The Rotation of the Non-Spinning Gyrostat 245 



greased. The motion of the stalk here is the same as with Kelvin's 

 trunnion rings and knife-edge gimbals, or his short length of elastic 

 wire, fixed to the stalk and the support; and it may be imitated 

 in the swaying motion of the body seated in a chair, where one 

 wall of the room is always faced ; the conical motion does not make 

 the body turn on the seat, not a music-stool. 



It would carry us too far to discuss the modification in ^ due 

 to these and other modes of suspension, such as Hookc's joint 

 and bevel-wheels. 



But so long as the inertia of the stalk may be ignored, there is 

 no modification in the d, ifj angles of the axle of the gyroscopic 

 wheel; but the angle ^ will require separate consideration. 



If however (/> is suppressed by clamping the wheel to the stalk, 

 or if the inertia of the stalk is taken into account, the motion is 

 hyperelliptic and intractable. 



The question quoted from the 1898 examination paper is a very 

 good specimen of many such, scattered anonymously in college 

 papers. Judging from the date, Dr Bennett ought to be able to 

 lift for us the veil of anonymity. 



A similar question by Maxwell in the Mathematical Tripos 

 1869, on vibration in its effect in causing a permanent deviation 

 in a pendulum, has proved useful, nearly 50 years later, in the 

 interpretation of compass deflection; and we have seen the deflec- 

 tion realised in an experiment devised by Mr C. C. Mason. 



Quaternions come in useful for the geometrical interpretation 

 of these questions on finite rotation ; as for instance in the resultant 

 rotation due to successive rotations of a spherical triangle through 

 the exterior angles. Then there is a theorem given by Dr W. Burnside 

 in the Messenger of Mathematics, xxiii., on the resultant screw dis- 

 placement due to two half turns about non-intersecting axes. 



The 1898 question, on the rotation that can be given to a body 

 on a smooth axle by a conical motion, may be quoted as an answer 

 to Aristotle's challenge — to rotate a smooth sphere — rjKia-ra Se 

 KivrjTiKov rj a<f)aipa St,a ro f^rjSev e^ecv opyavov irpo<i rrjv KLvrjaiv, 



To the preceding remarks of Sir George Greenhill, DrG. 3", Bennett 

 replies as follows: — 



From Sir George Greenhill's comments it is happily apparent 

 that the difference of four right angles between a result in his 

 Report and a result in my paper is not a miscalculation of either 

 but a discrepancy turning on a verbal ellipsis. My work has regard 

 only to the movement of the gyroscope relatively to ' fixed space ' ; 

 whereas his is limited to the movement of the gyroscope relatively 

 to the rotating stalk of his altazimuth suspension. 



