Equilibrium of Floating Bodies. 333 



that is, BM is equal to half the parameter of the parabola. 

 being the centre of gravity of the parabola JHF/, its height is 

 readily found to be f NF. Therefore for the whole height of 107. 

 the metacentre above the keel, we have 



10JVF 



Such is the height of the metacentre above the keel, on the 

 supposition that the vertical sections are all equal and parabolic, 

 which is nearly the case with respect to long track-boats. But 

 the figure of the keel in most vessels, fitted for sailing, approach- 

 es to a semi-ellipse, which is likewise the general form of a hori- 

 zontal section. Owing to these modifications, the metacentre is 

 found to be depressed about one fourth part, and consequently 

 its height above the centre of buoyancy will be 



3 3 ^ 2 



SJVF ~ 



In a ship, for example, whose water line is 40 feet, and the depth 

 of its immersed portion 15 feet, we shall have for the height of 

 the metacentre above the centre of buoyancy, 



But the centre of gravity of the immersed part is f 15 or 6 feet 

 below the water line. Hence the metacentre is 4 feet above 

 the water line. The ship will therefore float securely, so long 

 as the general centre of gravity is kept under that limit. In 



in which f NL represents the sum of the cubes of the perpendiculars 

 $Q, OJV, &c., of figure 50, these perpendiculars being taken at equal 

 distances, and so near to each other that the included portions of 

 the curve 0./V, OK, &c., may be considered as straight lines, the 

 common distance being denoted by e, and the bulk of the immersed 

 part of the vessel by b. The investigation of this formula is very 

 simple, and is omitted here merely on account of its length. See 

 Bezouf s Mecanique, art. 359. 



