EFFORT, ACCELERATION, AND VELOCITY DIAGRAMS 285 
An example of some interest is the relation between the space and 
time averages of the pressure on the piston of a steam-engine. The 
ordinary indicator diagram is a force-space diagram, and the mean 
found from it is a space average, and this space average is used 
Tn calculating the work done in the cylinder. In an indicator invented 
by Professor Ripper," the mean pressure is shown directly by a pointer on 
a dial; but this mean pressure is a 
time average, unless the proper cor- 
rection has been made by adjusting 
the instrument to convert the time 
average into the space average. The 
necessary correction to convert the 
time average into a space average will 
depend on the way in which the pres- 
sure varies in the cylinder. 
The following example will show 
the relation between the space and 
time averages of the pressure in a 
particular case. ABCD (Fig. 447) is 
the force-space diagram on a base AB, 
which represents the stroke of the 
piston (20 inches), The steam pressure ! 
is 150 Ibs. per square inch for the A 0 B 
first 4 inches of the stroke, after which Fic, 447. 
the pressure follows the law Pa= 600. 
Dividing the stroke into 10 equal parts of 2 inches each, and calculating 
the pressures at the middle points of these parts, the following table 
is constructed :— 
pie - - -150 --- = 
N------ 
a(space)| 1 3 5 7 9 11 13 15 17 19 
PP. .| 150 | 150 | 120 | 85°71 | 66°67 | 54°55 | 46°15 | 40 | 35°29 | 31°58 
from which the mean value of P is 78-0, and this is the space average 
of P. 
To find the time average of P, construct the semicircle A’9’B’, which 
represents the path of the crank pin for one stroke of the piston, and 
divide this into 10 equal arcs ; then assuming that the crank pin is moving 
with a uniform velocity, each of these arcs will be described by the crank 
pin in equal intervals of time. Bisect these equal arcs at the points 
1’, 3’, 5’, etc. Then, neglecting the effect of the obliquity of the con- 
necting-rod, the position of the piston, measured by its distance from the 
beginning of its stroke when the crank makes an angle @ with A’O, is 
x=10(1 —cos 6), and Piao Ee but is not greater than 150. 
Let the base AB now represent the time taken by the piston to make 
one stroke, and let it be divided into 10 equal parts, representing equal 
intervals of time, and let these be bisected at the points 1, 3, 5, ete. 
The points 1, 3, 5, ete., will be the positions of the pressure ordinates on 
* See the Proceedings of the Institution of Mechanical Engineers, 1899. 
