PROCEEDINGS OF THE POLYTECHNIC ASSOCIATION. 477 



" The subject of expansion, however, is so important, and at the same time 

 so mysterious to tyrof^ in steam science, that it would be unpardonable to 

 pass it over without offering such further familiar elucidations as may- 

 make the principle intelligible to unskillful persons ; and we believe these 

 explanations may be here introduced with greater propriety than at a more 

 advanced stage of our progress. 



"It is a well known law of pneumatics that the pressure of elastic fluids 

 varies inversely as the spaces into which they, are compressed; for exam- 

 ple, if a cubic foot of air be compressed into the space of half a cubic foot, 

 its elasticity will be increased from fifteen pounds the square inch to thirty 

 pounds the square inch, whereas if the volume be enlarged to two cubic 

 feet its elasticit}'^ will be diminished to seven and a half pounds per square 

 inch, just half the original pressure. The same law holds in all other pro- 

 portions, and with all other gases and vapors, provided their temperature 

 remain unchanged ; and if a steam valve be closed when the piston has 

 descended through one-fourth of the stroke, the steam within the cylinder, 

 at the end of the stroke, will just exert one-fourth the initial pressure. Let 

 the cylinder be supposed to be divided in its length into twenty equal 

 parts, and its diameter into ten equal parts. If now the piston be sup- 

 posed to descend through five of these divisions and the steam valve be 

 shut, the pressure represented at each subsequent position of the piston, if 

 computed by the well known laws of pneumatics, and which, if the initial 

 pressure be represented by 100, will give a pressure of 50 at the middle 

 of the siroke and 25 at the end of it. If this series be set off on the hori- 

 zontal lines they will give the Injperbolic curve; the area of the part exte- 

 rior to which represents the total efficacy of the stroke, and the interior 

 area represents the diminishing effect when cut off at one-fourth the stroke. 

 If the squares above the point where the steam is cut off be counted, they 

 will be found to amount to 50, and if those beneath be estimated they will 

 be found to amount to 68, and all these squares represent the power 

 exerted, so that while an amount of power represented by 50 has been 

 obtained by the expenditure of one-fourth of a cylinder full of steam, we 

 get an amount of power represented by 68 without any expenditure of 

 steam at all, merely by permitting the steam first used to expand into four 

 times its original volume. The efficacy of a given quantity of steam is 

 therefore more than doubled by expanding four times while the efficacy of 

 each stroke is made one-half less, and therefore, to carry out the expansive 

 system, the cylinders require to be larger in proportion to the extent to 

 which expansion is carried. Every one acquainted with simple arithmetic 

 can compute the terminal pressure of the steam when he knows the initial 

 pressure and the point of cutting off, and by the same process any pressure 

 intermediate. By setting down these pressures and taking their mean he 

 can easily determine the effect with tolerable accuracy of any particular 

 measure of expansion of which the mean pressure thus determined will be 

 the representative. And as a summary of the ascertained effects of expan- 

 sion will induce a more careful examination of the principle at a future 

 stage of our progress, we may here set down some of the most notorious. 

 Let the steam be stopped at one-half, its performance is multiplied 1.1 ; at 

 one third, 2.1; one-fourth, 2.4; one-fifth, 2.6; one-sixth, 2.8; one-seventh, 3.0, 



