PEOEESSOE OSBOENE EEYNOLDS ON EOLLIN G-EEICTION. 
165 
this, as has already been pointed out, is to make room for the lateral extension under 
the roller from C 'to D. From D towards B, therefore, and from C towards A the 
parallel lines are somewhat distorted, and at something less than their natural distance 
apart. From D to er vertical compression and lateral expansion is going on, and the 
lines are convex outwards. From er to P there is no slipping and the lines straighten. 
From P to fr, which is greater than the corresponding distance from P to er, there is no 
slipping, and at fr the lines are convex outwards. From/V to C vertical expansion 
and lateral contraction take place, so that the lines are all concave outwards. The 
lateral expansion from I) to er and the lateral contraction from fr to C can only take 
place hy the slipping of the india-rubber over the iron. Its extent is shown by the 
distance between the corresponding lines on the india-rubber and those on the iron, 
which latter have been set out equal to the distance between the lines on the rubber 
where greatest, namely from er to fr. 
The Actual and Apparent Slipping. 
Since there is no slipping at P, it is clear that the roller will roll through less than 
its geometrical distance, inasmuch as the geometrical distance between the lines on the 
plane at P is greater than their natural distance. Therefore the ostensible slipping 
will be equal to the difference between the intervals marked on the roller, and the 
initial distance between those on the rubber. The actual slipping, however, is equal to 
the difference between the intervals on the roller and the intervals on the rubber at D 
or C, which latter are less than the natural distance ; therefore the actual slipping is 
greater than the ostensible in proportion to the compression at C and X) ; and since this 
is increased by diminishing the coefficient of friction, such a diminution will affect the 
actual slipping in a greater degree than it affects the ostensible. This is in accordance 
with what has already been stated. 
India-rubber Holler. 
If the distance between the lines at P were exactly equal to the natural distance, 
then the roller would roll through its geometrical distance whatever might be the 
actual slipping. This is very nearly what actually takes place when an india-rubber 
roller rolls on an iron plane. 
In the case of an india-rubber roller on an india-rubber surface the lateral compression 
in the surface of the roller at D is greater than that in the plane, and the expansion at 
P is not so large, and hence there is slipping, and the roller will not accomplish its 
geometrical distance. 
In this explanation I have referred to india-rubber because it is much more easy to 
conceive the effects on it than on a hard substance like iron, the expansion and contraction 
of which is quite inappreciable to our senses ; the reasoning, however, applies equally 
well to all elastic substances, and is quite independent of their hardness or softness. 
That friction is sufficient to prevent the expansion of iron at a surface against which it 
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