376 ON THE MOTION OF 



A — r — x , a _ x — 1, 



(^i)= jj, (5i)= #, 



(^2)= •#, (52)= 5, 



(-93)= (.tf + 5 — C)r, (£3) = — (.#+5— C)r. 



(.#4) = #r, (54) = #r% 



(.-95)= (C— BY — m'£, (55)= (C<— .tf)r — /'£, 



(M) = — m' = —MgB,, : (56) = — /' = —MgA,. 



These equations may, by the elimination of u and o, be redu- 

 ced to two of the fourth order, of eleven terms each, no term 

 being wanting. They may be then completely integrated, 

 and after the determination of the value of the ten arbitrary 

 constants, eight of which are introduced by these equations 

 and two others by the equation W = o, the position of the 

 body and all the phenomena of the motion will be expressed 

 in terms of the sines and cosines of arcs proportional to the 

 time. The conditions of oscillatory motion will also be ex- 

 pressed by equations of limitation arising during the process 

 of determining the integrals. 



I shall conclude this paper with an application of the pre- 

 ceding formulas to the determination of the small oscillatory 

 motions of bodies of any figure, law of density, and areola of 

 contact, rolling with the three rotations on a surface which 

 from some slight asperity or other cause prevents entirely and 

 in all directions the sliding motion of the body, while in other 

 respects it leaves it free to rock, pitch and spin, with any 

 combination of these motions consistent with a small decli- 

 nation of the natural vertical of the body from the ver- 

 tical of equilibrium. I ought to remark that this motion, 

 although more resembling the actual oscillations of supported 

 bodies, differs from them materially in the circumstance that 

 the friction is supposed not to interfere with the motion round 

 the normal, whereas this cause undoubtedly cooperates with 

 the resisting medium to retard the horizontal rotation of the 



