15 



when depressed, s' when elevated, then when the limit is attained, 

 the air below the piston must be unable to open the piston valve, 

 and therefore its density is by the last equation 



s' 



When the piston is raised to the top of the barrel this becomes 



^Xl' + 'I^X {A+V); 



and as this elasticity, together with that of the receiver valve pre- 

 vent the air in the receiver from expanding, we have 



■ * = "' + ■!"'" +"11'' X (^H-")- 

 Smeaton's pump is therefore little better than Hooke's, as x can- 

 not be less than 0. 2, though as the remaining part of the expres- 

 sion is small it cannot much exceed it. If however the receiver 

 valve were opened mechanically as in the pump of Haas, it becomes 



far superior, for v' disappears in the expression, and | is always an 

 inconsiderable fraction. 



In Prince's machine, the piston in its ascent passes an aperture 

 communicating with the receiver, and the air which thus enters is 

 expelled in its descent through a discharging valve ; if this be 

 leather, it is evident that 



* = |x(^ + p). 



Mr. Kiernan has described an improvement of this in the 13th 

 volume of our Transactions, which consists in making the discharg- 

 ing valve metallic, and opening it by a simple but effective me- 

 chanism ; thus V disappears from the second factor, which however 

 is of little consequence, but s may be made scarcely perceptible ; 



