348 



or the axis describes a small circular cone, whose semi-angle = tan 

 / tt^^m^X ^.^^ ^ uniform angular velocity in a period 



\/ + sin ^^Oq 



Still more briefly, the same results may be arrived at by the consi- 

 deration of Poinsot's resultant couple ; for it is evident on inspection 

 that the axis M thus chosen is the axis of the resultant couple of all the 

 motion with which the gyroscope is started. jSTow, the axis and magni- 

 tude of the resultant couple remain fixed; therefore Jf is always this 

 axis, and G its moment, 



and since {Cn), the component of the resultant couple round the axis of 

 figure = G cos ^, it follows that 



Cn m a) sin 0o 

 cos c = const = -7^- = ^ or tan c = . 



Again, the component of the resultant couple round an axis in the plane 

 (XM) perpendicular to (X) = 6^ sin ^ = ^ sin ^ ^ , 



.^'^ - ^ = ^/^n^ + "^Oq , as before. 

 at A. 



The result in the unrestricted case may be thus recapitulated : — 

 If the axis of the gyroscope could be started in a position of absolute 

 rest, no angular motion being communicated to the axis either by the 

 earth or the experimenter, it must always continue so, pointing to the 

 same fixed star. When it is not so started, but the axis at the moment of 

 detachment has a velocity (7) in a given plane, it describes a circular 

 cone round a fixed line in space, the semi-angle of the cone being 



tan-i-^, 



m 



and the period of description 



27r 



y W2 +72 



"Wlien this starting velocity (7) is solely due to its connexion with the 

 earth before detachment, 7 = 0? sin 0^^ a quantity generally so small com- 

 pared to (w), that the minute arch described by the extremity of the 

 axis would appear an absolute point under the most powerful micro- 

 scope. 



