412 Hydrodynamics. 



4 



* : x p :: 34 : (a; p.) 



But the force which the water, comprehended between ZT and 

 RS, exerts in opposition to the exterior pressure of the air, is 

 measured by the height a x ; consequently, the elastic force 

 of the air in the space QOTZ, together with the weight of the 

 water between ZT and RS, will be expressed by 



Now in order that the water may always rise, this joint pressure 

 must be less than the weight of a column of water 34 feet high 

 by some variable quantity, which we will call y ; so that the fol- 

 lowing equation must always obtain, namely, 



34 (a; ) 



- - &- + a x = 34 y. 

 x 



The value of x deduced from this equation is ambiguous, being 

 thus expressed ; 



x = \a + \y V((i + ly) 2 34 p.) 



Now, when the water ceases to rise, y vanishes, and the 

 equation becomes x = | a \/ (-} a 2 34 p) ; of which the 

 two values are real, so long as a 2 is greater than 34 p. Hence 

 we conclude, that when one fourth of the square of the greatest 

 height oj the piston above the surface of the water in the reservoir is 

 gr .ater than 34 times the play of the piston, there are always two 

 points in the sucking pump where the water may stop in its motion ; 

 and the pump must be reputed bad when the lowest point to 

 which the piston can be brought is found between these two 

 points. 



But if 34 jo be greater than et 2 , the two values of #, when 

 y is supposed = 0, become imaginary; so that in a pump so 

 constructed it is impossible that y should vanish ; that is, the 

 pressure of the exterior air always prevails, and the water is 

 not arrested in its passage. Hence we conclude, secondly, that 

 in order that the sucking pu p may infallibly produce its effect^ the 

 square of half the greatest elevation of the piston above the wattr in 

 the reservoir must always b& less than 34 times the play of the piston. 



