214 OF FLOATATION AND THE SPECIFIC GRAVITY OF BODIES. 



Now, these pressures are manifestly equal to the weights of the 

 columns Gt and vu considered as fluid, and since the same may be 

 demonstrated with respect to every other portion of the immersed 

 surface, we therefore conclude, that the whole pressure upwards, is 

 equal to the sum of the weights of all the columns Gt, FU, &c. ; that 

 is, to the weight of a quantity of the fluid equal in magnitude to the 

 immersed part of the body ; hence the truth of the proposition is 

 manifest. 



COROL. From the principles demonstrated above, it follows, that 

 when a solid body floating on the surface of a fluid is in a state of 

 quiescence : 



The pressure downwards is equal to the buoyant effort ; 



that is, the weight of the floating body, is equal to the weight 



of a quantity of the fluid, whose magnitude is the same as 



that portion of the solid, which falls below the plane of 



floatation. 



PROBLEM XXXI. 



234. A cylindrical vessel of a given diameter, is filled to a 

 certain height with a fluid of known specific gravity, and a 

 spherical body of a given magnitude and substance is placed 

 in it : 



It is required to determine how high the fluid will rise in 

 consequence of the immersion of the spherical segment which 

 falls below the plane of floatation. 



Let ABCD represent a vertical section passing along the axis of a 

 cylindrical vessel, filled with an incompressible and 

 non-elastic fluid to the height ED, EF being the sur- 

 face of the fluid before the sphere whose diameter is 

 mn, is placed in it, and ab the surface after the 

 immersion of the segment tnu, the liquid rising to the 

 height A D. 



Then it is manifest from the nature of the problem, 

 that the spherical segment tvnwu, together with the quantity of fluid 

 in the vessel, must be equal to the capacity of the cylinder whose 

 diameter is DC, and perpendicular altitude D; for the fluid rises in 

 consequence of the immersion of the segment, and fills the spaces 

 atvE and buw^s all around the vessel ; we have therefore to calculate 



