272 HEIGHT WHICH WATER RISES IN ONE STROKE. 



in MB is greater than the atmospheric pressure. When 

 M is at A the pressure of the air = gp (h PC), and un- 

 less the valve V is opened before M arrives at H, the pressure 

 of the air in HB will 



= gp (k - PC) , 



which must be greater than gph, if the valve is to open, and 

 therefore A -AH must be greater than AB-PC. Hence, to 

 insure the opening of the valve while the surface is below 



B, we must have 



h-AU > AB-BC, (1) 



AH BC 

 -' 



i. e., the ratio of AH to HB must be at least as great as the 

 ratio of BC to h. This condition, although necessary in 

 every case, may not be sufficient. 



For, suppose that the surface of the water is at Q' when 

 the piston M is at A, in which case the pressure of the air 

 in AQ' = gp (h - Q'C). 



When the piston descends to H, the pressure in HQ' 



which must be greater than gph, if the valve is to open, and 

 therefore 



7*. AH > AQ'-Q'C. 



But the greatest value of AQ'-Q'C is AQ ; therefore 

 we must have 



A- AH > JAC 8 . (2) 



Since |AC 2 > AB-BC, unless B is the middle point of 

 AC, it follows that the condition in (2) includes the condi- 

 tion in (1), which is therefore in general insufficient. (See 

 Besant's Hydrostatics, p. 97.) 



