of floating Bodies, and the Stability of Ships. 67 
the solid floats permanently ; and when ds is greater than 
the equilibrium is that of instability; it follows 
fluent of AB si 
12 V 
that whenever 
fluent of A B i _ 
“ iTv “ 
fluent of AB si 
TzV 
ds, by resolving the equation 
d, one or more limits are obtained (depending 
on the dimensions and specific gravity of the solid), separat- 
ing the cases in which the solid floats with stability from . 
those in which the equilibrium is momentary and unstable. 
The limits here obtained evidently correspond to that species 
of equilibrium which has been denominated insensible, or the 
equilibrium of indifference. 
When the floating body is of uniform figure and dimen- 
sions, respecting the axis of motion, the expression here given 
for determining the stability or instability of floating will not 
involve any fluxional quantities, for in this case all the vertical 
sections which pass through the solid in a direction perpen- 
dicular to the axis are equal, and consequently the portions of 
those sections immersed under the fluid's surface are also 
equal ; if, therefore, the area of any one of these sections im- 
mersed under the fluid's surface be denoted by the letter D, 
the solid contents 'or volume immersed, corresponding to the 
length of the line will beD%; wherefore, in the preceding 
expression GZ = fluent of x 5 * * — ds, we have by substi- 
tution V = D%, and since AB is a constant or invariable 
quantity by the supposition, fluent ofABss5 ABs^ 
12 D z 
12 Dz 
AB 
— — : finally, therefore, in the case under consideration. 
we obtain GZ = 
A B x 5 
12D 
— ds . 
Kq 
