OF THE HYDROSTATIC BELLOWS. 143 



board, and by pouring more fluid into the tube, the moveable plane 

 A B and its incumbent load w, will be raised and kept in equilibrio 

 by the column of fluid in the tube ; and when the equilibrium obtains, 

 we infer, that : 



The iveight of the supporting column of fluid in the tube, is 

 to the weight upon the moveable plane, as the area of a section 

 of the tube, is to the area of the plane. 



This is manifest, for the fluid at i, the lowest point of the vertical 

 tube FEI, is pressed by a force varying as the altitude LI, and by the 

 nature of fluidity, this pressure is communicated horizontally to all 

 the particles in DC, and thence transmitted throughout the whole mass 

 of fluid in the bellows ; consequently, the pressure upwards on the 

 board AB, is equal to the weight of a column of the fluid, the diameter 

 of whose base is DC, and altitude LI or GD ; but the actual weight of 

 the fluid supported, is that of a column whose diameter is DC, and 

 altitude EI or AD. 



Hence, the weight which maintains the equilibrium, will be that of 

 a cylinder of fluid, whose base is A B and altitude A G ; consequently, 

 the weight w, placed upon the moveable plane of the bellows, since it 

 balances the column of fluid L E, is equivalent to the weight of a fluid 

 cylinder, whose section along the axis is ABHG. 



Put D rz AB or DC, the diameter of the cylindrical vessel or bellows, 

 d zz LM, the diameter of the vertical tube, 

 w ~ the weight upon the moveable plane, and 

 w'-=. the weight or pressure of the fluid in the column LE. 



Then, because by the principles of mensuration, the areas of circles 

 are to one another as the squares of their diameters ; the foregoing 

 inference gives 



w' : w : : d* : D 2 , 



and this, by equating the products of the extreme and mean terms, 

 becomes 



tfw'=:d*w. (113). 



Let both sides of this equation be divided by the quantity D 2 , which 

 is found in combination with the weight or pressure of the fluid in the 

 tube, and we shall obtain 



, d*w 

 W =^' (114). 



Here again, that singular property of non-elastic and incompressible 

 fluids becomes manifest, viz. 



