162 
PROFESSOR OSBORNE REYNOLDS ON ROLLING-FRICTION. 
sent, the intensity of the friction at the surface. As long as there is slipping the 
friction will be proportional to the pressure. Therefore from the edge of the surface 
to er the inclination will continually increase; and it will be greatest at er, for from 
this point inwards the tendency of the india-rubber to slip will obviously diminish until 
it vanishes at P. 
The distance of ex from P will not depend on the degree of compression, at all events 
so long as this is but small, for the tendency to extend laterally will be proportional 
to the intensity of the pressure ; and since the friction is proportional to the pressure, it 
will increase at all points in the same ratio as the forces tending to extend the rubber 
laterally. The distance of er from P will, however, obviously depend on the coefficient 
of friction. The greater this is the greater will be the region over which there is no 
slipping. 
By blackleading the india-rubber, therefore, we should change the shape of the section 
from that shown in fig. 4 to that shown in fig. 5, in which all the ends of the lines from 
er to the circumference are less inclined than the corresponding lines in fig. 4, and the 
intervals between them greater, showing that not only is the friction less and the area 
over which it acts greater, but that each point has also to slip through a greater 
distance. 
It is difficult to say how far these two latter effects will compensate for the former. 
We may, however, show that there must be some value of the coefficient of friction for 
which the work spent in overcoming the friction will be a maximum ; for when the 
coefficient was very great er would be at the circumference and there would be no 
slipping, and hence no work spent in friction; whereas if the coefficient were zero, 
er would be at P, and there would be no friction and consequently no work lost in over- 
coming it. Therefore the work spent in friction, which is a function of the coefficient 
of friction, is zero for two values of the variable ; and since it is positive for all inter- 
mediate values, it must pass through a maximum value. Hence for some position of 
er (for some particular coefficient of friction) the work spent in friction would be a 
maximum. What this value of the coefficient is it is impossible to say ; but it seems to 
be less than that between clean india-rubber and iron, and it may be less than that 
between blackleaded india-rubber and iron. This was shown by the experiments on 
rolling-friction. 
In considering these experiments, however, there is another thing to be taken into 
account besides the work spent in friction during compression, and that is the effect of 
friction during restitution ; for the action of a roller as it passes over the india-rubber 
will be first to compress it and then to allow it to expand again in a corresponding 
manner. 
The effect of Friction during Expansion. 
If, after the rubber has been compressed as shown in figs. 4 and 5, the surfaces 
gradually separate again, the shape of the lines will again change. The lines from 
P up to er will assume the same forms which they had at corresponding periods of the 
