7GG PEOFESSOB SYLVESTER ON THE MOTION OF A EIGID BODY 



Hence 



(AQ-M')[P]^=(AQ-M')J^-(AQ-M*)[F]' 



- {4pqM^ - (2q' - Spr)WA+ iqrMA')Q+U%qU - 2r A)'. 



Hence 



_ 2n«-(2pM-gA)n + M ( yM-2rA) . 

 L^J" -/An-M* 



and as the arm at which this couple acts is 



, /c«2 M . / 



An-M«^ 



MA 



the pressure 



p_2n«-(2pM-9A)fl+ ((?M«-2rMA) 



An-AP ' 



If we call the constant perpendicular from the centre and the radius vector to the 

 point of contact h and I respectively, and substitute for^^' ^f their respective values h^P, 



F 

 h\ we may express p as a function of h, I, and making this a maximum in respect to I, 



the least sufficient value of the coefficient of friction necessary to ensure rolling may he 



deduced in terms of 'the quantities ?; t'> -• 



Also if fl denote the angle between the axis of the couple J and the pole of the plane 

 PI, we have 



^ ' [J]' {h^P-h'')(^^rh'+(^f-Apr)hn''-{qh''-2r)'^y 



Or 



cos^= <2A^/^-(2;>A«-g)Z^+(gA^-2r)) 



It has been already seen how, by the method of confocal ellipsoids, the number of 

 constants entering into the question of the rotation of a rigid body about its centre of 

 gravity has virtually been reduced by a unit ; to render this important theory complete, 

 and to give it the fullest extension of which it is capable, a corresponding dynamical 

 theory of contrafocal ellipsoids remains to be developed, and might undoubtedly be 

 discussed by analogous geometrical methods ; but it will be found more expedient to 

 take up the subject afresh from a purely analytical point of view, and then the theorj- 

 will present itself in all its completeness under a single aspect. 



Calling a, j3, y the angles which the invariable axis makes with the principal axes of 

 the rotating body, we have the well-kno\vn equations 



Ao), _ BtOa Ccoo 



cosa=-j^) cos/3=^-^5 cos7=-j^ 



(immediate deductions from the self-obvious principle of the constancy of the couple 



