OF THE EQUILIBRIUM OF FLOATATION. 267 



question and the nature of the body. For instance, if the body be a 

 solid of revolution, and r the radius of the section coincident with the 

 plane of floatation ; then, the above equation becomes 



w^nrsfax, (206). 



where the symbol TT denotes the number 3. 1416, or four times the area 

 of a circle whose diameter is expressed by unity. 



327. The solution of the problem, however, may be effected inde- 

 pendently of the fluxional analysis, especially in all cases where the 

 floating body is symmetrical with respect to its axis ; for if it be in 

 the form of a right cylinder with its axis vertical, as in the annexed 

 diagram; then, the solution becomes an object of the greatest 

 simplicity ; for since the area of the horizontal section is constant, 

 the space through which the body moves will be the same, whether 

 the weight be added to it or subtracted from it. 



Let A B c D be a vertical section of a 

 cylinder, floating in equilibrio on a fluid 

 whose surface is GH, the axis mn being 

 perpendicular to the horizon, and sup- 

 pose the weight n to be placed on the 

 upper end of the cylinder ; it is obvious 

 that the equilibrium will then be de- 

 stroyed, and the body will continue to 

 descend, until it has displaced a quan- 

 tity of the fluid, whose weight is equal 



to that of the compound mass, consisting of the cylinder, together with 

 the applied body whose weight is TV; or it will continue to descend, until 

 the weight of the fluid displaced by the space IKFE is equal to n the 

 weight of the applied body ; in which case, the equilibrium will again 

 obtain, and the plane of floatation, which originally cut the cylinder 

 in E F, will now be transferred to i K. 



Again, on the other hand, if a,weight rv be subtracted from the 

 cylinder, supposed to be in a state of equilibrium with the plane of 

 floatation passing through EF, the body will then ascend, until the 

 weight of the fluid which rushes into its place becomes equal to the 

 weight subtracted, in which case the solid will again be in a state of 

 quiescence with the plane of floatation passing through a b. 



Put r ~ id or nd, the radius of the horizontal section, 

 m the magnitude of the space IKFE or EF#a, 

 x de or ec, the space through which the body is depressed 

 or elevated in consequence of the extraneous weight, 



