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XTI. Pseudo- Regular Precession, 

 By Sir G. Greenhill *. 



I^HIS is the gyroscopic motion described and illustrated 

 in Klein-Sommerfeld's Kreisel-Tkeorie, p. 209, where 

 a spinning top appears at first sight to be moving steadily in 

 uniform precession at a constant angle with the vertical, but 

 on closer scrutiny the axle is seen to be describing a crinkled 

 curve of small loops or waves; so that in this penultimate 

 state a realisation is obtained of a motion expressible by a 

 function which does not possess a differential coefficient, 

 a paradox fascinating to a certain school of pure mathe- 

 maticians. 



A pseudo-regular precession, although invisible, would 

 not feel impalpable to the analytical thumb passed over it, 

 which would detect a roughness. But in regular precession 

 the curve would feel quite smooth. 



In the investigation the axle may first be supposed moving 

 in perfect steadiness with no trepidation or nutation ; and 

 then to receive a small impulse, blow or couple, giving rise 

 to the pseudo-regular precession visible to the eye. 



1. Begin with a rapid spin about the axle, held fixed at a 

 constant inclination, taken at first as horizontal for simplicity, 

 in fig. 1. 



Fisr. I. 



For visible experimental illustration, it is convenient to 

 take a large (52-iiuh) bicycle wheel, mounted on a spindle 

 with ball bearings, and to prolong the spindle by sere wing- 

 on a stalk, at one end or both. 



The end of the stalk may be supported on the hand and 

 the wheel set in rotation by a swirl ; the hand accompanies 

 the wheel in the precession ; or else the point may be placed 

 in a cup fixed on the floor. 



When the axle is released from rest, it will start from a 

 cusp and sink down, then rise up again in a regular series of 

 loops or festoons ; so that to secure the uniform precession 



* Communicated by the Author. 



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