84 DR. JAMES BOTTOMLEY ON THE 



the equation will be 



v f d z y d z x~\ 



Or, more simply, if we suppose the cylinder to move as a 

 rigid body, 



K*.*£)-t,, ••-•'•• 0) 



M ,#* denoting the instantaneous moment of inertia. From 

 (i) and (3) we have the following equation : — 



M ^? + J( M - F S) =M ^ sina - • • (4) 



Also we have the following geometrical equation, since 



the space x is described by a circle of variable radius : — 



dx = yd6. Also the mass in contact with the plane will be 



bdcx, where c denotes thickness, b and d breadth and 



density. If we regard the rolling portion as a circular 



cylinder, its mass will be bdiry z } supposing it to have the 



same density as the unrolled portion. Let M be the whole 



mass, and It the initial radius, then M = 7rR 2 ^. Since 



the mass is constant, we obtain the equation 7ry z = 7rR z — cx; 



since y and x are the coordinates of the centre of gravity, 



v z 

 this equation gives a parabola as its locus. Also k z = — ; 



hence equation (4) may be written 

 d z x c I ' dxX* 2 



By integration we obtain 



