72 OF THE PARALLELOGRAM AND ITS INSCRIBED PARABOLA. 



Hence, the pressure on the plane in this case, is only two thirds of 

 what we found it to be in the foregoing case, where the vertex is in 

 contact with the surface of the fluid. 



With respect to the position of the centre of gravity in this case, it 

 is manifest, that the mode of discovering it, is similar to that which 

 we employed in the case immediately preceding, where the axis of the 

 parabola was supposed lo be vertical, and its s-ummit in contact with 

 the surface of the fluid ; it is therefore unnecessary to repeat the 

 investigation, but there is another condition of the figure remaining 

 to be considered, in which a knowledge of the position of the centre 

 of gravity becomes of more importance, as will readily appear from 

 the circumstances which present themselves in the solution of the 

 following problem. 



4. WHEN THE BASE OF THE PARABOLIC PLANE IS PERPENDICULAR TO 

 THE HORIZON, ITS AXIS HORIZONTAL, AND THE PRESSURE UPON 

 IT IS TO BE DETERMINED AS COMPARED WITH THAT UPON ITS 

 CIRCUMSCRIBING RECTANGULAR PARALLELOGRAM. 



PROBLEM XII. 



84. If a parabolic plane be perpendicularly immersed in an 

 incompressible fluid, in such a manner, that its base may be 

 vertical, and just in contact with the surface : 



It is required to determine the pressure upon it, and to 

 compare it with that upon its circumscribing rectangular 

 parallelogram. 



Let ABEF be a rectangular parallelogram immersed in a fluid, with 

 its plane perpendicular to the plane of the 

 horizon, and its upper side AB coincident 

 with the surface of the fluid in which it is 

 immersed. 



Bisect AF and BE in the points D and c; 

 join DC, and upon AF as a base, with the 

 corresponding axis DC, describe the parabola 

 ACF, touching AB the surface of the fluid in 

 the point A ; then is AC F the surface for which the pressure is required 

 to be investigated. 



Join BF, intersecting DC the axis of the parabola in the point n- r 

 then is n the centre of gravity of the rectangular parallelogram ABEF. 



