OF THE PARALLELOGRAM AND ITS INSCRIBED PARABOLA. 



hence, if the specific gravity of the fluid be expressed by unity, we get 



V = Thl. (35). 



COROL. It therefore appears, that the centre of gravity of the space, 

 included between any rectangular parallelogram and its inscribed 

 parabola, is situated in the axis, at the distance of three tenths of its 

 length from the vertex. 



The preceding investigation determines the place of the centre of 

 gravity to be at three tenths of the axis below a tangent line passing 

 through the vertex of the parabola, and that it is situated in the axis 

 is manifest; for the spaces A EC and BFC are equal, and they are 

 similarly and symmetrically placed with respect to the axis c D ; there 

 can therefore be no reason, why the centre of gravity should occur at 

 a point, which is nearer to the one than it is to the other ; it conse- 

 quently occurs at a point which is equally distant from both, and that 

 point must obviously be found in the axis of the figure. 



The above is a valuable proposition in the practice of bridge build- 

 ing ; for by it, we can readily assign the position of the centre of gravity 

 of the arch with all its balancing materials, and consequently, many im- 

 portant particulars respecting the weight and mechanical thrust, may 

 be determined and examined with the greatest facility : all of which will 

 be investigated and applied in our treatise on Hydraulic Architecture. 



3. WHEN THE PARABOLIC PLANE PERPENDICULARLY IMMERSED HAS 

 ITS BASE COINCIDENT WITH THE SURFACE OF THE FLUID. 



81 . What has hitherto been done under the present problem, applies 

 only to the case, in which the parabola is perpendicularly immersed in 

 the fluid, and having its vertex coincident with the surface ; but when 

 the parabolic plane is perpendicularly immersed, and having its base 

 coincident with the fluid surface, the circumstances will be something 

 different, as will become manifest from the following investigation. 



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

 its plane perpendicular to the plane of the 

 horizon, and having its upper side coincident 

 with the surface of the fluid ; and let A c B be 

 a parabola described upon the rectangular 

 plane, in such a manner, that its vertex may 

 be downwards, its axis vertical, and its base 

 in contact with the surface of the fluid in which 

 it is placed. 



JV C 



