14 



not be raised, and the exhaustion is at its limit, which Hmit is there- 

 fore given by an equation between these three forces. 



Let S be the space below the piston when raised to its greatest 

 height, s that remaining when it is depressed as much as possible, 

 A the altitude of the barometric column, v a column of mercury 

 the pressure of which would open the valve, and x that which 

 measures the density of the air in the receiver : then as in this con- 

 struction the piston valve (which is similar to the receiver valve) is 

 opened against the pressure of the atmosphere, the space s at the 

 termination of a stroke remains full of air whose density is A-{-v. 

 Let the piston be drawn up, and this air will expand till its density 

 , becomes, 



therefore 



The quantity v is found by experiment to be about 0. 2. and 

 therefore if the workmanship were so perfect as to leave no space 

 below the piston when depressed, this must be the limit of exhaus- 

 tion, corresponding to a rarefaction of 150 times. In fact, x never 

 is less than 0. 3, or the best of these machines do not exhaust 100 

 times. Smeaton made the first great improvement on this construc- 

 tion, by closing the top of the barrel with a collar of leather, 

 through which the piston rod works air-light, so that in its descent, 

 a partial vacuum is left above the piston, and on its return, the 

 ttir which had entered this through the piston valve, is discharged 

 into the atmosphere through a valve on the top of the pump. He 

 also altered the construction of the receiver valve, but I believe the 

 change was no improvement. Let 5' be the space above the piston 



