343 



PiEST Case. — The Axis is restricted to move in a given Plane. 



Let (P) be the trace of the polar line on the unit sphere, {JSFX) that 

 of the fixed plane ; (X) that of the axis 

 of the gyroscope ; or, to define it exactly, 

 of that end of the axis on looking down 

 which the rotation of the gyroscope 

 would appear contrary to the movement ^ 

 of the hands of a watch — that is, would 

 appear in the same direction as the / 

 earth's rotation. K /--^ 



Draw the arc perpendicular to 

 iVX, leti\^P=^, WX=(p; 



ji ^ ^ J ds d<p 



then cos 9 =co8 B cos 0, and = -rr - 



at at 



.-.by equation (4) 



^) ~ ^ ^^^^ ^ ~ ^"-^ ~ cos^)8 (cos V 



- cos 20o). 



(5) 



Such is the rigorous differential equation for determining the motion. 

 In its complete form it is unintegrable. 



If we confine ourselves to terms of the first order, and suppose the 

 axis of the gyroscope started at relative rest, it becomes 



dt 



= 2mw cos jS (cos 0 - cos 0o). 



The motion is therefore identical with that of a simple pendulum whose 

 length, I = ^ oscillating about the line (W). When the vibra- 



27r 



mco cos , 



tions are small, the period of a double vibration T = 



\/ m(o cos y3 



_ ^ Asec^ ^ jrf where T is a mean proportional between the earth's 



period of rotation and the gyroscope's. 



