81] 



FIXED AND ROLLING CONES. 



255 



Nfi 



and p 2 , the radii of curvature of the sections of the fixed and rolling 

 cones, the fourth , and consequently the whole motion, are determined. 

 This corresponds to the fact that the body 

 has. three degrees of freedom. 



If 01, Fig. 72, is the instantaneous 

 axis, OC and OC 2 the lines of centers of 

 curvature, the point I may be considered 

 to be travelling around the cones with the 



ds 

 velocity -^-t and about the lines 00 X and 



00 2 as axes with the angular velocities v, ^, 

 of which 03 is the resultant. This was 

 proved by Poinsot as follows. Let Fig. 74 

 represent a section of Fig. 72 a in the plane 



of the axes OC 1; 01, OC 2 . If r 19 r 2 are 



Pig. 74. 



the perpendiculars from I on 00 1? 00 2 , 



and v and u are the angles C^OI, 2 OJ, we have, considering absolute 



values only, 



10) 



dt "*^~' T dt 



1 ds l 



cosu. 



~^T = VCO$V, - -rr = II COS U . 



g^ dt Q 2 dt 



Inserting in 8') 



also since 



we have 

 12) 



09 = 4 U- COS U + V COS V, 



r, i\ , 



-~- = -r^i and vr, = 



*ITI 4t sm ^ 



It sin u = v sin v. 



sm v sm w 



That is, since the three axes are in the same plane, and G> is the 

 sum of the components of [i and v in its direction, while their 

 components in the perpendicular direction are equal and opposite, 

 a) is the resultant of ^ and v. Figs. 72 a, 72 b, 72 c show the three 

 cases, where the cones have external contact, where the fixed cone 

 is internal, and the rolling cone is internal, respectively. The 

 parallelogram construction is shown on the figures, and the direction 

 of rotation is shown by the arrows representing the vector rotations. 

 It will be noticed that in each case the arrow on the figure showing 

 the direction in which / is travelling around the rolling cone is 

 opposite to the direction of rotation /i about 2 . 



The rotation about OOj is known as precession. If both fixed 

 and rolling cones are cones of revolution, and /t, v, co constant, the 



