310 
PROFESSOR O. RETROLDS AND MR. W. H. MOORBT 
the end and beginning of the interval, divided by the time, is the time-mean moment 
of effort on the shaft. 
The possible limit of this error may be estimated when the maximum moment of 
momentum of the water is known as well as the minimum moment of resistance, and 
the minimum interval of time. 
Thus taking the limits to be 30 lbs. of water, with radius of gyration 0 66 foot 
at 300 revolutions a minute (< 14), the interval of running 3600 seconds, the moment 
of the load 400 ft.-lbs., the limit of the time-mean of change of moment of momentum 
of the water is 14/3 600, and this divided bv the mean moment of resistance erives 
as the limits of relative error, dz 0-00001. This is supposing the wdiole of the water 
to he absent at the beginning or end of the trial, while the actual difference never 
amounts to more than 2 or 3 lbs., so that the limits do not exceed 0-000001, which 
is neglected. 
The Angle-Mean of the Moment of Effort on the Shaft. 
7. As already pointed out in Art. 4, when both the angular velocity of, and the 
.moment of effort on, the shaft are subject to fluctuations of speed, the time-mean of 
the moment of effort may differ from the angle-mean. This applies to all brakes, but 
in hydraulic brakes, in which the resisi.ance is proportional to the square of the 
speed although lagging by an unknown interval, it becomes possible to estimate the 
possible limits of this error when the limits of fluctuation of speed are known. 
Taking m the angular velocity of the shaft and ojq the time-mean of the angular 
velocity, 2crojo the extreme differences of speed, and assuming the variation to be 
harmonic, 
0 ) = o)q [1 -f a~ cos 71 {t — Tj)}.(1), 
coq^ .[ 1 -1- -f 2a^ cos n (t — Tj) cos 2n {t — T)} . . (2). 
Then to a second approximation neglecting a®, if Th is the interval of lagging in 
the resistance and M the moment of resistance at the time t, 
M = Mo {1 -f- 2a3 cos n {t - T^ - T.j) -f cos 2n (^ - T^ - To)} . (3), 
where Mq is the time-mean of the moment of resistance. Also the rate at Avhich 
work is done with uniform velocity, is Mc^q, of which the mean is MqWq, and is the 
rate of wmrk as measured by the mean moment on the case, multiplied by the mean- 
angular velocity. 
