irror was provided, but by
the double reflection in the improved pattern, any accidental twisting
of the rod or handle produces no displacement of the images, since the
inclination of one mirror neutralizes the equal and opposite
inclination of the other. No cross line is required with the new
arrangement, since it is only necessary that the two images should
coincide.

[Illustration: 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 [theta], 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 [theta] 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:

[Illustration: 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