74 J. G. Barnard on the Motion of the noes 
An elimination of = mi between the two stead (4) and a 
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as m ‘the gyroscope. All the results and conclusions few lal 
60) would be d As, however, the centre of gravity, to 
which these angular motions are referred, is not a fixed point, 
but is itself constantly rising and falling as @ increases or di- 
minishes, the actual motion of the axis is of a more complicated 
racter. 
H Gi” (see fig. 2) is the 9. 
initial position of the axis of q 
the top, the motion of the cen- 4 
tre of gravity will consist in 
a are falling and rising 
thro the distance G@’= 
GK “ad G! G" —cosz, FG”) 
=7 (cos 9, —cos@) (in which 6, 
while the extremity of the axis 
or point, K, describes on the 
upporting surface and about 
the projection ” of the cen- 
curve a, 0, a, U', a'', &e., hav- 
ing cusps a, a’, &e., in the circle . 
described about @” with the ae 
radius G K'=7sine, and tangent, externally, to the > circle de- 
scribed with a radius Q" =z sin n6,. But as in the case of the 
roscope, these little undulations ‘speedily disappear thromge 
the ene. influence of friction and _ resistance ot the air, a 
the Sari of the top describes a circle, more or less mt 
about 
The par of the self-sustaining power of the top is identi- 
cal with that of the gyToscope ; the stl force due to” 
