J. G. Barnard on the Motion of the Gyroscope. 73 
But z (the height of the centre of gravity above the fixed plane) 
=~7y cos 6; hence 7d.cos@=—dz; and equation (1) gives 
: 2 
= (Ta +0). Substituting these values of Rand 7d.cos@ in 
the preceding equation, and integrating, we have 
dz20 + 
A(ey? +2?) + I(T +292) = (3.) 
From the 2d and 84d of equations (2) the equation (c) (of the 
8yToscope, p. 54) is deduced by an identical process. 
A (bvy,--ar,)-+ Cn cos 0=l, 
and a substitution in the two foregoing equations of the values 
of the cosines a and d, and of the angular velocities v, and vy, 
m terms of the angles , @ and w (see pp. 52, 58), and for 2 and 
2 ois y 
q, their values, —y cos, and ysin 6s and a determination of the 
constants, on the supposition of an initial inclination of the axis 
| 3 *, and of initial velocity of axial rotation », will give us for the 
7 equations of motion of the top: 
: dy Cn : 
PE hy MOE 
Sood F Wiakoy © vouedgmeeudaes | 
cougud? goer : d 62 
A ined te + ia) + Aan 20a 280 (om — cone) 
from which the angular motions of the top can be determined. 
| The first is identical with the first equation (4) for the gyroscope. 
4 ~The second differs from the second gyroscopic equation only in 
_ Containing in its first member the term My? sin 707 or its 
‘equivalent M =, expressing the living force of vertical transla- 
fon of the whole mass. % ie 
The second member (as in the corresponding equation for the 
- 8)TOscope) expresses the work of gravity, and the first term of 
the first. member 
“Tt motion, part of it is expended in vertical translation of the 
- Cehire of gtavity. The angular motion takes lace not (as in 
is  8Y?Oscope) about the point of support (whic in this case is 
ot fixed), but about the centre of gravity (to which the moments 
Of inertia 4 and B refer); and that centre, motionless horizon- 
tally > Moves vertically up and down, coincident with the small 
angular undulations of the axis through a space which 
ore and more minute as the rotary velocity n is greater. 
SECOND SERIES, VOL, XXV, NO. 73.—JAN., 1858. 
10 . 
