68 Mr. Atwood's Propositions determining the Positions 
In the subsequent pages, cases occur in which each of the 
preceding expressions are employed, not only to ascertain the 
laws of permanent and unstable equilibrium, but in develop- 
ing other properties relating to the subject. 
EFCD (fig. 3. ) represents a vertical section of an oblong solid 
or parallelepiped, placed on the surface of a fluid IABK, with 
one of the flat surfaces upward, or the line CE or FD vertical : 
this solid is moveable round an horizontal axis, which passes 
through the centre of gravity G, perpendicular to the plane 
ECDF. Let it be required to determine the limits, depend- 
ing on the dimensions and specific gravity of the solid, which 
separate the cases in which the solid will float permanently, 
from those in which it will overset ; through the centre of 
gravity G draw the line SGL parallel to CE or DF : let 
the height of the solid CE — c; let the base CD = «; also 
let the specific gravity of the solid be to that of the fluid on 
which it floats in the proportion of n to 1, or as SN to SL ; 
so that when it is placed on the fluid with the line SL verti- 
cal, it may sink to the depth SN ; let O be the centre of gra- 
vity of the part immersed : suppose the solid to be placed on 
the surface of the fluid with the line SL vertical ; then, since 
SN is the depth to which the solid sinks in the fluid, and SN 
is to SL as n to 1, it follow’s that SN == nc ; and consequently 
GO = — ; the area immersed ABCD = acn ; wherefore, 
to ascertain the perpendicular distance between the two ver- 
ticals which pass through the centres of gravity of the solid 
and of the part immersed, w'hen the solid is inclined through 
a very small angle, of which the sine is = 5 to radius 1, re- 
