
OF SINGLE-ACTING EXPANSIVE STEAM-ENGINES. 201 
Economy of Single-Acting Expansive Engines. 
(37.) By increasing the ratio of expansion in a Cornish engine, the quantity of 
steam required to perform a given duty is diminished; and the cost of fuel, and 
of the boilers, is lowered. But at the same time, as the cylinders and every part 
of the engine must be made larger, to admit of a greater expansion, the cost of 
the engine is increased. It thus becomes a problem of maxima and minima to 
determine what ratio of expansion ought to be adopted under given circum- 
stances, in order that the sum of the annual cost of fuel, and the interest of the 
capital employed in construction, may be the least possible, as compared with the 
work done. : 
That this problem may admit of a definite solution, the following five quanti- 
ties must be given :— 
P., the initial pressure in the cylinder. 
F, the resistance not depending on the useful load. 
tn, the amount of the length of the effective strokes made in unity of time. 
h, the annual cost of producing unity of weight of steam in unity of time, 
which consists of two parts; the price of fuel, and the interest of the cost of the 
boilers. 
k, the interest of the cost of the engine, per unit of area of piston. 
Hence the annual expenditure to be taken into consideration, reduced to unity 
of weight of steam, is 
h+k = =h+k _ 
And the useful effect of unity of weight of steam being 
V, (2, Z-Fs) 
The problem is to determine the ratio of expansion s, so that 
V, (P, Z—F s) 
hepa? 
ln 

shall be a maximum. 
Dividing the numerator of this fraction by V,P,, and the denominator by 
kV. 
3? both of which are constants in this problem, we find that it will be solved 
by making the ratio 
Z— 2 3] 
ies tity oat, 182) 
pv 
a maximum. 
The algebraical solution would be extremely complicated and tedious. The 
