158 
PROFESSOR OSBORNE REYNOLDS ON ROLLING-FRICTION. 
intervals of - 01 of an inch, and suppose that as the cylinder rolls across these lines the 
surface of the india-rubber extends so that the intervals become equal to *011 of an inch, 
closing again after the cylinder is past, then the cylinder will measure its circumference 
(so to speak) on the extended plane, and the actual distance rolled through when 
measured on the contracted surface will be one tenth less than the circumference. In 
the same way there would be an alteration in the distance rolled through if the surface 
of the roller extended or if either of the surfaces contracted. 
In the subsequent remarks I shall call the distance which the roller would roll 
through if there were no extension or contraction its geometrical distance. 
Since no material is perfectly hard, when a heavy roller rests on a surface the weight 
of the roller will cause it to indent the surface to a greater or less extent according to 
the softness of the latter; and in the same way the surface of the cylinder will be 
flattened at the point of contact in the manner shown in fig. 1. 
Fig. 1. 
This indentation and flattening will alter the lengths of the surfaces at the point of 
contact, and will therefore affect the progress of the roller. When a body of any shape 
is compressed in one direction it extends in the other directions ; hence the weight of 
the roller resting on the plane will, by compressing the material of the plane in a 
vertical direction, cause it to extend laterally at the point of contact, and thus the 
length of the surface which the cylinder actually rolls over would be greater than the 
length measured on the undisturbed plane. From this cause, therefore, the cylinder 
would roll through less than its geometrical distance. 
On the other hand, the surface of the roller would also be extended (squeezed out) 
in a similar manner by the pressure of the plane at the point of contact ; and hence the 
surface of the roller would be greater than its natural length, and this would cause the 
roller to roll through more than its geometrical distance. 
To a certain extent, therefore, the expansion of the surface of the roller would 
counteract the expansion of the plane ; and if the two were of the same material, then 
the one of these extensions would, if nothing interfered to prevent it, exactly counteract 
the other. But if the one was harder than the other, then the effect is that one would 
be least. Thus an iron cylinder rolling on an india-rubber plane would roll through 
