72 Mr. Atwood's Propositions determining the Positions 
position of equilibrium, are very small, that is, GZ = — B * - 
— ds, and applying this equation to the case under considera- 
tion, we obtain the following values ; AB 3 = 8a 3 x^; D = 
a % n; d = : making therefore ~ - BJ s - — ds , in 
v'z x 3 12 D 
order to obtain the limit, separating the cases of stability and 
instability of floating ; or, which is the same thing, making 
— = a x 3 _- ^ , 8 « } t ] ie following equation arises, — ^ — 
1 2 a 1 n V 2x3 3 
-■■■ ■ 8;t , or n = — = .28123 = the specific gravity, which 
V 2 X 3 3 2 
will cause the solid to float in the insensible equilibrium, and 
is therefore the limit separating the specific gravities which 
cause the solid to float with stability from those which produce 
the equilibrium of instability. It is collected from the general 
equation GZ = ds, or GZ = 
8 a? n's 
a X 3 — v"8 n . 
^2x3 * 
that when the specific gravity ( n ) is evanescent or very small, 
the solid will overset when placed on the fluid with an angle 
upward, because in this case the quantity ■ must necessa- 
1 1 J 12 a 71 
rily be less than - a x L or ds. When the specific gravity 
of the solid is to that of the fluid in the proportion of 9 to 32, 
the solid floats in the insensible equilibrium ; if therefore the 
specific gravity of the solid should be to that of the fluid in a 
less proportion than that of 9 to 32, the solid will overset ; 
but if the specific gravity of the solid exceeds that limit when 
placed on the fluid with the angle upward, or diagonal line 
EC vertical, it will float permanently in that position. 
