500 MR. S. S. HOUGH ON THE OSCILLATIONS OF A 
which are satisfied by 


(= mm 105 = — 
_ Suppose 6, = ¢e‘°'*? where ¢ is a real quantity ; we have as one solution 
Gh = Gh. Hy SS hg 9) 
Changing the sign of 7 wherever it occurs, another solution is given by 
0, = hemi ett9, 6, = — ie ~itetto 
and this corresponds to the value — o of Xd. . 
Combining these two solutions we get as the real motion corresponding to the root 
Ne == 1075 
0, = 2¢ cos (at + «), 6, = — 2¢ sin (at + e). 
Now @,, @, serve to determine the position of that principal axis which in the 
steady motion coincides with the axis of rotation, relatively to axes which are 
themselves revolving with angular velocity o. 
Let us consider the angular displacements relatively to fixed axes O€, On, O€ 
coincident with the position occupied by the moving axes at the time ¢=0; they 
are 
@, cos wt — 6, sin wt = 2¢ cos «, 
#, sin wf + 8, cos ot = — 2¢ sin ge, 
and these are constant quantities. Thus, the apparent oscillation which corresponds 
to the root \* = w*, consists of a small permanent displacement of the axis of rotation, 
and the system rotates as if rigid about an axis which does not accurately coincide 
with our axis Oz. It is obvious that if » be the angular velocity of rotation about 
this axis, the system and the moving axes Ox, Oy, Oz will return to their original 
positions after an interval 27/o, and, therefore, the system will appear to oscillate in 
a period 27/w relatively to these moving axes. 
It is easy to see that the fluid motion, indicated by the analysis, also consists of a 
motion of pure rotation. 
For when 6, = de**!*® 
Br 0 B,/7a” = — 2B,/Tw* = ide“. 
Therefore, from (34), 
ww, = wide faz + yz}, 
and from (6), 
