Mr. J. Bridge on the Gyroscope. 345 
d 
= = = —a(1+c)v, 
d 
=> e +a@(1+c)u, 
d¢' dy’ 
ad =u, = =v. 
dy! be i? #4 
Now ae =0, from the condition that the principal axis is to 
remain in the plane Aaa’. Therefore 
a d*¢' du 
yO; a Spa a and pea ae 
The acceleration produced by the force is therefore constant ; 
at least it may be so considered while @ remains small. If the 
4 db. s 
force a ceases to act, Wy or u, and therefore @ and oe will remain 
constant. In other words, if the principal axis of a solid of revo- 
lution is free to move in a given plane, it behaves nearly in the 
same manner under the action of forces, whether the body is in 
motion or not. 
- The Third Problem. 
Let Aaa’ be the plane fixed with respect to the earth in which 
the principal axis of the body is free to move. Then the effect 
of the earth’s rotation with velocity Q may be found by resolving 
it into— 
Q cosy about the normal to the plane; © sin y cos ¢! about 
the principal axis of the body, ¢! being reckoned from the 
extremity of the meridian line of the plane, which is moving 
in the plane with velocity O cosy; and Q sin y sin d! about 
an axis perpendicular to the other two. 
The effect of the last component alone need be considered. 
From this it appears that the forces which act on the body must 
be such as to cause the extremity of the principal axis to move 
upwards, that is, perpendicular to the plane of constraint, with 
a velocity 0 sin y sin ¢’. 
In the equations of the last problem, let « be 0, and 
dy 
a = sin y sin qd’. 
Then we have 
dq! du 
f =ou—2 cosy, a= —o(1+e)v, 
diy! dv 
v=o, Gaol +eu— 2; 
