66 Mr. Atwood’s Propositions determining the Positions 
to the whole length of the line % will be fluen ' of ^ B " 5 x ~ ; and 
therefore the distance of the common centre of gravity of 
the volume immersed in consequence of the inclination 
from the horizontal line passing through the point X, is 
fluent of a b * ' r ' . ; i n like manner the distance of the com- 
24 A 
mon centre of gravity of the volume, elevated above the sur- 
face by the inclination of the given plane, appears to be 
fluent of ab^ x s > x . an( j consequently the distance between the 
two centres of gravity measured on the horizontal line, or be 
(fig. 2.) = fluent 01 * x z : this value being substituted for b 
in the equation GZ = ds, we obtain the following re- 
sult, i. e. GZ = fluent ° f ^ B y x 5 x z — ds, which is a general ex- 
pression for ascertaining whether a solid, when placed on the 
surface of a fluid in a given position, will float permanently, or 
overset, the sine of the angle of inclination or s being assum- 
ed evanescent ; for, when fluei . t _ ( l ^ ^ x ■ s x g is greater than 
ds, the line of support OZ (fig. 2.) will be situated between 
the axis of motion, and the parts of the solid which are im- 
mersed by the inclination, in which case the solid will float 
permanently ; and when fluent * 5 . x * is less than ds, the 
line of support passing through the point % will be on the 
contrary side of the axis, and according to the preceding de- 
termination (page 64) the solid will in this case overset. 
Since, when the fluent of (fig. 2.) is greater than ds. 
