298 Mr. T. Tate on a A'ew Double-acting Air-pump 



the rod connecting them may be of any section consistent with 

 strength. The effective length of the stroke is equal to the 

 space between one side of the pipe R, and the corresponding end 

 face of the cylinder, or it is very nearly equal to one-half the 

 length of the cylinder. 



In an upward stroke, the air above the piston A is propelled 

 through the valve V into the atmosphere, while a vacuum is being 

 formed beneath the piston B. When the piston A strikes against 

 the top of the cylinder, the air from the receiver rushes through 

 the pipe 11 and diffuses itself through the lower half of the 

 cylinder. In a downward stroke, the air beneath the piston B 

 is propelled through the valve v into the atmosphere, while a 

 vacuum is being formed above the piston A ; and so on. It 

 will be observed that the double j)iston performs a double duty at 

 every single stroke; for while a vacuum is being formed on one 

 side of the cylinder by one piston, the other piston is propelling 

 air from the opposite side of the cylinder into the atmosphere. 



With the view of calculating the exhausting power of this air- 

 pump, let a = the volume of the air in the receiver ; b = the 

 volume of the air discharged at each stroke ; b■^ = the volume 

 of the air between the piston and the valve at the end of each 

 stroke ; E = the elasticity of the external air, measured by a 

 column of mercury; eE = the elasticity of the air required to 

 lift the valves; and n = the number of strokes : then we have 



E,(rt + i-f&i) = flE + 6,eE, and .-. E,= ^ '^ ; 



a -|- + 0] 



, n -m cE„_, + 6,eE , „ -, 



and generally E„= = — ^ — : hence we hnd 



° ^ ' a-i-b + bi ' 



E-f ^ Y e I ^'^^ fi 1 '—+ I ( ^ r'l 



''~\a + b + bj ^ a + b + b,V^a + b + b^^""^\a + b+bJ J 



= CTfTI,)Xi-*^,)n^- • • • • • '" 



This expression gives the elasticity of the air in the receiver at 

 the end of the nth stroke. 



When the number of strokes is indefinitely increased, or when 

 n = co, or what is equivalent in result, when a is very small as 

 compared with b, we have 



^. = |if|. (^) 



which expresses the limit of exliaustion. 



In the common pump, the limit of exhaustion for the cylinder 

 is the same as for the receiver of the new pump ; let E' be put 

 for the elasticity of the air in the cylinder at this limit, and E" 

 for the elasticity of the air in the receiver at the limit of ex- 



