110 Proceedings of the Royal Society of Edinburgh. [Sess. 
We have seen that approximately we may represent the retarding 
frictional force as proportional to the speed. On this assumption we may 
with comparative ease deduce the equations to the curves of fig. 3. 
For the stopping curve 
s— - ks 
. \ log s— - Jet + A . 
If y = maximum speed, and go = speed at time t, 
.-. to = ye~ kt (1) 
which evidently represents a curve of the form of the stopping curve in 
fig. 3. 
If Y be the velocity of the water entering through the orifice of area 
a, then the volume that enters per second is «Y, so that the momentum of 
the water before impact is paV 2 , say o-V 2 . Let s be the resolved velocity 
of the turbine blade in the direction of the impinging jet of water, then the 
total momentum transferred per second is 
<rV 2 - crVs = crV (V - s). 
It is otherwise evident that this term must be merely of the first degree in s 
from the fact that when the steady state has been attained, the driving force 
equals the frictional retardation, which has already been shown to be pro- 
portional to a + bs for large values of s. The equation of motion is therefore 
We may write it 
When 
s — o*Y ( Y — s) - Jcs 
S = crY 2 - s(Jc + crV) 
gg(fc+o-V)e _ crV 2 J d 
dt e( fc +o-V)* = YX — £(&+<rV)£ 
lc + o -V 
^ = 0 s = 0 
A = 
oY 2 
Je + aY 
( 2 ) 
which for different values of Y gives a system of curves of the type shown 
in fig. 3. 
The laws governing the rise and fall of speed of the turbine in cases of 
steady flow having thus been ascertained, it ought to be possible to predict 
the behaviour of the apparatus under any given condition of flow. In 
particular, if the flow fluctuates periodically with the time, the definite 
question presents itself, Do the ensuing imperfectly sympathetic fluctuations 
