the new Double-acting Air-pump with a Single Cylinder. 363 



and when n is taken very great, we find the work expended in 

 exhausting the receiver 



It will be observed, that the work in passing the exhausting 

 orifice has been neglected in this investigation. 



3, To find the work expended in exhausting the receiver of 

 the pump No. 3, and also No. 2 very nearly. 



In this case we have for the work of the nth. single stroke, 



\]„ = U + U^ + Uci + Uq- 



Here m, Mj, and Wg have the same values as in the preceding case, 

 and 



12^ ^ \a + b/ ^' a 12 \a + b/ °^ a ' 



,(„_,),(^)"-,„,.^,,]. 



And when n is very great, we find the work expended in ex- 

 hausting the receiver 



a + b(^ ^ 1 wv^\ \ jn /.x 



Now ^ — inr^ — exactly expresses the least possible work requi- 



site for completely exhausting the space « + i of air, and the 

 other parts of the expression give the work of resistances essen- 

 tially connected with all air-puuip pistons ; hence it follows, that 

 with this construction of pump, the exhaustion is performed 

 with the least possible expenditure of woi'k. 



Let a = 108, i = 12, E = 15, ?/;=-086,?; = 100,/=2|-^f = li 

 for the single piston, and 3:|-f-f = 2^ for the double piston, and 

 71 = 60; then by formula? (2), (3), and (4), we find the work in 

 each case as follows, viz. 950, 440, and 300. 



Houuslow, April 18, 1856. 



2B2 



