202 MR W. J. M. RANKINE ON THE POWER AND ECONOMY 
graphic solution, on the other hand, is very simple and rapid, and sufficiently 
accurate for all practical purposes ; and I have therefore adopted it. 
In the annexed diagram, Plate VIIL., fig. 1, the axis of abscissee, —XO+X, is 
graduated from O towards +X into divisions representing ratios of expansion, or 
values of s. The divisions of the axis of ordinates, O Y, represent values of Z. 
The curve marked “locus of Z,” is laid down from the third column of Table I. 
of the Appendix to the original paper, being applicable to initial pressures not 
exceeding four atmospheres. 
Through the origin O draw a straight line BOA, at such an inclination to 
—X0O+X that its ordinates are represented by = s. Then the ordinates measured 
1 
from this inclined line to the locus of Z represent the value of the numerator 
Lp of the ratio (62), corresponding to the various values of s. 
Take a point at C on the line BOA, whose abscissa, measured along O—X, re- 
presents — = Then the ordinates, measured from BOA, of any straight line 
drawn through C, vary proportionally to the denominator iv tsof the ratio (62). 
Through the point C, therefore, draw a straight line CT, touching the locus of 
Z: Then the ratio (62) is a maximum at the point of contact T, and the abscissa 
at that point represents the ratio of expansion required. 
Example. 
(38.) To exemplify this method, let us take the following data. 
Greatest pressure in the cylinder P,=20 Ib. per square inch, =2880 Ib. per 
square foot. 
The corresponding value of V, is 20°248 cubic feet per pound of steam. 
To obtain this initial pressure in the cylinder, it will be necessary to have 
a pressure of about 50 Ibs. per square inch in the boiler. 
F, resistance not depending on the useful load =2 Ib. per square inch, 
=288 Ib. per square foot, =) P,. 
Jn, amount of down strokes, =4800 feet per hour; being the average speed 
found to answer best in practice. 
To estimate /, the annual cost of producing one pound of steam per hour, I 
shall suppose that the engine works 6000 hours per annum; that the cost of 
fuel is one penny per 100 1b. of steam;* that the cost of boiler for each pound 
of steam per hour is 0-016 ton, at £27, =£0°432; and that the interest of capital 
is five per cent. per annum. Hence / is thus made up— 
* This estimate is made on the supposition that coals capable of producing nine times their 
weight of steam are worth about 16s. 9d. per ton. 
