The theory of stadia measurements is based on the two following equations: 1. i:o: :fa:fa 1 1 1 2. — + fa fa f in which ail' is the height of image or distance between the stadia wires, the height of object or the stadia reading on the nod, the and departure ordinates are self-explanatory. The single meridian distances are the de partures of the respective stations from the most westerly station. The double meridian distances of the respective courses are equal to the sum of the single meridian distances of the stations at each end of the respective courses. Each North and South area is equal to the product of the latitude of a course into its corresponding double meridian distance. The area of any polygon is equal to one-half the difference between the sum of the North and South areas. When the course and distance of any side of any polygon is lacking it is found as follows: Ascertain the difference between the north and south latitudes and the difference between the east and west departures. Divide the difference in departures by the difference in latitudes for the tangent of the bearing. Divide the difference in departures by the sine of the bearing found, for the distance.
Topographical In order to facilitate the selection of a route for a railroad, to properly locate an irrigation reservoir or dam for power purposes, or to correctly make a geological survey of a large tract it is generally necessary to make a topographical survey of the area under consideration. From the field notes of the survey a map is made showing the surface elevations with considerable particu larity by a system of contour lines at stated differences in level, and also showing all im portant objects, such as buildings, roads, canals, fences, streams, etc. There are several methods focal length of the lens. From the above equa tions and other considerations it can be shown that when the rod is held vertical at various points in the line of sight, the rod readings will be proportional to the respective distances from the rod to a point at a distance “P in front of the lens of the telescope. It can also be shown that the distance from the stadia rod to the centre of the transit is equal to the sum of three factors, namely, the rod reading, the dis tance and the distance from the lens to the centre of the transit, ac.x' But this is true only for horizontal sights, for when the rod is held upon an elevation, as a hill, then ((fa" becomes the slope distance and, moreover, the stadia wires will intercept a length on the rod greater than the slope distance. Therefore, if the stadia wires be adjusted so that they will intercept a space of one foot on the rod at a horizontal distance equal to c + f + 100 feet from the centre of the transit, then the true horizontal distance to any point above or below the hori zontal line will be found by the following equation: 3. D =-- 100 r cosiv + (c + f) cos v.
In which D = true distance horizontally. v = vertical angle.
c =centre of transit to object lens. f =focal length of lens.
The height is found by multiplying the hori zontal distance by the sine of the vertical angle. Tables are published .containing the factors for the various vertical angles.
In topographical field work where the area to be surveyed is considerable it is usual to first lay out a triangulation net work, all measurements being made with a tape or by triangulation from an accurately measured base line and the stadia measurements taken from the triangulation stations. In work of less magnitude it is not necessary to lay out a sys tem of triangles, but all measurements from station to station are carefully made and checked by cross-readings. The location of objects is quickly and accurately accomplished by sights from two different points of the measured line. In making a map of a topo graphical survey there are several methods em ployed. One requires the use of a T-square and brass protractor, another and a better method is by the use of a protractor sheet larger than the sheet upon which the drawing is to be made. After all distances are laid off and the heights indicated at the various points, contour lines are drawn in by proportioning the distance between points with reference to heights. Sometimes the contour lines are drawn in pencil to be afterward erased and the uhatchure° method of representing topography employed.
Mines chief purposes of an underground survey are to ascertain the amount of ore °in sight," to find the °pitch° and position of the °pay chute° with respect to the shaft and levels, to find the *dip° of the vein and also to ascertain and lay out the direc tion of connections commonly called oholings.° Moreover the laws of some States and countries require plans of the underground workings to be kept on file. Ventilation being one of the serious problems in mining—especially coal is generally necessary to make passageways or uholings° connecting the vari ous drifts and levels in such a way as to facili tate the circulation of fresh air throughout the mine. In order to make these connections a careful transit survey is necessary, the compass needle being unreliable in the presence of pipes, rails, ore cars, etc. While mine surveying does not call for the precision of a geodetic survey, cases arise requiring great skill and ingenuity. One of the chief difficulties is to transfer the bearing or azimuth of a surface line to the lower workings. This is especially difficult if the survey has to be carried down a deep ver tical or inclined shaft. In vertical shafts heavy plumb-bobs suspended in pails of water or mo lasses at the bottom of the shaft by means of wires reaching to the surface have been suc cessfully used. A transit with a secondary telescope, so attached to the extended axis of the primary telescope that it may be sighted vertically downward, is also used for this pur pose. In underground work it is necessary to note both vertical and horizontal angles and to make tape measurements on the slope or level as the circumstances may permit. Illuminated plumb-lines are used for fore and back sights and it is also necessary to hold a candle so as to illuminate the telescope cross-wires. Per manent pegs for future use are usually placed in the roof instead of in the floor of a drift and even then their position must be often checked for ground moves considerably in some mines.