210 Prof. F. Elgar. The Variation of 



rft 



WL= , and the volume of displacement of an unit of length of the 



Ufl 



pri8m=Z*Z. Therefore B'M=-^L But B'S=-, and MS is therefore 



i 'i 2* 



13 i 



equal to +-. If a curve LMN (fig. 3) be constructed whose 







FIG. 3. 



abscissas represent the various depths of flotation of the body, and 

 ordinates the corresponding heights of the point M, this will be the 

 curve of metacentres. OS is equal to the depth of flotation WA in 

 fig. 2 ; BS is equal to the corresponding height of the centre of 

 buoyancy B' above S ; and MS is equal to the height of the meta- 

 centre above S. It is obvious that the locus of the centre of buoy- 





ancy OB is a straight line whose equation is y=^- The equation 



a 



to the curve of metacentres is - +_=w, or 6x 2 12zy= Z> 2 . 



!-.< 2 



This curve is therefore an hyperbola whose asymptotes are the axis 

 OY, which corresponds with the lower side AD of the section of the 

 floating body in fig. 2, and the straight line OB, which is the locus 

 of the centres of buoyancy. The curves of metacentres for various 

 geometrical forms of floating bodies possess many interesting pro- 

 perties, but it is foreign to the purpose of this paper to enter upon a 

 full discussion of them. It may, however, be noted, as additional 

 illustrations of these, that the ordinate of the metacentric cnrve at 

 zero, i.e., the one corresponding with no draught of water, is the 

 radius of curvature of the transverse section of the floating body 

 at its lowest point. Thus, for a body of circular section the height 

 of the metacentre at the point where the draught vanishes is equal 

 to the radius of the circle ; for one the lowest point of whose section 

 is angular, it is zero ; and for one the bottom of whose section is 

 straight and is parallel to the water-line, it is infinite. 



When the body is completely immersed the metacentre is identical 



