Or THE EQUILIBRIUM OF FLOATATION. 271 



and finally, by extracting the square root and transposing, it is 



' = a V a '- (211). 



334. And the practical rule by which this equation is reduced, may 

 be expressed in words at length, in the following manner. 



RULE. Divide the weight which is proposed to be subtracted 

 from the paraboloid, by 1.5708 times the parameter of the 

 generating parabola drawn into the specific gravity of the 

 fluid, and subtract the quotient from the square of the dis- 

 tance of the vertex below the plane of floatation in theflrst 

 position of equilibrium ; then, from the said distance , subtract 

 the square root of the remainder for the quantity of ascent 

 required. 



335. If we take the data of the foregoing example, and proceed 

 according to the directions of the rule, we shall obtain 



x 1 = 12 4 / 144 __ _ ?? _ 1.35 inches, 

 y 1.5708X16X.03617 



exceeding the descent in the former example, by 0.13 of an inch. 



Numerous other examples akin to the above, respecting bodies of 

 various forms, and placed in different positions, might here be pro- 

 posed ; but since they are all resolvable by the general formula, 

 equation (206), we have thought proper to omit them. 



PROBLEM XLVIII. 



336. If a body which is symmetrical with respect to its verti- 

 cal axis, floats upon a fluid in a state of equilibrium : 



It is required to determine what weight must be placed 

 upon the body, so that it shall sink to a level with the surface 

 of the fluid, the specific gravities of the solid and the fluid, 

 together with the magnitude of the solid, being given. 



In the second example to the seventh proposition, we have advanced 

 principles of nearly a similar import to those required for the solution 

 of the present problem, yet nevertheless, we think a separate solution 

 in the present instance is necessary, since it can be somewhat differ- 

 ently represented ; for which purpose, 



Put m ~ the entire magnitude of the floating body, 



w its weight before the external body or weight is super- 

 added, 



