Sighting Gravity

Following is a description of the "gravity sextant", which is a 
simple (though not very practical) device for determining the 
exponent of a spherically symmetrical force law of the form 
F = 1/r^c.

The sextant (or perhaps "quadrant" would be more correct) consists 
of a right angled frame subtended by a protractor, similar to a 
normal sextant.  At the end of each arm of the right angled frame 
is a mass held by a spring in a slider.  Both of the sliders are 
parallel to one of the arms, as illustrated below:



Oriented so that the plane of the sextant contains the center of the 
field source, rotate the sextant until the displacements of the masses 
in the two sliders are equal, and then sight the field source on a 
line through the corner of the frame, and mark the angle on the 
protractor.

The angle "theta" between the line of sight and the slider direction 
is related to the exponent c in the force law according to the equation 

               tan^2(theta) + (c+1)tan(theta) - c = 0

For example, if the angle reads 29.3165 degrees, then we have c=2, so 
we know the field obeys an inverse square force law.  On the other 
hand, if the angle reads 22.5 degrees, then c=1, and the field obeys 
a simple inverse law.  If the angle reads 32.8524 degrees, then the 
field obeys an inverse cube law.

Actually, for any given c, there are two solutions of the above 
equation.  If we call the two solution angles theta1 and theta2, 
then it turns out that theta1 + theta2 = -45 degrees for all c 
(assuming both angles are measured from the same arm of the right-
angled frame).  The angle between the two "equilibrium orientations" 
varies from 45 degrees at c=0 to 135 degrees at c=inf.

Here's a brief table of the angles for various exponents in the 
force law:

          c       theta1       theta2      theta1-theta2
         ---     --------      ------      -------------
          0         0.0        -45.0           45.0
          0.1       4.8        -49.8           54.6
          1        22.5        -67.5           90.0
          2        29.3        -74.3          103.6
          3        32.8        -77.8          110.7
          4        35.0        -80.0          115.1
         10        40.2        -85.2          125.3
        100        44.4        -89.4          133.9

I find it interesting that, even though the field is spherically
symmetrical, there are "special angles of incidence" associated with 
each possible exponent.  Since c tends to equal 2 empirically, this 
suggests that the angles 29.3165.. and -74.3165... degrees have special
significance, although off hand I can't think of any natural physical 
process that would make use of these angles.