MOTION OF DEVELOPABLE CYLINDERS. 83 



examples we may take rolls of tape or ribbon. I suppose 

 the lamina to have thickness, but this thickness to be in- 

 definitely small. Approximately the motions may be de- 

 termined as follows : — Let a section of the cylinder be made 

 by a vertical plane through its centre of gravity. Take the 

 fixed point as origin. The external forces acting on the 

 mass are g sin a 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 — g cos a 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 un- 

 rolled 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 



d % x d z v 



the plane -^ and -^ vanish. If then T x denotes the 



tension due to the rolling mass and R x the resistance, the 

 ordinary equations of motion will give 



tm^=M 1 gsina-T l , (i) 



d z y 

 ^ m ^i =— Mucosa + U lf .... (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 : — 



d z x 

 M I ^=M^sin«-T IJ 



If we take moments about the centre of the rolling cylinder, 



