exactly in the vertical plane of the instrument, when in adjustment, there is an important source of error, which will affect any horizontal angles taken with the instru

ment.

Should the mounting of the telescope be of the transit theodolite pattern, the telescope bearings or trunnions may be a little worn, and this deviation of the turning axis from the horizontal may be the sole cause of error; in some of these instruments this may be rectified, or nearly so, by using the adjusting screws made for this purpose; in larger instruments of this kind a striding level may be used for checking this error, and the deduced level correction applied to all angles taken, on the principle adopted with transit instruments in astronomical observation. (Refer to Chapter on Route Surveys, article on the Corrections of the Transit Instrument.) Under other circumstances and with instruments of other patterns the source of error may lie in any part of the attachments; if the results are very marked, it is certainly best to reject the instrument entirely until altered by an instrument-maker.

To investigate the correction to be applied on account of this defect.

Let a the inclination of the bearing axis to the

=

horizon.

And 90°-a=the inclination to the horizon of the incorrect vertical plane of motion of the telescope.

Let the horizontally projected angle error in an angle observed on two objects differing in height,

=the vertical angle of elevation or depression,

in the same case.

then we have in the right-angled spherical triangle

or sin tan a tan

=a tan 0, as a and are very small,

where if is positive its value must be subtracted from, and if negative added to, the observed angle. In this formula ✪ will be positive when it is an angle of elevation, and negative when an angle of depression; a will be positive when the numbering is arranged from left to right, and the turning axis of the telescope is higher on the left than on the right, but negative when higher on the right than on the left; and will be positive when a and are of similar sign, and will be negative when they are of different sign.

Hence if we suppose C to be the true horizontal angle after correction, C' to be the observed horizontal angle obtained by two readings on the horizontal arc c, and c, and , and v, to be the two readings above and below zero on the vertical arc,

C will c1-c-a(tan v ̧+tan v1⁄2).

To obtain the value of the remaining unknown quantity a, let us suppose that, from observing on a plumb-line, the turning axis of the telescope is found to be higher on the left than the right, and a is positive when the angular readings are c1 and c2 as above; if we now take a second observation of angle on the same objects, after reversing the telescope and turning the upper plate round 180°, and let c, and c, be the two readings in this case; then the effect of reversal of the turning axis will be to make the second pair of readings

give an angle that will be as much too small as the first pair gave too large, hence

C will c2-c1+a (tan v, +tan v2)

and as also C=c1—c2—a (tan v1+tan v2)

we obtain from these practical observations, a value of a

which can be recorded as suitable to the instrument; from which the value of any error pa tan may be obtained for any subsequent observation in accordance with the circumstances above referred to.

For example. If a=+5 minutes; and a horizontal angle be observed when the first line of sight is directed at an angle of elevation of +5° 27', and the second line of sight is at a depression giving an angle -2° 51′, then =+5'. tan (+5° 27′)-5'. tan (—2° 51′)

=

+4749-2486=0′2263 minutes

As the difference in elevation has so important an effect on errors of this class, the index error of the vertical arc, if there be any, should be remembered and occasionally tested by levelling the telescope.

II. Repeating and Reflecting Instruments for angular measurements in any plane or direction. Borda's repeating circle has been preferred to the theodolite by the French and Swedish Great Trigonometrical surveyors, both on the score of its simplicity and portability and from mistaken ideas with regard to excessive accuracy as obtained by repetition.

The repeating circle used by Delambre had a diameter of about 7 inches, each of the two telescopes attached being about 24 inches long; the circle was

graduated to minutes from right to left around the whole circumference, on its upper side only, the telescopes were capable both of independent motion, and of motion with the circle by the use of clamping screws, and a stand was used to hold the instrument in any required plane.

The principle of repetition, or of observing the same angle an even number of times and then obtaining the mean was justly depended on for diminishing errors of graduation, and neutralising errors of observation; but it was also then imagined that by frequent repetition. up to 20 times, any angle might be determined within a second; as however the graduations only read to minutes, and the telescopes were not sufficiently powerful to discern or divide to a quarter of a minute, this seems impossible.

The method of repetition is as follows.

The instrument having been brought into the plane of the two objects X and Y, the upper telescope Pp is set to zero and directed to X, the lower telescope Qg is

directed to Y, and the two telescopes are then clamped, thus making one observation of the required angle: its value however cannot immediately be obtained as the lower side of the circle is not graduated. The instrument is then turned onwards in its present plane until Qq be directed to X, when Pp will fall into the position Rr; and the instrument is then fixed in position. The telescope Pp is then unclamped, removed from its position Rr, directed to Y, and clamped. The reading on the circle will now be double the value of the angle. The whole circle must then be turned until Pp points to X, and the previous process repeated any even number of times, the circle being always turned to the right through the arc pq, and the two telescopes alternately to the left through the arc rq, or double the former. The final reading is divided by the number of observations to obtain the mean.

The angle having been observed in an oblique plane, the altitudes or elevations of the two objects are then observed or obtained, and the oblique angle is reduced to the horizon by tables prepared for this purpose, or by calculation. (See pp. 52 and 59 for formulæ and example.)

The advantages of this principle of repetition are obtained in the Reflecting Repeating Circle of Troughton, an exceedingly convenient instrument for measuring astronomical altitudes and distances in arc; it must however be noticed that there is always a constant error which cannot be removed by any number of repetitions.

The Surveying- or Box-Sextant is used by surveyors, as a light substitute for a theodolite, for obtaining a few horizontal angles in a small survey, or for one or two casual altitudes, such as heights of buildings, and stationmarks; but it is not a favourite instrument with them,