Water-Pumps. 413 



516. This general rule may also be easily deduced geomet- 

 rically ; suppose the valve E to be placed at the surface RS of Fi 245 

 the water, the tube to be of a uniform bore, and YS to be the 

 height of a column of water whose pressure is equal to that of 

 the atmosphere ; that is, YS = 34 feet. Let the water be 

 raised by working to JV ; then the weight of the column of water 

 JV, together with the elasticity of the air above it, exactly balan- 

 ces the pressure of the atmosphere YS. But the elasticity of the 

 air in the space OM, (QO being the highest and CD the lowest 



DN 



situation of the piston,) is proportional to YS . ^r^,; and, conse- 



quently, in the case where the limit obtains, and the water rises no 



n/v/* 

 further, we shall have YS = NS+ YS. ~ Transposing JVS, 



we have 



YS JVS or FJV = YS . 



whence 



OJV : DJV : : YS : FJV; 



or, 



OJV DNorDO : OJV : : YS FJV or JVS : YS-, 

 consequently 



DO . YS = OJV . JVS. 



Hence we see, that if OS, the distance of the piston in its highest 

 position from the water, and DO the length of the half-stroke, or 

 the play of the piston, be given, there is a certain determinate 

 height, as SJV, to which the water can be raised by the difference 

 of the pressures of the exterior and interior air ; for YS is to be 

 considered as a constant quantity, and, of course, when DO is 

 given, OJV . JVS is given likewise. To ensure, therefore, the de- 

 livery of water by the pump, the stroke must be such that the 

 rectangle DO . YS shall be greater than any rectangle that can 

 be made of the parts of OS ; that is, greater than the square of 

 I OS, by a well known theorem. 



Hence we deduce a practical maxim of the same import as 

 the preceding, which is, that no sucking pump can raise water effect- 

 ually, unless the play of the piston in feet be greater than the square 

 of the greatest height of the piston, divided by 136. 



