218 OF FLOATATION AND THE SPECIFIC GRAVITY OF BODIES. 



base is ab and axis cf ; therefore, in order to calculate the solidity 

 of the segment, and that of the paraboloids DCE and acb, 



Put d n: me, the whole axis or height of the paraboloid, 

 p zn the parameter or latus rectum of the axis, 

 r zz: cr, cm or cw, the radius of the sphere, 

 s the specific gravity of the fluid in the vessel, 

 / zz the specific gravity of the floating body, and 

 x zz tc, the whole height to which the fluid ascends, n being 

 the part originally filled. 



By the principles of solid mensuration, the capacity or solidity of a 

 sphere, is equivalent to two thirds of that of its circumscribing cylin- 

 der ; consequently, the capacity of the floating sphere, is 



now, we have demonstrated in another place, that the magnitudes of 

 bodies, are inversely as their respective gravities ; hence we have for 

 that portion actually immersed, 



*:^: 4.1888,* : 



Again, by the principles of mensuration, the solidity of a paraboloid 

 is equal to one half the solidity of its circumscribing cylinder, and by 

 the property of the parabola, we have 



niA*~pd; 

 therefore, the capacity of the paraboloidal vessel, is 



3.1416X^^X^1-5708^^, 

 and consequently, the quantity of fluid in it is expressed by 



But the capacity, or the solid content of the paraboloid a c b, whose 

 axis is tc, becomes 



consequently, by addition and comparison, we have 



4.1888rV \.570Spd* 

 1 .57(% X* zz -- -f - , 

 s n 



and dividing all the terms by 1.5708, we get 



8rV pd* 

 **- = + - 



