates that the attraction has ceased. The

angle between any two lines will be correctly indicated by the difference of the bearings in spite of any local attraction, as both bearings will be equally affected by it. Compass surveys have been run a distance of 20 miles, coming in to 1 chain; but when any great accuracy is required the theodolite should be used.

If readings are taken sometimes from one end of the needle and sometimes from the other indiscriminately, mistakes are liable to occur. To prevent this, always point the north end of the compass box towards the object whose bearing is being observed, and always read the north end of the needle. This is also a more accurate method of procedure.

Variations of the Compass: Annual Variation.-The magnetic needle does not point to true north but to magnetic north. The magnetic north is not a fixed point, but varies with approximate regularity from year to year. The angular difference between true north and magnetic north is termed the variation. The maximum variation is about 25° east and west of true north. Near London in the year 1580 the magnetic variation was 11° 15′ E.; in 1657 the variation was o; in 1818 the maximum westerly variation of 24° 38′ was reached. Since then the variation has steadily decreased to 18° 10' in 1887. The average annual movement between the years 1580 and 1880 has been 8 minutes. At Edinburgh the variation is about 3° greater than at London. At Dublin the variation is about 3° 50' greater.

The magnetic variation thus differs at each different place and time, and can only be correctly determined by an astronomical observation for true north by one of the methods described in Chapter X. An approximate value of the magnetic variation. may be deduced from a map of the world showing the lines of equal magnetic variation with the amounts for a given year. By allowing an annual difference of 81 minutes since the date of the map the approximate magnetic variation may be deduced. Such a map may be found in "Hints to Travellers," published by the Royal Geographical Society. In the United Kingdom the approximate value of the magnetic variation may be deduced from the map prepared by Sir F. J. Evans, Hydrographer to the Admiralty. This map will be found in Philosophical Transactions,

vol. 162.

H

Diurnal Variation.—In addition to the regular annual variation there is also a diurnal variation more or less regular. Near London there is a total difference of magnetic variation of 10 minutes in any one day. About 8 A.M. the needle is at its furthest point east, and about 1 P.M. it reaches its furthest point west. At other times of the day the position of the needle varies between these points. About 10 A.M. the needle coincides with the mean position of the magnetic meridian, and again a little before 7 P.M.

The diurnal variation may amount to at places, and is always at least several minutes. It changes continually at the rate of 1 or 2 minutes per hour, and is greater in summer than in winter. The cause has been stated to be the influence of sunlight.

The magnetic needle is liable to violent and irregular disturbances which in extreme cases may amount to 1° or 2°. These magnetic storms appear to coincide with the appearance of the aurora borealis, earthquakes, and volcanic eruptions.

Notices of magnetic storms are given by the Magnetic Department at Greenwich Observatory specially for the benefit of mine surveyors.

It may be readily imagined from the above that compass surveys are not as a rule susceptible of very great accuracy. The value of compass work lies, however, in the fact that errors are not cumulative, each error being confined to its particular line. When the lines are very short, an error of even 10 minutes in bearing is often negligible in practice; 10 minutes on a length of 1 chain is about 2 in., which on a small scale of plotting is inappreciable.

Dip of Magnetic Needle. In the northern hemisphere the needle dips to the north, and in the southern hemisphere to the south. It is adjusted to a horizontal position by the instrument maker weighting one end.

Local Attractions.-The observer should be careful to see that there is nothing about the person which may attract the needle. Sometimes a steel band in the brim of a felt hat or steel

buttons have given trouble. The glass cover of the compass may be electrified by friction and attract the needle. The electricity may be discharged by touching it with the wet finger. Magnifying glasses with guttapercha frames become highly electrified by wiping the lens, and attract the needle. They should be of brass