95 
of floating Bodies , and the Stability of Ships. 
will float permanently with the axis vertical, or will overset, 
the plane of the base being supposed perpendicular to the axis: 
Let CED represent a plane section of this solid passing through 
the axis, which section will therefore be a parabola. Suppose 
the specific gravity to be such as causes the solid to sink to the 
depth FE. AIBHA represents a circular section of the solid 
which coincides with the fluid's surface ; draw any diameter 
HI, and the diameter AB perpendicular to HI. Through any 
point W, in the radius FH, draw the ordinate KM perpendi- 
cular to FH ; and suppose the solid to be moveable round an 
axis of motion parallel to the diameter HI ; put the parameter 
of the parabola =zp ; the length of the axis KE — a, FW =z; 
the specific gravity = n ; ^=3.14159; also let G be the centre 
of gravity of the solid, and O the centre of gravity of the part 
immersed. Then, since the volume immersed AEB is to the 
volume CED as AB x EF is to CD x EK, or as EF to EK ; 
and since the volume immersed AEB is to the volume CED as 
n to 1, it follows that as EF : EK = a 1 : : n to 1, and therefore 
EF = as/ ra, and FB = pa %/ n ; referring to the expression 
for determining the stability of floating bodies when the in- 
clinations from a position of equilibrium are very small, or 
fluent °f km % x _s ^ we jj aV e, applicable to the present 
case, the entire fluent of KM z = 37? x FB ; or, because FB + 
= p z a n, the fluent of KM £ = 3 wp* a n\ V or the volume 
immersed = — ; and since by the properties of the figure, 
GE = — and OE = 2 -l^L, we have GO = = d, 
these substitutions being made in the general value, 
— ds ; this quantity becomes = ~ 
12 v ^ J 12 x 7 r a z pn 
