40$ Hydrodynamics. 



r r 2 



o + f ' f , , : ,, , = density after 2 strokes, 



6 + r : r : : : - = density after three strokes ; 



and the nth power of the ratio - r , 



b + r 



or - = D, the density after n strokes. 



(6 + r) 



From which we may easily find the density after any number 

 of strokes of the piston necessary to rarefy the air a number of 

 times, or to give it a certain density D, the primitive density 

 being 1 . For the above equation, expressed logarithmically, is 



n 

 or 



n X (log. r log. (6 + r)) = log. p; 

 consequently 



log. D 



n = 



log. r log. (6 + r) ' 



in which expression D will be a fraction. If the number of times 

 which the air is rarefied be expressed by N, an integer, the 

 logarithmic equation will be 



log. N 



n = 



log. (6 + r) log. r 



A further reduction of the same theorem will furnish us 

 with the proportion between the capacities of the receiver and 

 barrel, when the air is rarefied to the density D' by a definite 

 number of strokes n of the piston. For since 



(6 + r) 



if we take the nth root of both members of the equation we shall 

 have 



