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 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 nothing at the edges. Thus de-
formation is allowed to go on between the surfaces after they have come in
contact, and is performed by the slipping of the one over the other.
116 ON ROLLING-FRICTION. [18
The slipping is performed against friction, and therefore gives rise to
resistance to the motion of the roller.
This resistance will obviously be proportional to the work spent in over-
coming the friction between the surfaces during a certain extent of motion ;
and at first sight it appears as if this would be proportional to the coefficient
of friction between these surfaces. When I first commenced this investigation
I was under the impression that such would be the case, and that by oiling
the surfaces the resistance to rolling might be considerably reduced. Finding
by experiment, however, that this was not the case, that although in certain
cases the effect of oiling or blackleading the surfaces does reduce the resist-
ance to rolling, yet this reduction is never great, and in some cases the effect
appeared to be reversed, it occurred to me that the friction would itself
modify the deformation which would otherwise take place after contact had
commenced, and by preventing slipping might diminish the work that
would otherwise have been spent.
The action of friction to prevent the deformation at any point of the
surfaces in contact will obviously depend on two things the magnitude of
the friction, and the force tending to slide the one surface over the other.
If P (Fig. 1, p. 114) be the point of greatest pressure, the possible friction
will gradually diminish with the pressure as the distance from P increases ;
whereas we may assume that the tendency of the one surface to slip over the
other will be nothing at P, and will gradually increase with the distance ; so
that for a certain distance the friction may be sufficient to prevent slipping
altogether, but beyond this distance slipping will go on in an increasing
The effect of oiling the surface would therefore be to diminish the region
of no slipping, and increase the area over which slipping goes on, as well
as the extent of slipping at each point. These effects would to a certain
extent counteract the advantage gained by the reduced coefficient of friction ;
and it may well be conceived that under certain circumstances they would
overbalance it, and that the oil would actually increase the resistance.
The effect which friction has upon the deformation beneath the roller, as
well as the general nature of this deformation, will be rendered clearer by
examining the effect of friction under circumstances of a less complicated
nature than those of rolling.
A Soft Bar between Hard Plates.
Let Fig. 2 represent the end or a section of a long rectangular bar of
india-rubber, or any elastic material, placed between two flat plates. Suppose
these plates to approach each other, compressing the india-rubber, which will
extend laterally. Now if there were no friction between the rubber and the
plates, then the surfaces in contact with the plates would extend in the same
proportion as the rest of the bar, and the section would preserve its recti-
linear form, as shown in Fig. 3.
With friction, however, the case would be different. The friction would
prevent the surface of the india-rubber expanding laterally to the same extent
as the rest of the bar, and the section would lose its rectilinear form, and bulge
out in the middle, as shown in Figs. 4 and 5.
118 ON ROLLING-FRICTION. [18
If we imagine the section of the bar to have been marked with a series of
lines (ee') initially vertical and at equal intervals apart, these lines will, when
the bar is compressed, assume the form shown in the figures.
If there were no friction, then, as shown in Fig. 3, the ends of these lines
would still be equidistant after compression ; but with friction the in-
tervals will not be all equal, but will vary according to their distance from
P, the middle of the section. Up to a certain distance (er) the friction will
be sufficient to prevent slipping ; and hence up to this point the ends of the
lines will preserve their original distance. From this point (er), however,
slipping will commence, and will go on increasing to the edge of the surface.
From this point, therefore, the distance between the ends of the lines will
With regard to the distribution of the pressure between the india-rubber
and the plates: Without friction this will obviously be uniform over the
whole surface. Friction, however, will not only increase the mean intensity
of the pressure, but will also alter its distribution, causing it to be greatest at
P and gradually diminish towards the edge.
The inclination of the ends of the lines ee' is caused by, and may be taken
to represent, 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 er 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, show-
ing 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
18] ON ROLLING-FRICTION. 119
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 overcoming it. There-
fore 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 of 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 compression ; but since that portion of the
surface which lies beyond er has been extended by the compression, it will
have to contract as the surfaces recede, and the friction of the surface will
oppose such contraction. Hence the lines, which during compression were
curved outwards, will gradually straighten and curve inwards, as shown in
Fig. 6. Those at the edges will take the form first, and then those nearer to
er, until the expansion has become complete.
The extent to which friction will deform the india-rubber during this
operation will obviously depend on the extent to which friction has allowed
the surfaces to expand during compression. The smaller the friction the
greater will be this expansion, and consequently the further they will have to
contract, and the greater will be the pressure under which contraction must
120 ON ROLLING-FRICTION. [18
take place. It is obvious, therefore, that the work spent in friction during
the recoil will increase up to a certain point as the coefficient of friction
diminishes ; and it would appear to be this increase which mainly balances
the advantage which is gained during compression by reducing the co-
It is evident that the action of friction to prevent contraction during
restitution, will tend to reduce not only the mean pressure, but also the whole
pressure, for exactly the same reason as by preventing expansion the friction
increases these pressures during compression. Therefore, for every distance
between the plates, after the curves become inclined inwards, the pressure on
the surface would be less than at the same distance with no friction, and in a
still greater degree than during compression with friction. We can see at
once, therefore, that of the work spent in compressing the material only a
part would be returned during restitution. The difference is what is spent in
overcoming the friction.
The Direction of the Friction.
In Figures 5 and 6 the direction of slipping is opposite on opposite sides
of P. If, however, we conceive one half of the bar, that towards A, to have
been compressed and to be expanding again, while the other half, that
towards B, is being compressed, and the distance between the plates which
hold both parts to be the same, we may imagine the plate AB to have been
first inclined towards A and then towards B so as to raise the end A. Then
the lines would assume the form shown in Fig. 7.
er p er
In this case we see that the slipping takes place in the same direction on
both sides of P, so that the top plate AB would slip backwards in direction
A over the india-rubber, while, on the other hand, the india-rubber would
slip forwards in the direction D over the lower plate.
The turning of the plate AB, which has been supposed to be going on in
Figure 7, represents very closely the action of a roller in compressing the
material beneath it ; and this case affords us an illustration of the way in
which the lateral extension of the material under the roller, or of the roller
itself, will, by causing slipping, alter the distance travelled by the roller. If
the roller be hard and the surface on which it rolls soft, then the top plate
AB may be taken to represent the roller, and, as has just been explained,
this slips back ; whereas if the roller be soft and the surface hard, then we
may take the india-rubber to represent the roller, and this slips forward.
A Continuous Surface.
It is clear that in the case of the bar shown in Fig. 7 the slipping will
diminish as the coefficient of friction increases. There is, however, an im-
portant difference between this case and that of a roller, in which it is not
the entire breadth of a bar that is compressed, but a portion of a continuous
surface ; for whatever lateral extension there may be immediately under the
roller must be compensated by a lateral compression immediately in front
and behind it. The greater the lateral extension under the roller, the greater
will be the lateral compression; and since the action of the roller is con-
tinually to change the one for the other, the one effect will to a certain
extent counteract the other ; so that in this case we need not expect to find
the diminution attended with a corresponding increase in the ostensible
slipping. This will be rendered clearer by examining these circumstances
as they affect rolling.
The Deformation caused by a Roller.
Fig. 8 may be taken to represent a section of an iron cylinder on an india-
rubber plane. The lines on the plane are supposed to represent lines initially
vertical and at equal distances apart. The motion of the roller is towards B.
P is the point of greatest compression directly below the centre of the roller ;
er and fr' limit the surfaces over which there is no slipping ; D is the point
at which contact commences, and G that at which it ceases.
The portions of the india-rubber immediately without G and D are laterally
compressed ; this, as has already been pointed out, is to make room for the
122 ON ROLLING- FRICTION. [18
lateral extension under the roller from G to D. From D towards B, therefore,
and from C towards A the parallel lines are somewhat distorted, and at some-
thing less than their natural distance apart. From D to er vertical compression
and lateral expansion are 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 fr' to G vertical expansion
and lateral contraction take place, so that the lines are all concave outwards.
The lateral expansion from D to er and the lateral contraction from fr' to G
can only take place by 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 //.
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 dis-
tance. 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
G, which latter are less than the natural distance ; therefore the actual
slipping is greater than the ostensible in proportion to the compression at G
and D ; 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
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
18] ON ROLLING-FRICTION. 123
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 is squeezed out is amply
proved by the fact that when a block of iron, hot or cold, is squeezed on an
anvil the iron bulges out in the middle, as shown in Fig. 4.
Experimental Verification of the Figures.
The figures which illustrate the foregoing remarks are not altogether ideal,
for they have been verified to a certain extent by experiments on india-
rubber ; for instance, by drawing vertical lines on the edge of a plate of india-
rubber, and then observing these lines as the roller passed along as near as
possible to this edge ; also by observing lines drawn in the same way on the
edge of an india-rubber roller. The effect of friction to prevent expansion,
shown in Figures 4 and 5, was verified by marking the surface of the india-
rubber under the plate AB with parallel lines in chalk, which left a mark on
the iron and showed how far there had been slipping. The figures are never-
theless intended rather to illustrate the nature of the slipping and various
effects than their extent, which latter must be judged of by the experimental
results which I now proceed to describe.
My first object in making these experiments was to ascertain if, and how,
oiling the surfaces in contact affected the resistance to rolling.
The apparatus employed consisted of a wooden slab or table-top supported
on three set-screws for legs, so that it could be tipped in any required
direction. On this table rested one of Whitworth's surface-plates. On the
surface-plate was placed a surveyor's level, which read divisions to the
thousandth of a foot on a staff erected at 50-feet distance ; also a bell-
glass covered another part of the surface-plate in the manner of the receiver
of an air-pump, which could be filled with oil from an aperture in the top.
This glass served either to protect the roller from dust or surround it with
oil, and thus prevent any surface-tension the oil might exert from affecting
The roller was of cast-iron, 6 inches in diameter and 2 inches thick, and
weighed about 14 Ibs. It was not cylindrical, for the edge was somewhat
rounded. Originally the roller was turned up so that the edge was curved
to a radius of 1 foot ; but subsequent grinding somewhat modified this
In the first instance the roller was turned up and polished in the ordinary
manner; but some preliminary experiments showed that the surface thus formed
124 ON ROLLING-FRICTION. [18
was far from perfect, as indeed was apparent when it was examined with a
magnifying-glass. The roller was therefore again turned, and ground very
carefully with Turkey-stone for several days, until the surface appeared
through the glass to be as perfect as the iron would allow ; there were still
some small pits, but these appeared to be in the iron itself.
The roller when thus finished was rolled on various surfaces. First of all
it was tried on the cast-iron surface-plate already mentioned ; but this surface,
which had been formed by scraping, was altogether too rough. Thus when
the roller was placed on the plate it immediately rolled into a hollow. Surfaces
were then formed by grinding two plates together with powdered Turkey-
stone. In this way the plates were made so true that the roller would remain
in any position, and would roll either way with an inclination of 1 in 5000, or
about 1 foot in a mile. It appeared impossible, however, to produce surfaces
altogether free from inequalities, which may be seen from the results of the
The Effect of Oiling the Surface.
In the first experiments the surface on which the roller was to roll was
brought into a level position, so that the roller when placed on it remained at
rest. A line of sights, consisting of a mark on the glass and a pin-hole in a
plate fixed at some distance, was then brought to bear on a mark on the top
of the roller, so that the least motion could be detected, and the position of
the roller could be recovered after it had been allowed to roll in one direction.
The level was then adjusted to read zero on the staff, and the table tipped
until the roller rolled off in one direction. The reading of the level was then
noted, and the same operation repeated in the opposite direction, the roller
having in the meantime been brought back into its former position. Sundry
observations were then taken with different points of the plane and roller in
contact. After a considerable number of observations had thus been taken
oil was poured into the glass until the roller was covered, and then the
observations were repeated. Table I. shows a series of such observations for
a surface of plate-glass both with and without oil. In these particular
experiments, however, the surface was simply oiled, it having been found by
experience that the effect was the same as when the glass was filled with oil.
It will be seen that in these experiments the advantage is slightly in favour
of the oiled glass.
The results contained in the second part of this Table were obtained by
starting the roller in one direction against the inclination of the plane with
just sufficient velocity to carry it up to a certain point, the inclination of the
plane being adjusted until it would roll back. In this way the advantage is
against the oil. This, however, I think is due to the surface-tension or
fluid-friction arising from the motion of the roller.
Cast-iron Roller on Plate-glass. (The distance of the Staff from the Object-
glass of the Level = 50 feet. The Divisions on the Scale = T n foot.)
There is a very marked difference between these inclinations and those
required to start the roller from rest, a difference which appears to exist with
all the materials tried, and which I think is only in part explained by the
roughness of the surface.
In these experiments with a surface of glass, the friction was so small that
the inequalities of the surface rendered the results very irregular and un-
certain. To obviate this a surface of box-wood cut across the grain was next
tried. This, being softer, allowed the roller to indent it more than the glass
and gave rise to greater friction, and hence the inequalities in the surface are
less apparent in the results, which are shown in Table II. These observations
were made in the same way as those with the glass, except that blacklead was
Cast-iron Roller on Box-wood.