PROFESSOR OSBORNE REYNOLDS ON ROLLING-FRICTION. 
159 
less than its geometrical distance ; whereas, inversely, an india-rubber roller on an iron 
plane would roll through more than its geometrical distance. 
These things actually take place. But there is, besides softness, another circumstance 
not hitherto mentioned which affects the lateral extension of the surface when com- 
pressed by the roller, viz. the shape of the surface. 
A little consideration will be sufficient to show that a curved indent in a flat surface 
null have a greater effect to extend the surface than a flat indent on a rounded surface. 
In the case of the rounded surface it will be seen that the effect of vertical compression 
to a certain extent counteracts the effect of lateral expansion ; whereas in the case of the 
flat surface these things are reversed, and the effort of the surrounding material to 
uphold that which is depressed will increase the lateral expansion. 
From this cause, therefore, even if the cylinder and the plane were made of the same 
material, there would still be a difference in the lateral extension of the surfaces at the 
point of contact depending on the smallness of the diameter of the cylinder, and this 
difference would still cause the cylinder to roll through less than its geometrical 
distance. 
If, instead of on a plane, the one cylinder rolled on another parallel cylinder under a 
force tending towards the centre, then, if the two cylinders were of the same material 
and them diameters were equal, they would roll through their geometrical distance ; 
but if the one was larger than the other, the largest would be most retarded. 
It appears, therefore, that there are two independent causes which affect the progress 
of a roller on a plane — the relative softness of the materials and the diameter of the 
roller. Of these the curvature of the roller always acts to retard its progress ; while 
the other (the relative softness) to retard or to accelerate according as the plane is softer 
than the cylinder, or vice versa. These two causes will therefore act in conjunction or 
in opposition, according to whether the roller is harder or softer than the plane. In the 
former case the roller will be retarded, whereas in the latter it will depend on the 
relation between the relative softness and the diameter of the cylinder whether its 
progress is greater than, less than, or equal to its geometrical progress. Thus an iron 
roller on an india-rubber plane will make less than its geometrical progress ; while an 
india-rubber cylinder on an iron plane will make more than, less than, or exactly its 
geometrical progress, according to the relation between its diameter and softness, or, 
what comes to the same thing, its weight, which conclusions are borne out by 
experiment. 
The Slipping. 
The lateral extension of the material, and the effect this has on the progress of the 
roller, causes slipping between the surface of the roller and that of the plane ; for the 
surface of the roller, owing to the indentation and flattening, really touches the surface 
of the plane over an area of some extent; and the pressure between these surfaces, 
which is greatest towards the middle of the area in which they touch, will shade off to 
