KK.Sf LTI N(i 1 ROM T II K T II K O U Y OF T II K (', Y U () S (' O I> E. 5 



instituting in these , , n for 0, rejecting all small quantities of the second order 

 onir which is /// ). and introducm" i and '/. fsee conations V*) 



Substituting in these ,, n for 0, rejecting a sma quanttes 

 (am-ing which is in 2 ), and introducing J and >. (see equations V?) 



. - 



y dt- 



14. SM Iff -.--+ m - 



<// \Xsin w ' sm o 



The integral of (18) (using k with its already given value) in (7) is 



l-'i. = (-in 



J ,A 



Substitute in (1 I) and integrate 



K;. 4= *,_ 



2^- V4 ; ;r J/.-s.uu 



Tlie coefficient ^ in (1C)) is identical with that of equation (9), = 3JK, showing 



ow 



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

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

 periodic ally to rot, moves along a prolate cycloid or even witJiout undulation, yet 

 the rate of g\ ration is unchanged. 



If u=90 and m= ' , and become zero for all values of <, and the body 



, ,l 



:\ rates liorizontally without nutation. 1 



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 the moment of the accelerating 

 (or i-ouplr) 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 expression ' . 



My 



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

 m "-' V J, is necessary. 



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

 locns, in the body, of the instantaneous axis), described about the axis of figure with the angle 



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



, i M'.iy 



angle u __ ( ( w hcn u is not 90, the centrifugal as well as the deflecting force affects the 



relation between m, n, and g). 



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

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

 instants of time at which the processional movements of the axis are identically the same. 



