328 Hydrostatics. 



445. To fix, in a few simple cases, the dimensions of the sol- 

 id and its relative density to that of the fluid required for a par- 

 ticular state of equilibrium, let the body in question be a ho- 

 mogeneous parallelepiped, placed vertically in the fluid ; and let 

 Fig.217. D* '*E be a section of this body through the axis parallel to one 

 of its faces. The solid will evidently sink till the immersed part 

 42 ' 7 - JVF shall be to the whole height AF, as its density is to the 

 density of the fluid ; and its centre of gravity and that of buoy- 

 ancy will be G and B, the middle points respectively of the axis 

 and of the depressed portion JVF. Suppose now that the body is 

 inclined a little, shifting its water-line from the position HN1 to 

 jff'JV/', the centre of buoyancy, changing from B to B', will des- 

 cribe a small arc of a circle, which for the extent under considera- 

 tion may be regarded as a straight line, and B' will be raised by 

 a quantity which will be to the altitude PO of the centre of grav- 

 ity of the triangle 7JV/', as the area of the rectangle JV/F is to 

 that of the triangle 7JV/', that is, as JVF : \ II'. Moreover the 

 horizontal motion of B will be to JVP or f JV7 in the same pro- 

 portion. Whence 



JVF : 177' : : f JV/ : BB> = ** X 7// . 



3 NF 



But, by similar triangles, 



A/7 v 77' 



: BM:: IP : JV7; 



accordingly we have, for the height of the metacentre above the 

 centre of buoyancy, 



Let ./-IF, the height of the parallelepiped, be denoted by /*, 

 its breadth or thickness HI by a, and its density or specific 

 gravity by A. When the metacentre coincides with the centre 

 of gravity, and the solid floats indifferently in any position, BM 



n r \TT? 



is equal to BG or to - ; that is, (water being the fluid 



in question,) since 1, the density of the fluid is to A, the density 

 ef the solid, as AFoY h is to JVF or A /*, we shall have 



