OF A VESSEL WHOSE ATHWART SECTIONS ARE VARIABLE. 
619 
^ =inclination to horizon of line about which the plane PQ is symmetrical. 
X =distance of section CAD, measured along the line whose projection is O, 
from the point where that line intersects the midship section. 
y =0(3. 
':=PQ. 
3/2 =RS. 
—hn-\-mg. 
A=KL. 
I = moment of inertia of plane PQ about axis O. 
A and B = moments of inertia of PQ about its principal axes. 
^ =weight of a cubic unit of water. 
Suppose the water actually displaced by the vessel to be, on the contrary, contained 
by it ; and conceive that which occupies the space QOS to pass into the space POR, 
the whole becoming solid. Let AH3 represent the corresponding elevation of the 
centre of gravity of the whole contained fluid. Then will AHa+AHg represent the 
total elevation of the centre of gravity of this fluid as it passes from the position it 
occupied when the vessel was vertical into the position PAQ. But this elevation is 
obviously the same as though the fluid had assumed the solid state in the vertical 
position of the body, and the latter had revolved with it, in that state, into its present 
position. It is therefore represented by KH— NH*; 
.*. AH3+AH3=KH-NH and AH3=KH-NH-AH3. 
Since, moreover, by the elevation of the fluid in QOS, whose v/eight is w, into the 
space OPR, and of its centre of gravity through {gm-{-hn), the centre of gravity of 
mass of fluid of which it forms a part, and whose weight is W, is raised through the 
space AH3; it follows, by a well-known property of the centre of gravity of a system'!', 
that 
W.AH3=zr;(gm+A«) ; 
.'. W(KH — NH — AH2)=w(gm4-Aw). 
But 
NH=KHcos KL=H 2 Cos ; 
.•.KH-NH=H2 vers^-fX, 
and 
mg-\-nh=z ; 
.*. W(H2 vers AH2) =wz ; 
.'. W. AH 2 =W(H 2 vers irz . (10.) 
* The line joining the centres of gravity of the vessel and its immersed part, in its vertical position, is 
parallel to the plane CAD, for it is perpendicular to the plane PQ, to whose intersection with the plane RS 
the plane CAD is perpendicular; GK=Hj and HK=H 2 . 
t PoNCELET, Mecanique Industrielle, partie. Art. 50, or Moseley’s Mechanical Principles of Engineering, 
Art. 59. 
4 K 2 
