OF THE PARALLELOGRAM AND ITS INSCRIBED PARABOLA. 813 



straight lines ma and HG, intersecting each other in the point G ; and 

 the point G thus determined, is the position of the centre of gravity 

 of the space comprehended between the semi-parabola and its circum- 

 scribing rectangular parallelogram. 



This method of determining the position of the centre of gravity of 

 the space comprehended between the curve and its circumscribing 

 parallelogram, will be illustrated and applied in all its generality, 

 when we come to treat on the subject of Hydraulic Architecture, to 

 which it more properly belongs ; and for this reason, we shall take 

 no further notice of it in this place, but proceed with our inquiry 

 respecting pressure, which is more immediately the object of our 

 research. 



7. METHOD OF DIVIDING A PARABOLIC PLANE PARALLEL TO ITS BASE, 

 SO THAT THE FLUID PRESSURES ON EACH PART MAY BE EQUAL 

 TO ONE ANOTHER. 



PROBLEM XIII. 



95. If a parabolic plane be immersed perpendicularly in an 

 incompressible fluid, in such a manner, that its vertex is j ust in 

 contact with the surface : 



It is required to determine at what distance from the 

 vertex, a straight line must be drawn parallel to the base, so 

 that the figure may be divided into two parts, on which the 

 pressures are equal to one another. 



Let ACB be the given parabola, immersed in the fluid after the 

 manner specified in the problem, and let aAEb be the circumscribing 

 rectangular parallelogram. 



Take cm for the distance from the vertex 

 through which the line of division passes, and 

 draw EF parallel to the base AB; then are 

 the spaces ABFE and ECF, the parts into 

 which the parabola is divided, and on which, 

 by the conditions of the problem, the pres- 

 sures are equal. 



Through the points E and F, the extremities of the line of division, 

 draw EC and FG? respectively parallel to CD the axis of the figure; 

 then is CEFC?, the rectangular parallelogram circumscribing the para- 

 bola ECF. 



G2 



