ON SEMICIRCULAR PLANES IMMERSED VERTICALLY. 165 



P' .7854ef X h - X --7 = .7854d h s. (128). 



\ n ' n -f- 1 



The cylindric surface in contact with the fluid after expansion, may 

 be expressed as follows, viz. 



but it has been shown above, that 



3.1416eM; 

 therefore, by substitution, we obtain 



n 

 and consequently, the pressure becomes 



If therefore, the equations (126) and (128) be compared with one 

 another, it will be found that the pressure is the same, and equal to 

 the weight of the fluid in both cases; but if the equations (127) and 

 (129) be compared, the pressure in the one case, is to that in the other, 

 in the ratio of n : n + 1 ; that is, 



p : p' : : n : n -\- I . 



PROBLEM XXVI. 



164. A semi-circular plane is vertically immersed in a fluid 

 whose density increases as the depth, and in such a manner, 

 that the horizontal diameter coincides with the upper surface of 

 the fluid : 



It is required to determine, on which chord parallel to the 

 horizon, the pressure is a maximum, or greater than the 

 pressure on any other chord. 



Let A BCD represent a vertical section of a mass of fluid, of which 

 AB is the surface, and whose density 

 varies directly as its depth ; and let jmr 

 aGFHB be the semi-circular plane im- 

 mersed in it, in such a manner, that 

 the horizontal diameter a b, coincides 

 with AB the upper surface. 



Let m be the point in the vertical 



