By way of example, let the diameter of the cylinder be 29.59 inches, and let the back pressure from all causes be 7 pounds uniformly throughout.  It will be represented by the line b_{1}, b_{2}, etc.  This quantity subtracted from the pressures p_{1}, p_{2}, etc., leaves the remainder (p-b) upon each ordinate, which remainder represents the net pressures which at that point may be applied to produce external power.

If, now, A is the area of the piston, then the external power (d W) produced between each ordinate is: 

To any convenient scale, upon each ordinate, set off the appropriate power as calculated by this equation (1).

A(p-b)dx
dW = --------------. (1.)
33,000

There will result the curve w, w, w, determining the power which at any point in the diagram is to be regarded as a gain, to be carried to the credit side of the account.

It is evident that, so long as the gains from expansion exceed the losses from expansion, it is profitable to proceed with expansion, but that expansion should cease at that point at which gains and losses just balance each other.

TO CALCULATE THE LOSSES.

The requisite data are furnished by the experiments conducted some years since by President D.M.  Greene, of Troy College, for the Bureau of Steam Engineering, U.S.  Navy.

According to these experiments, the heat which is lost per hour by radiation through a metallic plate of ordinary thickness, exposed to dry air upon one side and to the source of heat upon the other, for one degree difference in temperature, is as follows: 

Condition.  Heat units.

Naked...................................... 2.9330672
Covered with hair felt, 0.25 inch thick.... 1.0540710
"           "      0.50     "     .... 0.5728647
"           "      0.75     "     .... 0.4124625
"           "      1.00     "     .... 0.3070554
"           "      1.25     "     .... 0.2746387
"           "      1.50     "     .... 0.2507097

If now t’ = temperature of steam at the ordinate,
       t = temperature of the surrounding atmosphere,
       dS = surface of the cylinder included between each ordinate,
       k = that figure from the table satisfying the conditions,
then the power loss (dR) per minute will be: 

k    (t’-t)dS
dR = ( -- ) ----------.  (2)
60     33,000

To the same scale as the power gains, upon each ordinate, set off the appropriate power loss, as calculated by this equation (2).

There will result the curve r, r, r, which determines the power which at any point in the diagram is to be regarded as a loss, to be carried to the debit side of the account.  This curve of losses intersects the curve of gains at a point (it is evident) where each equals the other.

Therefore this is the point at which expansion should cease, and this absolute pressure is the economic terminal pressure, which determines the number of expansions profitable under the given conditions.