OF THE STABILITY OF FLOATING BODIES AND OF SHIPS. 365 



and the pressure of the fluid operates to restore the equilibrium ; in 

 this case, therefore, the equilibrium is that of stability. 



But when the line of support falls on the same side of the centre of 

 effort as the parts of the solid which are elevated in consequence of 

 the inclination ; then the length of the equilibrating lever is accounted 

 negative, and the equilibrium is that of instability. 



Hence it appears, that the stability of a floating body is positive, 

 nothing or negative, according as the metacentre is above, coincident 

 with, or below the centre of effort: these consequences, however, 

 will be more readily and more legitimately deduced from the general 

 formula which indicates the conditions of stability, and this formula 

 we shall shortly proceed to investigate. 



PROPOSITION XII. 



454. The common centre of gravity of any system of bodies 

 being given in position, if any one of these bodies be moved from 

 one part of the system to another, it is manifest, from the principles 

 of mechanics, that : 



The motion of the common centre of gravity, estimated in 

 any given direction, is to the motion of the body moved, 

 estimated in the same direction, as the weight of the said 

 body, is to the weight of the entire system. 



Therefore, by means of these propositions and the definitions that 

 precede them, the whole doctrine of the stability of floating bodies, 

 with the train of consequences which immediately flow from it, may 

 be easily and expeditiously deduced ; but in proceeding to develope 

 the laws on which the stability of floating depends, it will be con- 

 venient for the sake of simplicity, to consider the body as some 

 regular homogeneous solid, of uniform shape and dimensions through- 

 out the whole of its length ; for in that case, all the vertical transverse 

 sections will be figures precisely equal and similar to each other ; and 

 if the body be divided by a vertical plane passing along the axis of 

 motion, the two parts into which it is separated will be symmetri- 

 cally placed with respect to the dividing plane. 



This being premised, the principles upon which the stability of 

 floatation depends, will be determined by the resolution of the follow- 

 ing problem, in which all the transverse sections are trapezoids. 



