usual way, so that the bubble is at the centre of its run at the instant of reading the staff. When bisecting a bench mark, however, it is convenient to use the elevating screw and note the deviation of the bubble.

Value of one Division of the Bubble Tube.-This is found by sighting on to the levelling staff, whose distance from the instrument is carefully measured. The divisions of the bubble tube are numbered from the centre towards the ends, and the bubble being moved one division at a time, staff readings are taken for each position of the bubble. Readings need only be noted for extreme positions of the bubble, as central and intermediate positions are of little use in finding the mean value of one division.

Let E1

Then

= mean of all the north end readings of the bubble when run to the north end of tube.

Editto, for bubble at south end of tube.

F1 = mean of all the south end readings of the bubble when run to the north end of tube.

F= ditto, for bubble at south end of tube.

$1

= mean reading of staff for bubble at north end of tube. S, ditto, for bubble at south end.

d = distance of staff from instrument.

v = value of one division of bubble tube (sine of the angle of inclination) at unit distance.

The value of one division of a bubble tube should be constant, but is often altered by changes of temperature of the fastenings of the tube in its case.*

*

By experiments on the level of Ramsden's 3 ft. theodolite it was found that though at the ordinary temperature of 66° the value of one division was about 1 second, yet at 32° it was 5 seconds.

Correction for Inclination of Bubble to Upper Surfaces of Rings. This correction is determined in terms of divisions of the bubble tube. It is to be found by reversing the bubble tube on the telescope and taking readings in both positions. An odd number of observations should be made.

Let Ni

N

= mean of north end readings for bubble tube direct.

where i is the inclination of the bubble tube to the upper surface of the rings in terms of divisions of the bubble tube.

Correction for Collimation Line. This is found by reading the staff with the telescope normal, and then with the telescope inverted, i.c., rotated 180°, about its own axis.

Let s1 = staff reading for telescope normal.

inverted. d = distance of staff from instrument.

where is the correction for unit distance. The correction for any distance is therefore cx distance.

Correction for Inequality in the Size of the Rings.-This is determined by reversing the bubble on the rings, and also reversing the telescope in the wyes. The following is an example of the method of ascertaining this correction :—

Therefore the bubble moves 2.29 divisions towards the object glass when the telescope is reversed in the wyes. This is evidently twice the difference in the rings, and the angle between the axis of a cone and its slant side being half the apex angle, therefore the line of sight makes an angle with the tops of the rings of one quarter of 2.29 or 0.57 division of the bubble tube. In this case the eye end ring is the smaller, and therefore when the upper surfaces of the rings are level the line of sight is depressed. This correction is termed the "pivot correction," and alters only with unequal wear of the rings.

The angular value of one division of the bubble tube and the inequality of the size of the pivot rings need only be determined once each season. The constant for the stadia hairs which is used for the distances as well as the absolute length of the levelling staff are usually also determined once each season. The inclination of the bubble tube to the upper surfaces of the rings and the collimation line correction are, however, determined daily, at the beginning and at the end of each day's work.

Final Correction. If D is the difference between the sum of the back sights and the sum of the fore sights, or vice versâ, then the final correction is

D{c + v (i + p)}

where is the collimation line correction, v sin of angle of inclination corresponding to one division of bubble tube, equation 1, page 191, i the correction for inclination of bubble to upper surfaces of rings, the pivot correction.

INSTRUCTIONS FOR PRECISE SPIRIT LEVELLING UNDER THE MISSISSIPPI RIVER COMMISSION.*

1. Before commencing operations the constants of the instruments will be determined. The most important of these is the value of one division of the level tube. This can best be determined by means of a level trier. It can also

be determined in the field as follows:

Set up the instrument firmly, if possible mounting it on a wooden post, or better still, on a stone pier. Set up a rod in its tripod at such a distance that it can be distinctly read through the telescope. The distance should be at least 50 metres, or if the air is very still, 100 metres, and should be carefully measured. Adjust the instrument carefully, taking such length of bubble in the level tube that its ends will be about the middle or tenth graduated line on each side. Direct the telescope to the rod, and by means of the elevation screw cause the bubble to run to near one end of the level.

Carefully note the position of the three wires on the rod and the reading of the level. Now by means of the elevation screw cause the bubble to run to near the other end of the tube, and note the reading of the wire and bubble as before. One result for value of one division of level can then be obtained.

This operation should be repeated ten times.

The elevation of the rod should be changed occasionally between sets, in order to avoid estimating the same part of the same centimetre on the rod. It will be sufficient to run the bubble five divisions each side of its central position. If distance from instrument to rod

d, d'= distance through which eye and object ends of bubble move when run from near eye end to near object end

d+d'

r,

2

= amount of displacement of bubble between two readings

r'=corresponding means of three thread readings on rod, and

7= value of one division of level in seconds of arc.

2. With the value of one division of the level, tables will be constructed showing the correction to be applied to a rod reading for an observed inclination of the level and for a distance determined by interval between extreme threads. If the level bubble is well ground, equal displacements of the bubble, say of two divisions, will correspond to equal displacements on the rod.

3. Before using the level or determining its value, the fastening of the tube in its case should be examined. One end should be clamped down just tight enough to prevent the tube from moving easily, but not tight enough to strain the glass. The other should be lightly clamped so that the tube may be free to expand and contract with temperature changes. The cotton packing at the ends should not exert a lateral strain on the tube. All level tubes will be numbered, and have their numbers marked on them.

4. In order to determine the inequality in the telescope rings, the instruments should be mounted on a stone pier or other firm support and carefully levelled. The level should be carefully adjusted, and the instrument clamped to prevent its moving in azimuth. Now with the eyepiece of the telescope over the elevating screw, note the reading of the bubble when level is set on telescope, both in direct and reversed position. Now reverse the telescope in the wyes and read the level as before. Several sets of observations should be made.

Let b, b'inclination of telescope as denoted by means of level readings with telescope direct and reversed, then the inequality of rings—

b-b'
4

Sixteen determinations of the value of p of two instruments in use on the Lake Survey gave probable errors of ±0.046" and ±0.041′′. The inequality may be expressed in seconds of arc if desired, but for purposes of computation is best expressed in terms of level divisions, as it can then be combined directly with the error of adjustment of level.

5. The centring of the object glass will be examined. This may be done as follows:-Draw out the eyepiece until the threads are no longer visible. Direct the telescope on some well-defined object, and when looking at it rotate the telescope on its wyes.

If the object remains steady, the object glass is sufficiently well centred. Should the object appear unsteady, the fault can only be remedied by a maker. The objective should be firmly screwed into the telescope.

6. The values of the wire intervals will be determined as follows:-Set up a rod at carefully measured distances of 10, 20, 30 to 100 metres from the instrument. Read the rod ten times at each distance. The rod may be altered in elevation, the level may be caused to change and the telescope may be rotated 180° (reversed) in order to change the position of the threads on the rod.

Taking the mean of the ten observed differences of readings of the extreme threads at each station occupied by the rod, a table will be constructed giving in metres the distance of the rod from the instrument for any observed difference of reading between extreme wires.

7. Unless the rods used have been previously compared with some known standard, they will be compared with each other and their relative lengths determined. This may be done by establishing two fixed points or two foot plates at equal distances from the instrument and differing in elevation about 2.7