Watir-Pumps. 411 



piston. To understand this, suppose that the water has been 

 actually raised to 7 1 , and that the situation of the piston in the Fig.245. 

 figure is the lowest which can be given to it ; and, for greater 

 simplicity, suppose that the pump is of the same internal diame- 

 ter throughout. It is obvious that the air comprised in the space 

 CDTZ is of the same density and elasticity as the exterior air (at 

 least leaving out of consideration the weight of the valve L, and 

 the friction attending its motion ;) for if its spring were less, the 

 water would rise higher than ZT, and if it were greater, it would 

 raise the valve L, and mix with the exterior air till both became 

 of the same density. Suppose now that the play of the piston, 

 or the distance through which it is raised at each stroke, is /JO; 

 then when the base CD is raised to OQ, the air which previously 

 occupied the space CDTZ will tend to expand and fill the space 

 QOTZ ; and, if the water did not rise, would actually be so ex- 

 panded. Its elastic force would then be less than that of the 

 natural air, in the ratio of CDTZ to QOTZ, or of DT to OT. 

 If, therefore, this elastic force, together with the weight of the col- 46g 

 umn of water whose height is ZR, constitute a pressure equal to 

 that of the atmosphere, or equal to the weight of a column of 

 water of the same base, and at a mean 34 feet in height, there 

 will be an equilibrium, and the water will not rise further ; if 

 this joint pressure is greater than that of 34 feet of water, the 

 water cannot be retained so high ; but if it is less than a col- 

 umn of 34 feet, the water will continue to rise in the pump. 



515. From these considerations we may readily investigate 

 a general theorem. 



Let a be the altitude or vertical distance from the point O to 

 the surface RS of the water in the reservoir,^=OZ), the play of 

 the piston, and a: the distance OT\ then we have DT = x p, 

 and S7 1 , the height of the point T, will be a x. Since the air 

 contained in CDTZ has the same density and elasticity as the 

 exterior air, its force may be measured by a column of water of 

 the same base Z T and 34 feet high ; and because when this air 

 is so expanded as to fill the space QOTZ, the elastic force will be 

 less in the ratio of DT to O 7 1 , we shall have (rejecting the base of 

 the column, as equally affecting every part of the process) this 

 latter force expressed by the fourth term of this proportion, 



