566.—In adjusting the lines, webs, or points to a given subtense, the anallatic lens may be moved to give more or less angular displacement or magnification of the image. Greater accuracy is obtainable when the staff is held normal to the line of sight instead of vertical. If the staff be held incorrectly in the inclined position at great angles of elevation or depression, the resulting error is very much smaller than in the case of an equal variation of the staff from the true vertical position. When adjustment is made upon a distant stadium at small angles of elevation or depression, the subtense of the small arc will vary so little from a tangent to one of its radii that the one or other may be taken without sensible error. The plan originally proposed by Green of placing a sight tube through the stadium at right angles to its face, as a means of keeping it in the chord of the arc, is as good as any other, but is more cumbersome than that described art. 561. If the vertical stadium be preferred, this may be set up by the small level, [Fig. 109], p. 163.
567.—It is well to note that with the anallatic telescope the stadium must not be so near that the rays from the object-glass do not cross in front of the anallatic lens or the subtense will appear much increased, so that there is a fixed nearness at which this form of telescope can be used, say 50 feet. For this reason engineers generally prefer an ordinary telescope, making use of the addition of a constant. The author also prefers the plain telescope, as being more correct according to his experiments where the constant is correctly allowed. There are many advantages in the use of a plain open telescope instead of the anallatic telescope for tacheometers, among them the following may be mentioned. More light reaches the eye because there are fewer lenses; there is no intermediate lens requiring adjustment and which becomes dirty and bedewed and is inaccessible for cleaning, and for the same dimensions of the telescope greater power can be obtained. A larger telescope and of higher power is of great advantage in subtense measurement, but the full advantage is not obtained in the anallatic telescope. The idea which appears to be still common that an ordinary open telescope will not give accurate results at all distances by means of stadia readings, plus the distance of the anterior principal focus of the object-glass from the axis of the instrument, is entirely erroneous. When a staff is held at any distance in front of the object-glass of an open telescope, an inverted image of the staff is formed at the conjugate focus which subtends an angle at the corresponding nodal point of the lens, equal to that subtended by the staff at the other nodal point. If a diaphragm with two stadia points or webs be placed at this conjugate focus the ratio i/f′ = the ratio l/D; where i is the space between the stadia points, l the height on the staff which these points appear to intercept when viewed through the eye-piece accurately focussed on them, D the distance of the staff from the object-glass, and f′ the distance of the diaphragm from the object-glass. Now in this equation i is a fixed space, l is the observed height on the staff, and both f′ and D are variables, of which it is desired to find the value of D. From the laws of optics it is also known that 1/f′ + 1/D = 1/F where F is the principal focal length of the lens. Therefore f′ = FD/(D - F) for all values of D. Substituting this value of f′ in equation (1) we get i × (D - F)/FD = l/D; and multiplying both sides by D, i × (D - F)/F = l. ∴ D - F = (F/i)l and D = (F/i)l + F which is true for all distances. But this distance is measured from the object-glass, and the distance S required by the surveyor is that from the axis of the instrument, and it is therefore necessary to add that of the object-glass from the axis d. ∴ S = D + d = (F/i)l + F + d, and F + d is the constant of the instrument = c. ∴ S = (F/i)l + c.
When the range is greater than that at which the divisions of an ordinary levelling staff can be clearly read with the stadia points, target stadia rods or targets fixed to a levelling staff are used. It is usual to use plain targets fixed with their centre lines at exactly 10 or 20 feet apart or other convenient distance, and the angle subtended by these is measured by a micrometer diaphragm. The reviser, in conjunction with Mr. C. W. Scott, B.A.I., A.M.I.C.E., has designed a micrometer diaphragm which has been proved to give very accurate results. It is made to revolve, so that either horizontal or vertical stadia rods may be measured, and it is fitted with fine fixed platino-iridium points, which are much more satisfactory than webs or lines engraved on glass. These are fixed on one side of the diaphragm, two each 1/200 part of the principal focal length of the object-glass above and below the axial point. On the other side of the diaphragm is a movable point which can be traversed over the fixed points by a micrometer screw, every complete turn of which moves the point over a distance equal to 1/1000th of the principal focal distance and the head of the micrometer being divided into 100 parts, it reads to the one hundred-thousandth part of the same; while a small star-wheel records the number of complete revolutions, five of which cover the space between any two of the fixed points. In using this micrometer with say a 10-foot target, let the lower target cross-bar be clamped to the level staff at 2 feet, and the upper target cross-bar at 12 feet. Direct the axial point to the centre between the targets at 7 feet and read the angle, then bring the nearest fixed point to the top or bottom mark by means of the tangent screw, and bring the micrometer point to the other mark by the micrometer screw. The micrometer reading is the reading on the divided head plus the hundredths indicated on the star-wheel plus 500 for each included complete space between the fixed points. See whether the micrometer reads up or down, and set the fixed point to the lower or upper mark on target accordingly. To obtain the distance from the axis of instrument, divide 100,000, multiplied by the length of the target by the micrometer reading, and add the constant of the instrument S = 100,000l/x + c where x is the micrometer reading, and l the length of the target. The tacheometer which the author has lately made has a plain open telescope, but this is of the same size as that used upon the Porro system, and consequently it gives much more light and better definition.
568.—Tacheometers consist essentially of any form of theodolite that is provided with means for reading distances by its telescope. Stadia work is simply another name for tacheometry, which is derived from the Greek tacheos (quickly), and metreo (I measure), and signifies the art of measuring rapidly. The graduation of the arcs and circles of these instruments is sometimes made upon the centesimal system, the circle reading 400 grades, which are subdivided to half grades to read with the vernier or micrometer to centigrade minutes of ·01 grade. The centesimal system facilitates calculation, and permits a free use of a logarithmic slide rule of a special kind. In France, where working with this system at one time became more general, we have very complete centesimal trigonometrical tables adapted to the tacheometer published in stereotype,[30] but it has not gained favour, and very few instruments are now so divided. A compromise which has found a certain amount of favour is the decimal division of the ordinary degree of 90 to the quadrant; this greatly facilitates the calculation compared with what is necessary with the sexagesimal division into minutes and seconds and the reading of the verniers is much simpler and less liable to errors. Moreover, the mental conversion of the sexagesimal division into decimals of the same degree is much simpler than the conversion into the centesimal degree of 100 to the quadrant. Any instrument divided sexagesimally can be converted by simply changing the vernier if the divisions on the limb are degrees or half degrees. The theodolite, [Fig. 169], the author made specially for a tacheometer. Any theodolite may be converted into a tacheometer by fitting it with a subtense diaphragm. A modern tacheometer should be a high-class theodolite in which every possible refinement is included.
569.—The tacheometer, although manufactured for many years for export, has been very little used in this country. The instrument to be described, shown Fig. 249, is the author's latest pattern. It is made with sexagesimal division or ingrades, to read by the verniers to 20″ or to centigrade minutes. The telescope is of much larger and of higher power than that of the ordinary theodolite. For a 6-inch instrument the telescope is of 11 inches focus, with an object-glass of 1¾ inches aperture. The eye-pieces are of the Ramsden form of powers 18 and 25. The points in the diaphragm are set to cut 100 divisions of the stadium at 100 units + constant of the measurement intended to be taken, links, feet, or metres. This precludes distant measures, say of over 15 chains, where a 16-feet stadium is used, but they are made adjustable so that they may be set, if desired, for any other subtense, although this is not recommended. It is doubtful whether the subtense method can be considered as reliable at a distance of over 1500 links; or at any rate we must assume that much greater accuracy can be obtained by dividing distances greater than this into two by an intermediate station for observation, independently of the additional convenience of having the staff-holder within easy distance of communication.
Fig. 249.—Stanley's 6-inch tacheometer.
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570.—Where points are not used in the diaphragm or where lines are preferred, these may be divided upon glass in fine lines as [Fig. 246]; or spider webs may be used, but these are more difficult to set exact for stadia.
571.—Stadium.—Any accurate levelling staff will answer for the stadium, but the ordinary Sopwith, [Fig. 99], is slightly confusing. A more open reading is generally recommended—that shown [Fig. 102], p. 155, which the author designed for the purpose, answers perfectly. It is better to read the stadium low, as there is less vibration; but it is not often possible or at any time advisable to read it from the bottom—1 foot up is generally most convenient. Readings are taken and recorded of each subtense web, or point, separately, and the difference of reading subtracted for the subtense of tangent. With a point diaphragm for taking the subtense angle a fair certainty of accuracy of measurement of distance within ·002 may be assured, which is much nearer than can be attained by average chaining, taking six times the labour.