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nitely small. Approximately the motions may be deter- 
mined as follows Let a section of the cylinder be made 
by a vertical plane through its centre of gravity. Take 
fixed point as origin. The external forces acting on the 
mass are gsina and the tension at the fixed point, these act 
along the plane, which may be taken for the direction of 
the axis of x; perpendicular to the plane the external forces 
are — gcosct and the resistance of the plane — these acting 
in the direction of the axis of y. Of the whole tension at 
the fixed point a portion will be due to the unrolled part of 
the cylinder in contact with the plane, also of the whole 
resistance of the plane a portion will be due to the unrolled 
portion. For each particle in contact with the plane 
d 2 v 
and vanishes. If then T x denotes the tension due to the 
rolling mass and It the resistance, the ordinary equations 
of motion will give 
(ffidC 
S ^ a -Mtf S ina- Tl 
a) 
- Mucosa + Rj ..... .... 
(2) 
The summation extending to all the particles of the rolling 
mass. If we suppose it to move as a rigid body these 
equations may be written 
M i^ = M^sina-T : 
= - Mucosa + Rj 
If we take moments about the centre of the rolling cylinder 
the equation will be 
/ d?y d?x ) 
2m| \ = T ^ 
Or, more simply, if we suppose the cylinder to move as a 
rigid body, 
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