RESULTING FROM THE THEORY OF THE GYROSCOPE. 5 



Substituting in these o — u for 0, rejecting all small quantities of the second order 

 (among which is nr), and introducing 3 and Z (see equations '2) 



13. -^^2u(smcj—23 P»A— 4i^V- 



14. 



9 dt- \ ' \^ 



^^—9-? \^ urn 



^=23 I 



(It ' S'A. sin (J sin o 

 The integral of (13) (using k with its already given value) in (7) is 



15. '^'^-, — A ^^'^ " —ml sur a:^ 



Substitute in (14) and integrate 



16. ^= JL Z:<_( i^_ "L_) sin 2Jd 



23^ \4;J^ 2A:Sin(j/ 



The coefficient < in (16) is identical with that of equation (9), — ~ , showing 



that although the character of tlie gyratory motion is altered, and the axis of 

 figure, instead of moving on a common cycloid (which forms cusps) and coming 

 periodically to rest, moves along a prolate cycloid or even ivlthout undulation, yet 

 the rate of gyration is unchanged. 



If (j=90° and m =j ?'-. « a"d -- become zero for all values of t, and the body 



gyrates horizontally Avitliout nutation.^ 



In all that precedes, the revolving body has been supposed retained by a fixed 

 point in its axis of figure, but not at its centre of gravity, while the accelerating 

 force, being gravity itself, acts through that centre. 



If, instead, the fixed point by which the body is retained is the centre of gravity, 

 and the accelerating or disturbing forces any other whatever (provided their direc- 

 tion is invariable and their resultant acts through a fixed point of the axis), the 



* This is the case referred to in the preceding paragraph in which tlie moment of the accelerating 

 force (or couple) is equal to that of (what I have styled in the work before referred to) the "deflect- 

 ing force," which has for its value the expres-sion - — -. 



That this case should arise, a determinate relation between m, n, and g, expressed by the equation 



mn My . 



=- '-, )s necessary. 



9 G 

 When this relation exists, the movement niav be represented by the rolling of a conical surface (the 

 locus, in the body, of the instantaneous axis), described aljout the axis of figure with the angle 



(approximately) equal to ~— , upon another, all of the elements of which make, with the vertical, 



the angle o — —J- (when u is not 90'^, the centrifugal as well as tlie deflecting force affects tiie 



relation between m, n, and [/). 



But no such relation is essential to the gyration expressed by (11) ; and. in the case of the preces- 

 sicm of the equinoxes, the supposition of rolling cones is not realized. There are, probably, no two 

 instants of time at which the precessional movements of the axis arc identically the same. 



