106 Proceedings of the Koyal Society of Edinburgh. [Sess. 
continually immersed in the oil, and since this was done the calibration has 
remained constant. The following table gives the results : — 
j Steady speed in 
revs./sec.=Y 1 . 
' 
Velocity of the 
water per sec. = Y. 
(In arbitrary units.) 
Steady speed in 
revs. /sec. =V 1 . 
Yelocity of the 
water per sec. = V. 
(In arbitrary units.) 
13-9 
•0276 
11-4 
•0228 
138 
•0275 
10-6 
•0213 
13-7 
•0274 
95 
0195 
13-5 
•0270 
8*5 
•0174 
13*4 
•0268 
7-4 
•0151 
13 2 
•0262 
5-9 
•0125 
12-9 
•0256 
4-6 
•0100 
12-4 
•0248 
36 
•0074 
11-9 
•0238 
2-4 
•0049 
On plotting these results (fig. 2) it appears that the steady speed is 
proportional to the velocity of discharge. If Y denote the velocity of the 
water, and V 1 that of the paddle-wheel, then Y 1 = <xV. It may be noted 
that with this form of turbine the efficiency, defined as the fraction of the 
energy of the water abstracted by the wheel — and then lost in friction — 
varies inversely as the velocity of the nozzle discharge. If a be the cross 
section of the nozzle, then the mass of water discharged per second is paV, 
and its kinetic energy JpaV 3 . The retarding frictional forces at the bear- 
ings, etc., being roughly kV lf as will be shown later, the work done against 
friction per second is /rY 1 2 , and the efficiency is given by 
KV* _ 2 a*k 
^paV 3 pa Y 
As a first step in the discussion of the problem of the effects of lag it is 
evidently necessary to determine how the speed of the turbine varies as the 
speed of the water is made to fluctuate in any given way. A continuous 
change in the velocity of the water can be considered as made up of a 
great number of small sudden changes, between each of which the flow 
is momentarily steady. This reduces the problem to that of ascertaining 
how the speed of the wheel running at its terminal velocity for any given 
rate of discharge varies to the terminal velocity for any other given rate 
of discharge, when the water flow is abruptly changed from the one rate to 
the other. It was sufficient for our immediate purpose, however, as will 
be apparent later, to find (1) how the speed falls to zero as a function 
of the time, when the water, previously full on, is abruptly shut com- 
pletely off; and (2) to find how the speed increases from zero to the 
