of floating Bodies , and the Stability of Ships. 69 
ferring to the general expression * GZ = — ds, we 
obtain the following values AB — a, D = acn, d = and 
therefore GZ = — — - s x c : by making the distance 
GZ = o, we obtain an equation expressing the relation of 
the dimensions and specific gravity of the solid, when the 
equilibrium becomes insensible, that is, when the centres of 
gravity of the solid and of the part immersed remain in the 
same vertical line, however the value of s or the sine of the 
inclination from the upright position may be altered, pro- 
vided it is always very small ; making, therefore, 
__ * x c -cn we |j ave n 1 — 6c n — — a 2 and n % — n— — 
which gives „=±. + x /±—g r , or « = 1 = 
from whence the following inference is obtained, i. e. in all 
cases whenever ~r is less than that is, whenever the height 
of the solid c bears to the base a a greater proportion than that 
of \/ 2 to s/ 3 > two values may be assigned to the specific gra- 
vity of the solid, each of which will cause it to float in the in- 
sensible equilibrium : thus, suppose the height c to be to the 
base a in the proportion of equality : to ascertain the two limit- 
ing specific gravities, by referring to the preceding solution, 
and making c — a, we obtain n~\ — v/-- or n — x 
\/ th at is n—\— .28868 = .21132, 
or n — \ -f .28868 = .78868. 
# When the angle KGS in fig. 2. is evanescent, the line GZ vanishes: this being 
the case represented by fig. 3, the'point Z coincides with the point G. 
