60 Mr. Atwood's Propositions determining the Positions 
angle of inclination EGO when EO is the radius), there will 
remain the line RT or GZ, which is therefore the distance 
between the vertical lines GO, SZT, passing through the 
centres of gravity G and Q, as determined by the construction. 
Let the whole volume of the immersed part of the solid be 
denoted by the letter V ; suppose the space NXP, or volume 
immersed in consequence of the inclination, to be A ; make 
GO = d ; and the sine of the angle of inclination KGS (to 
radius 1) = s ; also make be = b. Then since by the propo- 
sition ; as b : ET : : V : A, it appears that ET = ; 
And since as ER:EG = GO::so is s : i, we obtain 
ER = ds; 
Wherefore RT= ET — ER=y- ds = GZ. 
This result is founded on a supposition that the figure of 
the floating solid is uniform in respect of the axis of motion ; 
if the solid should be of an irregular form, the construction 
and demonstration will be precisely the same as in the pre- 
ceding case, the following particulars being attended to ; the 
volume, or space immersed in consequence of the inclination, 
will no longer be represented by the area NXP, but must be 
obtained by a calculation founded on the shape and dimen- 
sions of the said volume ; moreover the centres of gravity of 
the volumes PXN, IXW, will not now correspond with the 
centres of gravity of the areas PXN, IXW, and must there- 
fore be obtained from the known rules, or from methods of 
approximation by which the position of the centre of gravity 
is determined in solid bodies. 
The angle of inclination KGS is given by the supposition, 
and the solid contents of the equal volumes denoted by IXW, 
NXP, with the distance be of the centres of gravity a and d » 
