164 OF THE PRESSURE OF FLUIDS OF VARIABLE DENSITY 



Put P zz: the pressure on the bottom DC, by the fluid in its original 



state, 



p zz: the corresponding pressure on the cylindric surface, 

 P'zz: the pressure on the bottom dc, by the fluid after expansion, 

 p' zz: the corresponding pressure on the cylindric surface, 

 s zz: the specific gravity of the fluid before expansion, 

 s' z= the specific gravity of the fluid after expansion, 

 a zz: the area of the base or bottom of the vessel in both cases ; 



<j> and $' the cylindric surfaces, and 

 zz: the part of its bulk by which the fluid is increased. 



Then, since d denotes the diameter of the bottom, the area accord- 

 ing to the principles of mensuration, becomes 



and the pressure exerted by the fluid in its original state, is 



(126). 



Again, according to the principles of mensuration, the cylindric 

 surface in contact with the fluid before expansion, is 



and consequently, the pressure upon it, is 

 /;zz:3.1416d/iXjAXszz:1.5708dA 2 s. (127). 



Now, it is manifest, that since the diameter of the vessel is the 

 same both before and after the expansion of the fluid, the capacity 

 and the altitude must vary directly as each other; consequently, 

 because the capacity or bulk is increased by ^th part of itself, it 

 follows, that the altitude is increased in the same proportion ; there- 

 fore we have 



but when the weight of the fluid remains the same, the density, and 

 consequently the specific gravity, varies inversely as the magnitude. 



The specific gravity of the fluid, after it has expanded by reason 

 of an increase of temperature, is therefore, 



*J> ') .ft ........-' 



n n-j-l ' 



hence, the pressure on the bottom of the vessel, after the fluid has 

 increased by expansion, becomes 



P' = .7864dAV; that is, 



'Vv 



