OF FLOATATION AND THE SPECIFIC GRAVITY OF BODIES. 215 



the spherical segment tvmuu, and the cylinders EFCD and a ben, 

 for which purpose, 



Put d zz ED, the height to which the vessel is originally filled with 



the fluid, 



$ == DC, the diameter of the cylindrical vessel, 

 r zz ct, or cv, the radius of the sphere, 

 s = the specific grav ; ty of the fluid in the vessel, 

 s' nr the specific gravity of the floating body, and 

 x D, the height to which the fluid rises on the immersion of 



the spheric segment. 



Then, since by the principles of mensuration, the solid content or 

 capacity of a sphere, is equal to two thirds of that of its circumscribing 

 cylinder, it follows, that the capacity of the sphere mvnw, is ex- 

 pressed by 



3.1416r 2 x2rXf=: 4.1888^; 



but^we have elsewhere demonstrated, that the magnitudes of bodies 

 are inversely as their specific gravities ; consequently, the magnitude of 

 the part immersed, is determined by the following analogy, viz. 



.:.':-. 4.1888^: 



Now, as we have already observed, the quantity of fluid in the 



vessel at first, is 



.7854X^X^7854^, 



and the capacity of the cylinder formed by the fluid and the spherical 



segment, is 



consequently, by addition, we shall have 



.7854 tf x = .7854^-f H^^i . 



s 



and therefore, if all the terms of this equation be divided by the 

 quantity .78543% we shall obtain 



16rV 

 ' 



(177). 



Or if the height to which the vessel is originally filled, be subtracted 

 from both sides of the above expression, the increase of height in con- 

 sequence of the immersion of the spheric segment, becomes 



, 16rV 



d=x 3?7' 

 where &'=: a the increase of height. 



