An elementary treatment of the theory of spinning tops and gyroscopic motion, by Harold Crabtree.

THE SLEEPING TOP

127

If K' = ', and only in this case, equation (v) becomes
02 = K'{1 + cos (2Xt + 2()}
= 2K' cos2 (X2t + 6);.. 0 = /2K' cos (X2t + ),
and the motion is simple harmonic.
In this case, since K'2 - Ik'2 = y202,
we must have either 00 = 0 or y = 0, which gives A = 0.
The latter condition gives, from (iii), = -= 2A.
Hence if K'==c' the top is either initially vertical or is so
displaced that the azimuthal velocity communicated to it is X1.
The first of these two cases has already been considered in Art.
106; the second is clearly of the type discussed in Art. 134, a in
this case being small. The path of the head H is a simple wave
of period 7r. In Art. 140 it is shown that the path can in
general be produced by the combined effect of two waves whose
27r
periods are ~27, but it can be drawn without reference to
these considerations by the method employed in the following
Article.
139. The path in space of the head of the top as seen from
above. We have seen (Art. 137) that 0 never becomes zero, but
oscillates between two limits a. and a2.
If a graph be drawn such that the ordinates represent 02 and
the abscissae the time, then equation (v) shows that the resulting curve is a simple wave of period -.
2
If now we draw a second graph such that each ordinate is the
square root of the corresponding one in the first curve, the new
graph will give the values of 0 for all time. This second curve
is clearly flatter than the first, but repeats after the same period.
Now the value of /, if described with the mean value of A, is
proportional to the time; hence we may also regard this curve
as representing 0 in terms of the angular distance described
with the mean value 4, and we are thus enabled to realise
the general appearance of the path in space of the head H of
the top.
The dotted curve sketched in Fig. 75 shows the path described
on the assumption that 4 has its mean value throughout, while
the continuous curve gives the actual path. The following considerations will show how they are connected.