LXI. A STADIA DIAGRAM. By MORRIS K. TRUMBULL, Jun. M. W. S. E. The "Stadia" method of making a typographical survey has attained popularity among engineers, not because there is an interesting practical theory involved, but for the reason that it is decidedly quick, accurate and comprehensive; while the notes taken in the field are, after their "reduction," admirably adapted to office use. Since the field notes, as taken, require reduction before the map can be plotted, any method that will facilitate their reduction, so as to give results within the degree of accuracy, required, is welcomed. When an engineer has three or more stadia parties constantly at work making a survey, the reduction of the notes is no small item. With the end in view, therefore, of cutting down the amount of time required on this detail of the organized work, the accompanying Stadia Diagram was designed. In giving a brief description of the methods used in its construction, it must be stated that every point plotted was taken from the "Stadia Tables," compiled and published by Messrs. Alfred Noble and Wm. T. Casgrain. The values found in these tables were computed from the following formulæ, deduced by Professor S. W. Robinson: (a) h: R' —(B--c—f) Sin 2 V+(c+f) Sin V 2R R' (b) d (B-c in which R R' Any reading of the stadia for which the horizontal distance and difference of elevation are to be obtained. B=Length of a measured base. R= Reading of the stadia on that base. V Angle of elevation or depression. c Distance from center of instrument to center of object glass of the telescope. f-Principal focal distance of the object glass. h-Difference of elevation corresponding to a reading R' and angle V. d-Horizontal distance corresponding to a reading R' and angle V. For the computation of the tables, the following values were assigned to B, R and (c+f) There is nothing new in the plotting of the inclined straight line curves (which in this discussion, we will call vertical angle curves.) They represent vertical angles, from o° to 8°, and distances between zero and 1,600 feet. In plotting one of them, for say 2° 20', the differences of elevation due to this vertical angle and rod readings of 100, 200, 300 feet, etc., were taken from the table. Rod readings were plotted as abscissæ, and their corresponding differences of elevation as ordinates. These points were connected and gave the vertical angle curve for 2° 20'. The same operation was performed for every angle appearing on the diagram; as the diagram then stood, only differences of elevation could be determined from it. After the delineation had been completed thus far, it was desired to incorporate upon the same sheet a method of obtaining the horizontal correction for distance; and not only that this correction should be obtained from the same sheet, but that it might be determined at the same time as the difference of elevation. Whatever may be the scale of the maps to be plotted it is almost invariably sufficient to give the results of "side shots" to the nearest tenth of a foot for elevation and to the nearest foot for distance. The latter criterion of accuracy especially aided the attempt to secure simplicity in the construction of the diagram and at the same time gave assurance that no reading would be in error by more than one-half foot. The aim of the design was to enable the computer to tell at a glance what the horizontal correction should be, as he is determining the difference of elevation. The outcome of a careful study of the means of formulating such a device was to divide the diagram into zones, within each of which the correction to be applied to the reading stadia to give the correct horizontal distance, would be the same. With the diagram thus divided it is evident that the correction for horizontal distance could be observed instantaneously when taking out the difference of elevation. Since it was only sought to obtain the correct distance to the nearest foot, the number of zones would not be excessive and the lines separating the zones would not confuse the diagram. The method of drawing these separatng lines is now to be explained. pon any vertical angle curve a point may be fixed where the horizontal correction for distance is +0.5 ft; another where it is0.5 ft; another where it is-1.5 ft., and so on. Taking, for example, the vertical angle curve for 2, these points correspond with the following rod readings, or, expressing the various terms as in the general formulæ and remembering that (d-R') the horizontal correction for distance, we have from the tables for V -2°: These points being plotted on the diagram, the correction (to the nearest foot) will be +1.0 ft. when R'<345; o when_R' >345 and <727; -1 ft. when R'>727 and 1111, etc. This would enable the computer to apply instantaneously the proper correction to the rod reading to give true horizontal distance when V2°. A similar set of points was determined and plotted for other values of V; a curve was then drawn through the points where (d-R') =+0.5 ft.; another curve through the points where (d-R')=-0.5 ft.; another where (d-R)-1.5 ft., and so on. These curves divide the diagram into the zones desired. For any point on the diagram falling in the zone which lies between the curve where (d-R')+0.5 ft. and the curve where (d--R') -0.5 ft. it is evident that the reading equals the true horizontal distance within less than 2 ft.; in the diagram this zone is marked o on the left hand vertical angle curve and "No Change" in the lower margin. For any point in the diagram falling in the zone which lies between the curve where (d-R')=-0.5 ft. and the curve where (d-R')=-1.5 ft. the reading reduced by 1 ft. will give the true horizontal distance within less than 2 ft.; in the diagram this zone is marked -I (the correction) on the left hand vertical angle curve,and also in the lower and right hand margins. The succeeding zones are marked in a similar manner. In the zone falling below the correction curve passing through the points where (d-R')=+0.5 ft., the reading must obviously be increased by 1, or more, to give true horizontal distance. By the formulæ it will appear that the maximum correction occurs when Vo and Ro and in this case (d-R')+1.4; hence at all points below the correction curve of +0.5 ft. the readings are to be increased by unity to give the true horizontal distance within the required limit. This zone is marked +1 on the left hand vertical curve and in the lower margin. The use of the diagram will be illustrated by the following examples: 1. Let it be required to find the horizontal distance and difference of elevation when V 2°22′ and R'=-452 feet. Enter the diagram at the bottom with 452 feet. Follow up the vertical for this distance (estimating the 2 feet) till it intersects the straight line curve for 2°22' (this angle may be estimated). It will be noted at a glance that this intersection falls in the zone labeled No change. Therefore the cor rect horizontal distance is 452 feet. Then follow across horizontally and it will be seen that the difference of elevation is 18.7 feet. 2. For V 4°45' and R' 810 feet the diagram shows that h mean readings obtained at the respective stations are added, anu the total divided by the summation of one hundred foot intervals determined. This method distributes (c+f) among the several determined distances, and is the one that has been most used in the past for obtaining the unit value for graduation of stadia boards. The value so obtained is laid out successively upon the face of the board, and these spaces are then subdivided into tenths or smaller subdivisions and marked with such symbols as the engineer prefers. In the second application of Robinson's formulæ to the graduation of stadia boards, the condition is first laid down that the reading shall be the horizontal distance, at some given distance from the instrument on level ground. The distance usually taken is If perfect observations could be made at this distance the unit for graduating the stadia boards could be determined satisfactorily on a base line of this length; unless, however, the telescope is of unusual power and the air very clear better observations can be made at a less distance; if taken at such less distance account must be taken of the (c+f) factor, because the distances and readings are not in exact ratio. The formulæ, or preferably the tables based on them, furnish readily the data for laying out a base line with hubs at such intervals that the readings on them will be in simple ratio to the readings on the 1,000 ft. base and can be used with equal theoretical and greater practical accuracy for determining the unit for graduation. These intervals must be calculated before the field observations for determining the unit can be commenced. The preliminary computations are as follows: Say that (c+f) for the instrument is 1.4 feet, and it is desired to have the set of boards read 1,000 feet upon a measured base of 1,000 feet; that is to say, for 1,000 feet RB. With these values it will be found, in applying the formula for distance, that when the rod reading is 100 feet, the "actual distance" (d) of the rod from the center of the instrument is 101.26 feet. Likewise, when R' is 200 feet, the actual distance of the rod from the center of the instrument is 201.12 feet. Calculating the values of d for successive rod readings of 300, 400 feet, etc., the actual distances are found to be 300.98, 400.84, etc., making each interval, with the exception of the first one 100 -0.14 feet. = The first interval is 100+1.4-0.14 feet, as will readily be perceived upon solving the formula for distance using R' o, whence d 1.4 feet, or (c+f). It will be observed that 0.14 is just onetenth of the (c+f) constant, thus this amount, 1.4 feet, is equally distributed over the successive intervals of the base line. Consequently, the base will be staked out as follows: Calling the point at the distance 101.26 feet, Sta. B1, that at the distance 201.12 feet, Sta. B, etc. Now, just as was desired, points have been secured upon which to hold the level rod for the purpose of measuring the distance intercepted on the rod between the two stadia wires of the transit; if the intercepts are measured correctly the intercept at B2, B3, etc., will be exact multiples of the intercept at B,. Let these intercepts be designated I,, 12, 13, 14, etc.; then, if the work is etc., and this constant quotient is the unit for graduation desired. These preliminary calculations being made and the base line staked out as described, the party proceeds to the field equipped as for the previous method, and the instrument is set up at the zero end of the base line. Five or six shots are taken on the level-rod held vertically on Sta. B, (two targets being used as before). These readings should agree; if they do not agree exactly, but differ by small amounts, a mean may be taken as the reading on that station; if the divergence is too large a new set of readings should be taken. The rod is then held on station B2 making the same number of observations as upon Sta. B. the corresponding ratio determined at Sta. B1. The same is done upon Stations B, B., B,, etc., and including Sta. B1, which latter it will be remembered is exactly 1,000 feet from the center of the instrument. 10. If the test should be made on a day when the atmospheric conditions are such as to cause poor definition in reading the rod at the last station or two, then it may be found advisable to omit the observations at these points. Manifestly this does not affect the principle. If it should be found that the several determinations 12 1 I, I, |