/. G, Barnard on the Gyroscope. 61 



The second term is due to the circular motion of the axis 

 about this centre, and, combined with the corresponding A^alues 

 of w, constitutes what may be called the nutation. 

 , These cycloidal undulations are so minute — succeed each other 



with such rapidity, (with the high degrees of velocity usually 

 given to the gyroscope,) that they are entirely lost to the eye, 

 and the axis seems to maintain an unvarying elevation and move 

 around the vertical with a uniform slow motion. 



It is in omitting to take into account these minute undulations 



that nearly all popular explanations fail. They fail, in the first 



place, because they substitute, in the place of the real phenome- 



4 non, one which is purely imaginary and inexplicable^ since it is 



in direct variance with fact and the laws of nature ; — and they 

 fail, because these undulations — (great or small, according as the 

 impressed rotation is small or great) furnish the only true clue 

 to an understanding of the subject 



The fact is, that the phenomenon exhibited by the gyroscope 

 which is so striking, and for which explanations are so much 

 sought, is only fx particular and extreme phase of the motion ex- 

 pressed by equations (6) and (7) — that the self-sustaining power 

 IS not absolute^ but one of degree — that however minute the axial 

 ^ rotation may be, the body never will fall quite to the vertical;— 



[^ however great, it cannot sustain itself without any depression. 



I have exhibited the undulations, as they exist with high veloci- 

 ties, — wben they become minute and nearly true cycloids ; with 

 low velocities, they would occupy (horizontally) a larger portion 

 of the arc of a semi-circle, and reach downward approximating, 

 more or less nearly, to contact with the vertical : and, finally^ 

 when tbe rotary velocity is zero, their cusps are in diametrically 

 opposite points of the horizontal circle, while the curves resolve 

 themselves into vertical circular arcs which coincide with each 

 other, and the vibration of the pendulum is exhibited. All 

 these varieties of motion, of which that of the pendulum is one 

 %^ extreme phase and the gyroscopic another, are embraced in 



equations (6) and (7) and exhibited by varying ^ from to high 

 values, though, (wanting general integrals to these equations) 

 we cannot determine, except in these extreme cases, the exact 

 elements of the undulations. The minimum value of may 

 however always be determined by equation (8). 



If we scrutinize the meaning of equations (6) and (7), it will 

 be found that they represent, the first, the horizontal angular 

 component of the velocity of a point at units distauce from 0, 

 and the second, the actual velocity of such point.* 



* In .more general terms equations (4) express, the first, that the moment of the 



quantity of motion about the fixed vertical axis Oz remains always constant : the 

 ^ 9teond that the living forces generated in the body (over and above the impressed 

 axial rotation) are exactly what is due to gravity through the height, L 

 - Both are expressions of truths that might have been anticipated ; for gravity 

 camiot increase the moment of the auantitv of motion about an axu naralld ta 



