Surveying and Levelling Instruments, Theoretically and Practically Described. / For construction, qualities, selection, preservation, adjustments, and uses; with other apparatus and appliances used by civil engineers and surveyors in the field.

by William Ford Stanley

557.—Measuring Distances by the Ordinary Telescope by Measurement of its Focal Image.—When we apply a refracting telescope to measure a subtense angle by webs fixed in the diaphragm, vision is not direct as in the scheme Fig. 239, but subject to bending caused by the refractive quality of the lens, [art. 58], the telescopic focus varying with the distance from the staff. Thus with a 12-inch telescope there will be a difference of about ·25 inch in the focus, whether the staff is held at 50 or 500 links from the telescope; and this difference of focus is equal to a difference of base or cotangent between the points A and C in the last figure, so that these distances do not remain proportional to the fixed unit of the tangent or stadium. It is important to go carefully into this subject of the use of subtense webs in the ordinary telescope, as the necessary correction does not appear to have been recognised by English writers on instruments, and no doubt this is the principal reason that subtense measurement has not been more practised in this country.

558.—At the commencement of the last century, Riechenbach, a Bavarian engineer, pointed out a method still in use on the Continent. The author is indebted to the kindness of Lord Rayleigh for the following demonstration of Riechenbach's formula:—

Fig. 240.—Subtense diagram.

Larger image

Let Fig. 240 AB = s, ab = i, OA = d, Ob = r. O is the optical centre of the object-glass; ab a pair of webs at variable distance r from O according to telescopic focus; f focus for parallel rays. Then by similar triangles s/d = i/r or d = rs/i, r is found by optical laws to vary in the proportion of 1/r + 1/d = 1/f. We may therefore eliminate the variable r by substituting its value r = fd/(d - f), by which we find d = sf/i + f, which gives the true correction; and the distance from the axis of the instrument will be d = sf/i + f + c where c is the constant distance of the object-glass from the axis of the instrument. It is usual to place the vertical axis of a theodolite central between the object-glass and the diaphragm at solar focus, so that the constant c becomes f/2.

It is seen that sf/i represents the direct subtense, whereas the refraction, which is a constant, gives f and the position of the object-glass f/2. Riechenbach's formula being true for parallel rays is evidently also true for any subtense with refraction for the staff at any distance. We may therefore adopt a plus constant of 1½f, which added to the apparent subtense is found to produce no error. Thus with a telescope of 1 foot solar focus, and using the decimal system of notation, as before mentioned, if a stadium or distinct scale be placed at 301½ feet distance from the centre of the instrument, and the webs or points of the diaphragm be adjusted to read 3 feet = 300 divisions, every subtense may afterwards be taken as number of divisions read + 1½ feet for distance in feet. If the subtense is to be taken in links or metres the dividing of the stadium will be to these measures, but the constant remains the same 1½ feet always.

559.—When the line of sight is inclined from the horizon and the stadium is held erect—a convenient method commonly followed upon the Continent—the reading becomes in excess of the true reading, in the ratio of the cosine of the angle of the stadium, represented by a line tangent to the sight-line subtended to the foot of the stadium, as shown in the following diagram.

Fig. 241.—Diagram of vertical stadium on an incline.