82 DR. JAMES BOTTOMLEY ON THE 



ness when the lines are parallel, and W z when they cross, 

 we shall have 



W,_ \—inr 



~W Z ~~ (i—nr) 2 ' 



In both cases the lines are supposed to be so thin as to be 

 individually imperceptible. 



Again, suppose an engraver to cover a square white area 

 with black circular spots which touch one another, the 

 spots being very numerous and individually imperceptible, 

 so that we receive the impression of a grey surface. If 

 beyond this stage he exercise his skill in diminishing the 

 area of the spots indefinitely and increasing their number 

 so that they still fulfil the condition of touching, it will 

 make no difference in the intensity of the tint ; for the 

 area of the spots is always the same, and equal to that of 

 the circle that can be inscribed in a square. 



IX. On the Motion of Developable Cylinders. 

 By James Bottomley, B.A., D.Sc, F.C.S. 



Eead before the Physical and Mathematical Section, 

 March 29, 1881. 



I have not seen in any works on Dynamics the following 

 problem treated : — A perfectly flexible surface is rolled up 

 so as to form a cylinder ; the external edge is fixed, and 

 the cylinder is allowed to roll down an inclined plane 

 under the action of gravity : to determine the motion. As 



