612 
THE DYNAMICAL STABILITY OF SHIPS. 
7- Let a body be conceived to float, acted upon by no other forces than its weight 
W, and the upward pressure of the water (equal to its weight) ; which forces may be 
conceived to be applied respectively to the centre of gravity of the body and to the 
centre of gravity of the displaced fluid ; and let it be supposed to be subjected to the 
action of a third force whose direction is parallel to the surface of the fluid. Let AHj 
represent the vertical displacement of the centre of gravity of the body thereby pro- 
duced*, and AH 2 that of the centre of gravity of its immersed part. Let moreover 
the volume of the immersed part be conceived to remain unaltered'!' whilst the body 
is in the act of displacement. If each centre of gravity be assumed to ascend, the 
work of the weight of the body will be represented by — W.AHj, and that of the 
upward pressure of the fluid by -j-W.AHg, the negative sign being taken in the 
former case, because the force acts in a direction opposite to that in which the point 
of application is moved, and the positive sign in the latter, because it acts in the 
same direction, so that the aggregate work Swg (see equation 1.) of the forces which 
constituted the equilibrium of the body. in the state from which it has been disturbed 
is represented by 
-W.AH1H-W.AH2. 
If the centre of gravity of the body or of the displaced fluid descends (a property 
which will be found to characterize a large class of vessels), AH^ in the one case, 
and AH 2 in the other, must be taken with the negative sign, since the weight of the 
body will be applied in the same direction, and the pressure of the fluid in an opposite 
direction to that in which their respective points of application are moved. More- 
over, the system put in motion includes, with the floating body, the particles of the 
fluid displaced by it as it changes its position, so that if the weight of any element of 
the floating body be represented by w^, and of the fluid by Wo, and if their velocities 
be and ^ 2 , the whole vis viva is represented by 
9 ’ ^^9 ^ 
and we have by equation 1, 
U((l) - W(AH,-AHJ=i ■tw.vl 
(4.) 
In the extreme position into which the body is made to roll and in which '2WiV\=.Q, 
U(«)=W.(AH,-AH,)+i2«),«’, (5.) 
* 
* When, a floating body is so made to incline from any one position into any other as that the volume of 
fluid displaced by it may in the one position be equal to that in the other, its centre of gravity is also vertically 
displaced ; for if this be not the case, the perpendicular distance of the centre of gravity of the body from its 
plane of flotation must remain unchanged, and the form of that portion of its surface, which is subject to im- 
mersion, must be determined geometrically by this condition ; but by the supposition the form of the body is 
undetermined. It is remarkable what currency has been given to the error, that whilst a vessel is rolling or 
pitching its centre of gravity remains at rest. I should not otherwise have thought this note necessary, 
t It will be shown that this supposition is only approximately true. 
