368 OF THE STABILITY OF FLOATING BODIES AND OF SHIPS. 



the principles of mechanics, it will act with the same energy at what- 

 soever point of the line gm it may be applied. 



Through the point n and parallel to hi the line of floatation, draw 

 nz cutting the vertical line gm in the point z, and through G the 

 centre of gravity of the whole space abed, draw or perpendicular 

 and GS parallel to nz, and let k be the point in which the lines ef 

 and hi intersect one another; then, as we have stated above, the 

 pressure of the fluid will have the same effect to turn the body round 

 its axis, whether it be applied at the point g or the point s ; we shall 

 therefore suppose it to be applied at the point s, in which case GS 

 will represent the point of the lever, at whose extremity the pressure 

 of the fluid acts to restore the body to its original state of equilibrium, 

 or to urge it farther from it. 



Since the effect of the fluid's pressure, acting in the direction of 

 the vertical line which passes through g the centre of buoyancy, has 

 no dependence on the absolute position of that point, but on the 

 horizontal distance between the vertical lines rG and gm ; it follows, 

 that in the actual determination of the positions which bodies assume 

 on the surface of a fluid, and their stability of floating, the situation 

 of the centre of buoyancy in the inclined position is not required, for 

 the horizontal distance between the vertical lines which pass through 

 that centre and the centre of effort, is sufficient for obtaining every 

 particular in the doctrine of floatation. 



Bisect the sides of the triangles h ke and/Az in the points u, v and 

 w, a:, and draw the straight lines ku, ev and kw, ix intersecting two 

 and two in the points / and o ; then are / and o the points thus deter- 

 mined, respectively the centres of gravity of the triangles hke and 



fki. 



Through the points I and o draw the straight lines ly and of, 

 respectively perpendicular to hi the line of floatation, corresponding 

 to the inclined position of the body ; then is yt the horizontal distance 

 through which the centre of gravity of the triangle hke has moved in 

 consequence of the inclination ; therefore, by the principle announced 

 in Proposition XII., we obtain 



area efcd : area hke : : yt : nz. 



It is easy to comprehend in what manner the proposition cited above 

 applies to the case in question ; for we may assume the area efcd as 

 a system of bodies, of which the common centre of gravity is n. One 

 of the bodies composing this system, viz. the triangular area hke, 

 conceived to be concentrated in the point /, is transferred, in conse- 

 quence of the inclination from the point I to the point o, in which the 



