114 
Proceedings of the Royal Society of Edinburgh. [Sess. 
Let R' = number of revolutions that ought to be described 
R' = - 
o-V 2 
o-Y 2 
k + vV 
since is the terminal velocity. 
Hence the fractional overrunning is 
Vt-P 2 h)j 
R - R' cr Y[e-(*+o‘ v ) i i — e~(fc+<rV)t 2 J y(e~**4 _ e -kt s ) 
R' k(k + o-V)(t 2 - ^) Y k(t 2 -t 1 ) 
_ — L) approximately = , 
Y (t 2 - tj) FF J YN ’ 
where F = interval during which water is “ off,” 
N = interval during which water is “ on,” 
Y = velocity of water. 
This would indicate that there is always overrunning, which increases 
directly with the time during which the tap is off, and inversely with the 
time during which it is on, and with the velocity of the water. 
Experiments were performed by means of the apparatus to give 
interrupted flow already described. The times. t x and t 2 were determined 
from the graphs in fig. 3, and the apparatus adjusted to give this period. 
The number of revolutions which the paddle-wheel ought to describe if 
the starting and stopping curves obtained were accurate, was measured by 
means of a planimeter, and the number of revolutions actually described 
during a complete period was noted, and compared with them. From these 
data the percentage overrunning was calculated and tabulated as follows : — 
R. 
(Observed.) 
R. 
(Calculated.) 
R'. 
K 
secs. 
F. 
secs. 
Terminal 
velocity. 
Percentage 
overrunning. 
179 
178 
132 
10 
5 
13-2 
36 
183 
183 
119 
10 
7-2 
1 19 
54 
165 
164 
106 
10 
8-2 
10-6 
56 
167 
167 
85 
10 
17*5 
8-5 
96 
282 
284 
132 
10 
2R2 
13*2 
113 
260 
259 
119 
10 
24*2 
11-9 
119 
235 
234 
106 
10 
27-2 
10-6 
122 
208 
207 
85 
10 
41*5 
8-5 
145 
220 
220 
82 
6*2 
22-5 
13*2 
169 
235 
236 
105 
88 
22-5 
11*9 
124 
297 
298 
173 
16-3 
22-5 
10-6 
72 
223 
225 
63 
4-8 
33 
13*2 
252 
234 
236 
77 
6*5 
33 
11*9 
203 
250 
257 
108 
10-2 
33 
10-6 
131 
If we compare the values of R in the first two columns it is evident 
that except in one case they never differ by more than 1 per cent. The 
