In the above forumulæ, c has been neglected.

The

By making the staff-reading of the centre thread equal to the height of the telescope axis above ground, the reduction of a to ground is avoided. distance and difference of elevation will be obtained in the units of the staff graduation; that is, feet, links, or metres. The vertical position of the staff is usually employed only.

Auxiliary methods for calculating cos. 2 and sin. 2z :

A variety of methods, such as diagrams, slide-rules, and tables, are used for that purpose. In practice, tables, such as Prof. Jordan's "Tachymeter Tafeln," are of great advantage. These tables, although published in German, may of course be used for any unit of length, as the argument with which D and ▲ are taken out is simply k 1, for k 100. Should have any other value, it is only necessary to multiply the tabular values by the ratio

k 100.

ACCURACY OF STADIA MEASUREMENTS.

If

The error of distance increases proportionally to the distance. m (D) is the mean error of the distance, m (1) the mean error of 1, we have, neglecting c,

The error of the staff-reading will be of course proportional to the distance, that is, m (1) = Dò; and (21) becomes m (D) = k D1, or

From (21) and (22) we see that it is of advantage to make k small. The value of k=100 is adapted for convenience of calculation. The

smaller we make k, the longer must be the staff. If k=100, a staff of 10 feet length will be subtended between the two outer threads at a distance of 100 feet. If we want to use a staff of such length for a greater distance, we must take the reading on one outer thread and the central one, for which k would be 200. The limit of a 10-feet staff would be, therefore, 2000 feet. This of course pre-supposes that the telescope is powerful enough to read the graduations.

If we succeed in bringing the error of sighting down to = 1", the accuracy according to (22) would be 0.05. In practice, however, such a degree of accuracy is almost impossible except under highly favourable conditions. The average accuracy of stadia work is about 0-25%, which for topographic purposes is quite near enough.

MEASUREMENT OF DISTANCES BY TANGENT SCREWS.

If in Fig. 5 we consider V and e joined rigidly together at K, which in the case of the theodolite represents the centre of the horizontal axis of the telescope, a complete turn of the tangent screw will move the

screw through a distancea, and will cover on a graduated staff a distance l. Moving through the same amount 7, another part will be traversed over by the horizontal thread. From the Fig. 5 we have

To use the tangent screw for the measurement of distances, the axis of the screw must be at right angles to e at every position of the telescope. To fulfil this condition, it is necessary that--

1st. The point of the screw must be centric, and must work on a

well-polished and perfectly plane surface.

2nd. The bearings of the horizontal axis must be closed.

3rd. The adjusting screw of the horizontal axis must be on the side opposite to that of the tangent screw.

If the 1st and 2nd conditions are not fulfilled, the point of the screw will bore into the lever e, and will therefore lift and lower that lever, especially if the point is eccentric. If 3 is not fulfilled, the constant proportion a:e will be changed by any loosening of the adjusting screws. Likewise, if the screw ends in a plane surface instead of a point, the lever e will be shortened or lengthened. The point of the screw should, therefore, be hardened.

DETERMINATION OF THE CONSTANT.

The ratio a e-that is, the distance through which the screw is moved by one revolution to the length of the lever is the constant upon which the measurement depends. The constant is best determined by holding the staff at different distances and observing the space on the staff over which the horizontal thread moves for one revolution of the screw. Each observation will give a value of the constant.

The mean of all values may then be taken as the final value of C. It is best to distribute the observations over several threads of the screw, as then, from the different values of C obtained, an insight into the condition of the screw is gained.

Method of Observing.

After the instrument is set up, the horizontal wire is set on the graduation of the staff, corresponding to the height of the telescope axis above ground, and the vertical anglez is read on the circle. The screw is then turned through a number of revolutions n, and the graduation bisected by the horizontal thread is noted.

The advantage of using the tangent screw in the above manner is twofold, namely, by using complete revolutions of the screw only, the periodic errors are eliminated, and by distributing the observations over several threads, the progressive small errors (regular and irregular ones) are partly eliminated.

ACCURACY OF MEASUREMENTS.

Providing the screw fulfils the required conditions, the errors affecting the results are—

Change of constant C = d C.

Error of reading index mark of screw head m (r).

Error of reading staff graduations

The corresponding errors of D are found by

d D=ld C

d D= ±C m (r)

d D= ±C m (r)

(r).

(6)

in which "d" equals the day of the month. The times require also the correction in minutes

C 0.16383 L.

=

in which " L" equals the longitude east of Greenwich, to be expressed

in hours.

The elongation, and prime vertical stars are for latitude 34 degs. ; the times tabulated will be within ten minutes for any latitude in the colony. For the formulæ and notation, readers are referred to Vol. XII, No. 12.

The foregoing table is arranged according to the powers of "t." The formula of interpolation, illustrated by a numerical example, will be found in Vol. XII. No. 12.

The sidereal time at mean noon Greenwich equals

As in previous formula "d" equals the day of the month

Jupiter. The largest planet crosses the meridian at 8 hours 42 min. on the 10th, with a southern declination of 197 degs. ; polar semi-diameter 19"-9; horizontal parallax 1"9. On the 20th he crosses the meridian at 8 hours 1 min. He crosses the meridian on the 30th at 7 hours 17 minutes, with a southern declination of 19.7 degs. ; polar semi-diameter 18"-9; horizontal parallax 18. This planet will be apparently stationary on the 29th at 10 hours.

The following satellite phenomena will take place before midnight :

JUPITER'S SATELLITES.

Saturn. This planet crosses the meridian at 10 hours 46 min. on the 10th, with a southern declination of 22-5 degs.; polar semidiameter 85; horizontal parallax 1"0. He crosses the meridian on the 20th at 10 hours 3 min. On the 30th he crosses the meridian at