450 SECTIONAL TRANSACTIONS.—A. 
Mr. C. H. H. FranKLin.—Demonstration of the orbits of a spherically free 
pendulum (12.30). 
A pendulum which is free to swing over a segment of a sphere has possible 
paths which vary from a rotation resembling an engine governor ball to the 
swing of a simple pendulum in a plane ; which plane appears to rotate very 
slowly, but really indicates the rotation of the earth beneath it (the Foucault 
pendulum effect). 
The intermediate orbits between the circular and the linear paths are 
ellipses, which latter precess at a rate dependent on the maximum and 
minimum angles of the pendulum and corresponding ratio of major and 
minor elliptical axes. This precession of the ellipse produces a pattern 
which corresponds to that produced by two opposed rotations having 
a ratio near unity (such as 100: 101, with appropriate amplitude) ; which 
may be drawn by harmonograph, etc. Virtually, the pendulum behaves as 
if it had two frequencies ; which is in agreement with the facts that the 
effective length of a pendulum swinging in a circle is cos « L, where « is the 
angle of the swing maintained and L is the length of the pendulum, and in 
tracing a single ellipse a maximum and minimum angle and corresponding 
minimum and maximum effective length are reached twice. 
It will be seen that the greater the angle reached, the greater the variation 
of effective length of the pendulum, and it is found that the rate of precession 
increases rapidly as the angle increases. 
Also, if the ratio of major and minor axes of the ellipse is high (which, in 
the limit, becoming ©, means the pendulum swinging in a plane), the rate 
of precession is small, in the limiting case becoming o. 
And as the ratio of the major and minor axes approaches 1, the rate of 
precession becomes a maximum for the angles involved. 
When the pendulum is capable of swinging over a hemisphere, and is 
oriented to maximum and minimum angles of 180° and 90° (which is a 2: 1 
ellipse), the vertical projection of the orbit becomes a 5-loop figure, corre- 
sponding to the 3:2 figure with opposed rotations as produced by the 
twin-elliptic pendulum, etc., with appropriate amplitudes. 
If the pendulum is taken above the equator of its sphere of rotation, 
a further reduction in the number of possible loops in the orbit appears to 
occur, but this is deceptive, because there is now a loop above and a loop 
below the equator, which correspond. Also the figuring can now be 
dependent on velocity, the angle of initial swing not necessarily deciding 
the pattern, as in small spherical angle orbits. 
If the speed of projection of the pendulum is high, the orbit tends to 
become a great circle on its sphere, which orbit precesses in the same 
direction as would be the case with a gyroscope. 
It follows that if the orbit is nearly in the horizontal plane, precession 
occurs rapidly for any given mean velocity of pendulum ; and as approach- 
ing the vertical plane precession becomes relatively slow, it should, in the 
limit, become zero again when the pendulum completely rotates in a vertical 
plane. 
The rate of precession for any given inclination of great circle orbit now 
depends on the velocity of the pendulum bob, in the same way that the rate 
of precession of an unbalanced gyroscope at a given slope depends on the 
velocity of rotation. 
AFTERNOON. 
Visit to University College, Nottingham. 
