FIG. 2.—OUTLINE OF INSTRUMENT SHOWING THE PATH OF THE DIRECT AND OF THE REFLECTED RAY.
The dotted line A B represents the direct ray, and the line A C D the reflected one. Fig. 3 shows the different geometrical and trigonometrical elements of the curve, which can be read upon the various scales, or to which the instrument may be set. An observer standing at C sights the point B directly and the point A by reflection. A staff being set up at each point, he will see them simultaneously, and in coincidence if the instrument be properly set for the curve. If any intermediate position be taken up on the curve, both A and B will be seen in coincidence. If the two rods do not appear superimposed, the operator must move to the right or the left until this is the case. The instrument will then be over a point in the curve. Any number of points at any regular or irregular distances along the curve can thus be set out. One of the simplest elements which can be taken as a datum is the ratio of the length of the chord to the radius, AB/AO, Fig. 3. This being given, the value of the ratio is found on the straight scale on the body of the instrument, and the curved plate is moved until the beveled edge cuts the scale at the desired point. The figure of this curve is a polar curve, whose equation is r = a ± b sin. 2 θ, where a is the distance from the zero graduation to the axis of the mirror, and b is the length of the scale from zero to 2, and θ is the inclination of the mirror. In the perspective view, Fig. 1, the curved edge cuts the scale at 1. The instrument being thus set, the following elements may be read either directly on the scales or by simple arithmetical calculation:
FIG. 3
The radius = 1.
AB, the chord, read direct on the straight scale.
AFB, the length of the arc, read direct on the back or under surface of the plate.
FH, the versed sine, read direct on the curved scale.
ACB, the angle in the segment, read direct on the graduated edge.
EAB, the angle between the chord and the tangent, read direct on the graduated edge.
GAB, the tangential angle = 180 deg. - ACB.
AOB, the angle at the center = 2GAB.
AGB, the angle between the tangents = 180 deg. - AOB.
OAB, the angle between the chord and the radius = EAB - 90 deg.
AH2 GF = —— - FH. HO
| AH2 | ||
| GF = | —— | - FH. |
| HO |
The foregoing elements are contained in a very simple diagram, Fig. 4, which is engraved on the instrument, together with the following references:
B = 180 deg. - A.
C = 2B.
D = 180 deg. - C.