BETWEEN SURFACES MOVING AT LOW SPEEDS. 
515 
velocity of the disk became less. Fig. 5 shows a short portion of the paper strip on 
which the dotted curve formed by the deposited drops of ink has been completed by a 
line drawn through the dots. The portion marked 8 was the first to be traced by the 
siphon ; then 7, which was traced one complete revolution later, and so on down to 1, 
which was the final portion of the curve traced just as the disk was coming to rest. P 
is the point at which the siphon first crossed the central position after the motion of 
the disk had ceased. 
When an unguent was used both bearings were well supplied with it just before the 
disk was set revolving ; and when necessary the supply was kept up during the revolu- 
tion. The total time during which the motion of the disk lasted never exceeded one 
minute. 
The pendulum was next allowed to come to rest, and while the disk was slowly turned 
by hand, the central line was traced out by the siphon. The strip of paper was then 
cut across at one place and removed from the disk. 
The distances from P (fig. 5) to the successive points at which the curve crossed the 
central line were measured by means of a rule graduated to six-hundredths of a foot, 
and the differences between the successive values were found. These differences, which 
may be called As, are the distances described by a point in the circumference of the 
A-Q 
disk during a half beat, At, of the pendulum. ^ is the mean velocity during the time 
At, and this mean is the actual velocity at the middle of the time At, provided that the 
acceleration is uniform during that time. Now even if the force due to friction were 
to change very considerably with changes in the velocity, this force, and therefore the 
acceleration, would be sensibly constant during the short interval At, during which the 
velocity undergoes exceedingly little change. Hence we are completely justified in 
A Q 
taking the successive values of ^ as accurately representing the velocities at times dif- 
fering by At ; and since At is constant, the successive values of As are proportional to 
these velocities. 
A curve was next drawn, as O A B in fig. 6, in which the ordinates were the succes- 
sive values of As, and the abscissae differed by the constant quantity A^. This curve 
expressed the velocity as a function of the time, and would be a straight line when the 
acceleration was uniform. When the acceleration was greater at low than at high 
velocities, the curve would be convex upwards, as shown in the figure, and the accelera- 
tion at any point, such as A, would be proportional to the tangent of the inclination of 
the tangent at that point, or 
This means of finding the value of the acceleration at any point, in which the method 
of tangents was only once used, was obviously much more accurate than if the first curve 
plotted had been simply one connecting the distances moved through by the disk with 
the times, and the method of tangents had been applied to that curve in order to 
enable a second curve to be drawn connecting the velocities with the times. 
