228 OF FLOATATION AND THE SPECIFIC GRAVITY OF BODIES. 



Now, the pressure upwards on the bottom of the attached cylindri- 

 cal body, according to the principle of the first proposition, is 



//=.78543V(J + a?) ; 



but in the case of an equilibrium, or when the body has attained a 

 state of quiescence, the pressure upwards is exactly equal to the 

 downward tendency ; consequently, by comparison, we have 



.78543V (l + x) = .7854 { (3* d*) s' x + Vis} ; 

 therefore, by suppressing the common factor .7854, and transposing, 



we get 



3V (I + x} (a 2 d 2 ) s' x = V Is, and this, 



by expanding and collecting the terms, becomes d?s'x=i&l(s s'), 

 and finally, by division, we obtain x zz: ~- f . (183) 



262. The following practical rule, drawn out in words at length, 

 will serve for the reduction of the equation. 



RULE. Multiply the difference between the greater and 

 less specific gravities, by the square of the diameter of the 

 attached body drawn into its length ; then divide the product 

 by the square of the diameter of the hollow cylinder, drawn 

 into the specific gravity of the fluid, and the quotient will 

 be the distance below the surface of the fluid at which the 

 body rests. 



263. EXAMPLE. A cylinder of lignum vitse, whose diameter is 

 8 inches, length 36 inches, and specific gravity 1.327, is attached to 

 the lower end of a hollow tube, whose diameter is 3 inches, in such a 

 manner that no fluid can enter ; now, supposing the body, and the 

 hollow cylinder to which it is attached, to be placed in a vessel full of 

 water, it is required to determine, at what distance below the surface 

 of the fluid the body will become quiescent ? 



Here, by operating according to the rule, we obtain 



8 2 X36(1.327 1.000) 

 ,= - ( - -' = 83.707 inches, 



Hence it appears, that a cylinder of lignum vitae of the proposed 

 dimensions, will sink to the depth of 83.707 inches, or very nearly 

 7 feet below the surface of the water, before the upward pressure 

 becomes an equipoise for its downward tendency ; and this being 

 added to the three feet which it is in length, gives 10 feet for the 

 depth of the body of water, necessary for admitting an equilibrium, 

 under the specified conditions of magnitude and attachment. 



