OF THE STABILITY OF FLOATING BODIES AND OF SHIPS. 367 



We are now to suppose, that by the application of some external 

 force, the solid revolves about its axis of motion until it comes unto 

 the position represented by abed, in which state the equilibrium does 

 not obtain. 



Here it is manifest that PQ, the axis of the section which was 

 vertical in the first instance, is transferred, in consequence of the 

 inclination, into the position pq ; and in like manner, the line EF, which 

 before was horizontal, is transferred into the oblique position ef, and hi 

 is now the line of floatation, or as it is otherwise called, the water line. 



Since the absolute weight of the body remains unaltered, whatever 

 may be the position of floating, the area of that portion of the section 

 which is immersed below the surface of the fluid, must also be inva- 

 riable ; it therefore follows, that the areas hied and EFCD are equal 

 to one another ; but the space efcd is equal to EFC D, hence the spaces 

 hied and efcd are each of them equal to EFCD; they are therefore 

 equal to one another, and consequently, the extant triangle hke is 

 equal to the immersed triangle fh i. 



On pq the axis of the section, set off GH equal to Gg, the distance 

 between the centre of effort and centre of buoyancy in the original 

 position of equilibrium ; then it is manifest, that in consequence of the 

 inclination, the point g, which is the centre of gravity of the space 

 EFCD, will be transferred to the point n, which is the centre of gravity 

 of the equal space efcd; and the pressure of the fluid would act upon 

 the body in the direction of a vertical line passing through 71, if efcd 

 were the portion of the section immersed under the fluid's surface ; 

 but this is not the case, for in consequence of the inclination, the 

 triangle fki, which was before above the fluid's surface, is now 

 depressed under it, and in like manner the triangle hke, which was 

 previously under the surface, is now elevated above it. 



It is therefore obvious from Proposition XII, that by transferring the 

 triangle hke into the position fki, the point n, which is the centre 

 of gravity of the space efcd, must partake of a corresponding motion 

 and in the same direction ; that is, the point n must move towards 

 those parts of the body that have become more immersed in conse- 

 quence of the inclination, until it settles in g the centre of gravity of 

 the immersed volume hied. 



Through # the centre of gravity of the immersed part hied, draw 

 ym perpendicular to hi the line of floatation, and meeting pq the 

 axis of the section in the point m ; then is m the metacentre, and the 

 pressure of the fluid will act in the direction of the vertical line g'm, 

 with a force precisely equal to the body's weight ; and according to 



