178 OF THE PRESSURE OF UNMIXABLE FLUIDS OF DIFFERENT DENSITIES 



Then, the pressures and the specific gravities being denoted as 



before, by the letters p,p',p", and s, s', s", we shall obtain as follows. 



According to the principles of mensuration, the circumference of 



the vessel is expressed by 3. 1416 D; consequently the several surfaces, 



estimated in order upwards, are 



3.1416Drf; 3.1416Dd', and 3.1416D<f ; 



and the corresponding pressures, are 

 j?=1.5708Dd 2 *; p' = 1.5708 D<T s 1 , and p" =1 1.5708 D d" 2 s" ; 



and the total pressure is 



P' z= (p 4-p' 4-p") = 1 .5708 D (d*s 4- d'*s' 4- d n s"). (137). 

 But the area of the base is expressed by .7854 D 8 ; consequently, 

 the equation numbered (136) becomes 



P = .7854 D 9 (d s 4- d's' 4- d"s"), 



and the pressures on the base, and on the upright surface of the vessel, 

 are to one another as D (ds 4- d's' 4- d" s") : 2 (d*s 4- d'*s'+ d"*s". 



PROPOSITION III. 



183. If two fluids of different densities or specific gravities, 

 communicate with one another through a bent tube or otherwise, 

 and remain in a state of equilibrium : 



The perpendicular altitudes of these fluids above their com- 

 mon surface, will vary inversely as their specific gravities. 



Let ABCD be a tube, through the bent arms of which, two fluids of 

 different specific gravities, communicate 

 with one another in the common surface 

 a b, and suppose the horizontal plane E F 

 to pass through the surface of communi- 

 cation. 



Take G the centre of gravity of the 

 plane EF, and through o draw the vertical 

 line GW, meeting Am and DTI respectively 



in the points m and n ; then, because the lines Am and DTI are parallel 

 to the horizon, TKG and no are the perpendicular depths of the plane 

 EF, below the surfaces of the fluids at A and D. 



Now it is manifest, that since the part of the plane ab, which is 

 contiguous to the common surface of the fluids, is sustained in its 

 place by the downward pressure of the lighter fluid in A b, and by the 



