Approximate forms of stadia formulas

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Distance and elevation for inclined sights

Intermediate Engineering

Horizontal Distance (Subtraction Scale).

APPROXIMATE FORMS OF STADIA FORMULAS. Because of the errors common in stadia surveying, it has been found that approximate stadia formulas are precise enough for most stadia work If you will refer again to figures 8-5 and 8-6, you will notice that it is customary to hold the stadia rod plumb rather than inclined at right angles to the line of sight. Failure to hold the rod plumb introduces an error causing the observed readings to be longer than the true readings. Another error inherent in stadia surveying is caused by the unequal refraction of light rays in the layers of air close to the earths surface. The refraction error is smallest when the day is cloudy or during the early morning or late afternoon hours on a sunny day. Unequal refraction, also, causes the observed readings to be longer than the true readings. Figure 8-7.-Stadia arc (multiplier type).

Figure 8-7 - Stadia arc (Multiplier type)

Figure 8-8.-Stadia arc (horizontal scale subtraction type).

To compensate for these errors, topographers often regard the instrument constant as zero in stadia surveying of ordinary precision, even if the instrument has an externally focusing telescope. In this way, the last terms in the stadia formulas for inclined sights vanish; that is, become zero. Then the approximate expressions for horizontal and vertical distance are

BEAMAN STADIA ARC. The Beaman stadia stadia circle. These arcs are used to determine distances and differences in elevation by stadia without using vertical angles and without using tables or diagrams. A stadia arc has no vernier, but readings are indicated by index marks.

The stadia arc shown in figure 8-7 is the multiplier stadia arc (the vertical index is at zero); that is, the observed stadia interval is multiplied by the Hor stadia arc reading to get the horizontal distance; or the stadia interval is multiplied by the Vert stadia arc reading to obtain the vertical distance from the center of the instrument to the point sighted on the rod This vertical distance, combined with the HI and the rod reading, will give the difference in elevation between the instrument station and the point where the rod is held

The stadia arc, as shown is figure 8-8, is called the horizontal scale subtraction stadia arc (the vertical index is at 50). The use of the Beaman stadia arc to obtain a horizontal distance and difference in elevation is explained in the following sections.