_Answer._ This can be determined approximately by the following method: In the first place, divide the vacuum line (fig. 39) into any number of equal divisions, say six. From the points of division, 4, 8, 12, 16 and 20, which in this case correspond with the points which represent inches of the stroke, draw perpendicular lines, which will divide the indicator diagram into six divisions. It is obvious that during the time the steam is working full stroke the pressure is uniformly 100 lbs. absolute. While the piston is moving from 8 to 12 inches the pressure falls from 100 to 66²⁄₃ lbs., so that at 10 inches we have very nearly the average pressure during the period named. So from 12 to 16, 16 to 20 and 20 to 24 the average is nearly 57.1, 44.4 and 36.3 lbs., respectively. Now, BY ADDING TOGETHER THE PRESSURES IN THE MIDDLE OF EACH ONE OF A NUMBER OF EQUAL DIVISIONS OF THE STROKE AND DIVIDING BY THE NUMBER OF DIVISIONS, WE WILL OBTAIN APPROXIMATELY THE AVERAGE ABSOLUTE PRESSURE DURING THE WHOLE STROKE. TO GET THE AVERAGE EFFECTIVE PRESSURE, DEDUCT THE ATMOSPHERIC PRESSURE FROM THE RESULT. The calculation would in the above case be as follows: 100 lbs. 100 „ 80 „ 57.1 44.4 36.3 ----- 6)417.8 ----- 69.6 = Average absolute pressure. 15 ----- 54.6 = Average effective pressure. A more accurate way of calculating the average or mean pressure, as it is called, when steam is used expansively, and the one which is usually employed, is to DIVIDE THE LENGTH OF THE PISTON’S STROKE IN INCHES BY THE NUMBER OF INCHES AT WHICH THE STEAM IS CUT OFF: THE QUOTIENT IS THE RATIO OF EXPANSION. GET THE HYPERBOLIC LOGARITHM OF THE RATIO OF EXPANSION FROM THE TABLE OF LOGARITHMS IN THE APPENDIX, ADD 1 TO IT, AND DIVIDE THE SUM BY THE RATIO OF EXPANSION AND MULTIPLY THE QUOTIENT BY THE MEAN ABSOLUTE STEAM PRESSURE IN THE CYLINDER DURING ITS ADMISSION. THE RESULT WILL BE THE MEAN ABSOLUTE PRESSURE DURING THE STROKE. TO GET THE MEAN EFFECTIVE PRESSURE, DEDUCT THE ATMOSPHERIC PRESSURE. The calculation for the above example would be as follows: 24 -- = 3 = Ratio of expansion. 8 1.0986 + 1 ---------- × 100 = 69.95 = Mean absolute pressure. 3 69.95 - 15 = 54.95 = Mean effective pressure. The table of hyperbolic logarithms given in the appendix will be needed in calculating the mean pressure of steam used expansively: QUESTION 58. _What advantages result from using steam expansively?_ _Answer._ There is a very important saving in the amount of steam required to do a given amount of work, and the strains and shocks which are produced by the rapid motion of the piston and other reciprocating and revolving parts of the engine are very much diminished by allowing the steam to expand, and thus become reduced in pressure during the latter part of the stroke. QUESTION 59. _How is steam saved by using it expansively?_ _Answer._ Less steam is required when it is used expansively: