of floating Bodies , and the Stability of Ships. icg 
or counterpoise the force of stability, there will arise the equa- 
tion M x SG = W x GZ. 
In the particular case, when the angles at which a floating 
solid is inclined from the position of equilibrium are very small, 
the line GZ (fig. 2.) has been found * = fluent of x z xs _ ds: 
in which expression z is a small portion of a line drawn co- 
incident with the fluid’s surface, and parallel to the axis of 
motion ; AB is the breadth of the solid at the water’s surface, 
corresponding to the line % parallel to the axis ; V is the total 
displacement or volume immersed ; d is the distance GO ; and 
s the sine of the small angle of inclination from the position of 
equilibrium. Respecting this expression it is observable, that 
since (1 ' lent of .T|’ -* = ET, (fig. 2.) and d = OG = EG, 
it follows that = ES nd fluent _ d _ GS ; 
which quantity is invariably the same whatever may be the 
inclination of the floating body from the position of equili- 
brium, provided that inclination is very small ; that is, the 
point S is immoveable in respect of the point G, while the 
floating body revolves through any different small angles 
round the axis, passing through the centre of gravity G in a 
direction perpendicular to the plane ADHB. Since, therefore, 
the measure of stability GZ x W is x it — ds x W 
and y; x 8 — d = GS, (fig. 2.) it follows that the mea- 
sure of stability = W x SG x s, agreeing with the value which 
Euler has deduced by other methods for expressing the sta- 
bility of vessels when the angles of inclination are evanescent.-)- 
* Page 66. 
t Theorie complette de la Construction des Vaisseaux , chap, viii. 
