108 
Proceedings of the Royal Society of Edinburgh. [Sess. 
terminal velocity, when the water previously shut off, with the wheel 
at rest, is suddenly turned on to any given rate of discharge. 
Accordingly, a series of experiments was undertaken in which the tap 
was turned full on, and the turbine wheel allowed to attain its maximum 
velocity, which takes place in about two minutes ; then suddenly the water 
was turned off, and by means of a chronograph the intervals of time 
between successive ten revolutions were recorded until the wheel gradually 
came to rest on account of its own friction. A curve was then constructed 
by plotting a number of revolutions against time. When the experiment 
was repeated, it was found that the new points lay exactly on the same 
curve ; and that the latter was so smooth that it was quite possible to 
obtain the first and second gradient curves with accuracy, viz. speed against 
time, and acceleration (i.e. the frictional retardation) against time. The 
curve giving speed against time is shown in fig. 3, and may for convenience 
be termed the “ stopping ” curve. 
To deduce the law connecting the retardation between the friction and 
the velocity it was only necessary to plot “ acceleration ” against “ speed ” 
as obtained from the two gradient curves. The result is represented in 
fig. 1, where for speeds less than about six revolutions per second 
“ acceleration ” is approximately proportional to “ speed,” while for greater 
values it is also linear, but of the form a + bs, a and b being constants. 
A suitable formula representing this would take the shape 
s = a{\ - e~ ks ) + bs, 
where for small values of s, 
s — {p + ah)s, 
while for larger values 
S = CL + OS . 
To obtain information concerning the increase of speed of the paddle- 
wheel from rest under a steady flow of water a set of experiments was 
carried out similar to that conducted to determine the stopping curve. The 
wheel being at rest, the tap was suddenly turned on to some definite angle, and 
the intervals of time that elapsed between successive ten revolutions as the 
speed of the wheel increased up to its terminal velocity were recorded by 
the chronograph. Once more a smooth and regular graph was obtained 
for “ revolutions ” against “time,” so that the speed-time curve was easily 
derived. A separate curve had to be obtained, of course, for each angle of 
the tap, and therefore in fig. 3 a sheaf of curves appears, spreading out 
from the origin and finally running asymptotically to lines parallel to the 
time axis. 
