position; but in some there is a special appliance fitted to the lower part of the instrument, by means of which a slight motion may be given to it in either direction, thus ensuring greater exactitude in this respect, as well as saving time in getting the required position. A further advantage may be obtained by having the instrument fitted with an eyepiece-micrometer, as it then can be used for direct observation of distance in telemetry; besides that greater exactitude is afforded in astronomical observations for time, latitude, &c.; this has been already noticed in the section on telemetry.

The setting up of a theodolite, which consists in getting it into position, levelling it, and focussing the eye and object lenses of the telescope; and the observation of an angle, which consists in directing the telescope suitably focussed first to one object and then to another, and taking the difference of the two corresponding readings on the horizontal graduated limb, are two very simple operations best learnt by imitation.

There are, however, two very common sources of error that require attention. Ist. Although the focussing may have been carefully attended to, and thus all parallax impossible, yet the cross-wires of the diaphragm may be slightly out of their proper position, which is the exact axis of the telescope. To test this, observe on any well-defined distant object, and then either roll the telescope half round in its Y's, or reverse it laterally in its bearings (according to its construction) and observe again; should the intersection of the cross-wires not be exactly on the same distant point in each case, and this be proved the second time by careful repetition, the cross-wires must be moved by means of their respective

pairs of diaphragm screws, until they are perfectly correct and stand the test. Or, if it be preferred to leave the cross-wires unmoved, each angle taken must be twice observed, the second time with the telescope reversed; the mean will be freed from this error. 2nd. If there happen to be any inaccuracy of division in the verniers or the graduation of the horizontal plate, the error from this cause may be nearly eliminated by Borda's method of repetition, which distributes the effect of these errors all round the circle, or throughout the length of the vernier, and nearly neutralises them. It is thus. Having observed an angle in the usual way from zero, and noted the position of the index, clamp the two horizontal plates firmly together, loosen the lower plate, and turn them, still fastened together, until the line of sight coincides with the direction first taken, or first side of the angle; now clamp the lower plate and loosen the upper, turning it until the telescope sights again the second side of the angle; next clamp the upper plate also, and notice the position of the index, which will then give a reading double the required angle as first observed. This repetition should be carried on to any number of times until the one plate has nearly made one, two, or more complete turns with regard to the other; the number of turns being noted, multiplied by 360°, and added to the angle finally read. The total divided by the number of observations of the angle gives a mean angle very nearly freed from this error as well as from the first of the two following sources of error; and, if necessary on the score of collimation error, the same process can be repeated with the telescope reversed.

A theodolite, either badly made or damaged from

wear, is liable to two additional sources of error, on account of which a correction must be applied to any angles observed with it.

1. From want of perfect parallelism of the horizontal plates.

2. From the vertical limb not moving exactly in a vertical plane.

The Horizontal Limb.-To test the accuracy of the horizontal limb of the instrument, level the horizontal plates very carefully, for four positions of the telescope, across, reversed, and reversed across, as well as in the original position. Do the same again, and in each case where needful bring the bubble of the telescope-level to the centre of its run, half by means of the tangent screw of the vertical and half by means of a pair of the thumbscrews; now, if both the plates be loosened and turned completely round, the bubble will retain its position, provided that the horizontal plates are perfectly parallel ; otherwise the axis is faulty, and an instrument-maker is necessary to put it right. In the meantime, however, the instrument may be used if the following correction be made on this account.

Let ZO, figure 5, represent the position of the vertical axis of the upper plate EABD, and EA'B'D' the position of the lower plate; then if OA and OB be the two positions of the index when an angle is observed, the angle A'OB' will be read instead of AOB. Let the values of these be represented by C' and C, and the zenith distances A'Z and B'Z by and , and since the angle included between the two planes ZOA' and ZOB'C, we have

Again, supposing that when the index is at E the zenith distance ZE happens to be exactly a right angle; let the angles EOA=D, EOB=D', EOA'=8, EOB' then since sin 90°= 1, and cos 90°=0,

=

But as in the triangles AEA', BEB', A and B are

hence also sin AA'= cos ; and sin BB=cos (';

Hence putting AA=0 and BB'=0′, and substituting, the original formula becomes

a formula adapted for logarithmic computation. The value of 0, with the aid of which any other value ' may be easily deduced, can be obtained thus.

D

Make the upper plate truly horizontal, and sight some distant object with the telescope; turn the lower plate through 180°, and then the upper plate with the telescope back again through 180°, the resulting deviation of the line of sight will be double the angle of error AOA'. Repeat this proceeding around the whole of the circle, until the exact position is found where the error becomes a maximum, which is supposed to be at FOF', and obtain a value of FF' at this point; then since the arcs are small

FF: BB: AA' :: sin C+ C': sin C': sin C; hence 0=FF' sin C, and 'FF' sin C'.

If the instrument happens to be so faulty in this respect as to render it necessary to reduce this error not for a single case, but for a large amount of pressing work that may utilise any part of the circle; it is better to start from a position where d=0, and calculate a tabular set of values of ' for position of every 5 degrees all round the circle up to the maximum, which may be interpolated by proportion; the formula for correcting the observed angle will then be

2. The Vertical Limb may not move exactly in a vertical plane, either from having the bearing axis of the telescope inclined to the horizon, or from any of the attachments or supports being ill-made, damaged, or bent. To test this, adjust the instrument very carefully in every respect, and observe in the vertical plane of the instrument on any well-defined object, as the top of a steeple, or a star, and its reflection in still water or in an artificial horizon; if the reflected image is not