of floating Bodies , and the Stability of Ships. 61 
estimated in the direction of the horizontal line AB, having 
been determined, let the volume NXP be put = A ; and be = 
b ; the other quantities signifying as before ; the perpendicu- 
lar distance GZ = ~ — ds, will be known. It is to be observed, 
that this proposition in general is equally applicable to hetero- 
geneal bodies as to those which are homogeneal. 
By this proposition the stability of vessels, and other bodies 
floating on a fluid's surface, at any angle of inclination, from 
a given position of equilibrium, is obtained. For the measure 
of the stability is precisely a force equal to the fluid's pres- 
sure; that is, equal to the vessel's weight,* applied perpendi- 
cularly at the distance GZ from the axis of motion, to incline 
the solid round that axis. 
From the same proposition, the different positions assumed 
by bodies which float freely on a fluid's surface, may be as- 
certained ; in some cases most easily by geometrical construc- 
tion ; in others, by analytical investigation. It has been al- 
ready observed, that to ascertain the various positions in which 
a body will float permanently on the surface of a fluid, it is 
necessary first, to have given the ratio of the specific gravities* 
In order to fix the proportion of the part immersed to the 
whole ; and secondly, the several positions are to be ascer- 
tained in which the solid may rest on the surface of a fluid, 
so that the centres of gravity of the solid and of the part im- 
mersed maybe in the same vertical line. The general expression 
for the line RT (fig. 2.) or GZ, is GZ — — ds ; by put- 
ting this quantity ~ — ds= o, an equation arises, from which 
one or more values of s will be obtained — the sine of the angle 
through which the solid has been inclined from a position of 
* The weight of a vessel implies the weight of the ship and lading. 
