OF THE POSITIONS OF EQUILIBRIUM. 357 



downwards, in such a manner, that its base or extreme ordinate is 

 entirely above the surface ; it is required to determine the position of 

 the body when in a state of equilibrium, the parameter of the parabolic 

 section being 16 inches, the axis 40 inches, and the specific gravity of 

 the floating solid, to that of the supporting fluid as 1 to 2 ? 



In this example there are given, p 16 inches, am 40 inches, and 

 s 0.5, the specific gravity of water being unity ; therefore, by sub- 

 stitution, we get from equation (281) 



* ^ 1.6(6X40 5X16 6x40/0125) 3.75 inches. 



And the positions of equilibrium corresponding to this value of g, 

 are as represented in the subjoined diagrams, and the following is the 

 method of construction. 



With the parameter or latus rectum equal to 16 inches, and the 

 subcontrary axes DC and dc each equal to 40 inches, describe the 

 parabolas ADB and adb ; from c the middle of the base and towards 

 the depressed part of the figure, set off cr equal to 3.75 inches, the 

 computed value of z ; through the point r, draw rm parallel to the 

 axis CD, and meeting the curve in the point m ; draw the tangent WE, 

 and on the diameter mr, set off mn equal to 25.19 inches, the value 

 of x as obtained by the reduction of equation (278) ; then through the 

 point n, and parallel to the tangent ?WE, draw the straight line IK, 

 which will coincide with the surface of the fluid, and cut the parabolas 

 ADB and adb in F, H and/", h the extremities of the lines of floata- 

 tion, corresponding to the positions of equilibrium which we have 

 exhibited in the diagrams. 



We must now endeavour to prove, that the positions in which we 

 have represented the body are those of equilibrium; and for this 

 purpose, we must inquire if the equation (279) is satisfied by the sub- 

 stitution of the computed values of x and z ; for when that is the 

 case, the line which joins the centres of gravity of the whole section 

 and the immersed part of it, is perpendicular to the surface of the fluid. 



