351 



and leaving out of consideration the rotation of the earth, its motion 

 would be that of an oscillation in a vertical plane, determined by the 

 equation 



When the starting position of the axis is but slightly inclined to the 

 vertical, and the oscillations are small. 



the period of vibration = /j5 . / 



^C- 



i 



C-A 



^ 5i minutes, nearly, 

 C-A 



a motion far more rapid than in this case (i. e., when the gyroscope is 

 placed in its frame without spin) could arise from the earth's rotation. 



3°. In the preceding analysis the problem discussed has had a purely 

 theoretical significance, the rings which realize the conditions proposed 

 being left out of consideration. How will their inertia modify the 

 results ? In the first two cases treated there is no difficulty in includ- 

 ing them in the moving system. Suppose in Case I. the axis confined 

 to a plane by rendering immoveable the outer ring ; let CiAi be the 

 moments of inertia of the inner ring round an axis perpendicular to its 

 plane, and an axis in its plane ; applying the equation of relative vis 

 viva to the whole moving system, the equation which replaces (5) will 

 be 



^ ] - U jo ^ • " ^ " 



A ^ C - A 



-j — — — cos (cos ^0 - cos 20o) . 



A + Ai 



If we compare this with equation (5), it is evident that, omitting terms in 

 iy^^), the only change to be made in the solution of that case is to suppose 

 {m) to represent 



^ \ instead of as before. 



^A +A,/ \A 



Again, the axis may be restricted to a right circular cone (as in Case IL), 

 by connecting together the two rings, their planes being set making 

 with each other an angle (as) equal to the angular radius of the required 

 cone, and leaving the exterior ring free to revolve round one of its own 

 diameters. Neglecting terms in (w^), the results already obtained hold, 

 supposing (m) now to stand for 



Cn sin 



A sin + A2 + Ax cos + Ci sin ' 



