SOLIDS ON »i id \« l> 



ces applied to the problem of any solid body rolling on any 

 given surface, 



3T-+- k(d? -+- dk ' -+- dk ■) - & = ./. 



./ being another arbitrary constant, and .1/= i . 



It is evident, moreover, thai the same triplet furnishes the 

 relation 



I x — Vij ■+- IV: = n 



When the body is in a state of permanent equilibrium upon 

 the surface of support, the velocities and accelerations of 

 -i\ elements of its position are nought, and the six equations 

 ni motion give n> 



L — a . A y — M z — o, 



-V = b", L,z — Xx = o, 



N, = c : M, _ L,y — o 



The firsl equations express that the direction of the aormal i* 

 vertical, the others thai it passes through the centri of gravity 

 In genera] we may observe, thai the equations of the motion 

 oJ rotation are in fact the equations of the normal al the poinl 

 of contact, and that the distance of the aormal from the centre 

 of gravity i- at all times equal to 



' r -i" 4-n 



i! : 



so thai \ l' —l—W) represents the ( diet which the plan 

 reaction untile body has in producing tin- motion of rotation. 

 The line which passes through tin- centre of gravity and any 

 inn- of the points li of the surface mi which it maybe balan- 

 is ii"t in general a principal axis; but a- the preceding 

 formulas ar< independenl of the position of these axes, w< i" 



