64 Mr. Atwood's Propositions determining the Positions 
the consequence of which is an equilibrium of stability ; and 
whenever = ET is less than ds = ER, the point q, and 
the line of support q%, will be on the contrary side of the axis, 
causing an equilibrium of instability to take place.* The equa- 
tion, therefore, GZ = ds, applied to any particular 
case, will always decide whether the equilibrium in which a 
solid is placed on the surface of a fluid is stable and permanent, 
or whether it is only momentary and unstable, provided the va- 
lue of s, or the sine of the angle of inclination from the given 
position of equilibrium, be assumed evanescent; since the solid 
either continues to float permanently, or will overset, accord- 
ing to circumstances which take place while it is inclined 
from its position of equilibrium through the smallest angle. 
The application of the condition just mentioned will cause the 
general expression to assume a form suited to this particular 
case, which is in the next place to be attended to. 
Referring to (fig. 2.), ADHB represents a vertical section 
of a floating body, passing in a direction perpendicular to 
the axis of motion ; suppose another section to be drawn 
parallel to the former, and extremely near to it ; these two 
planes will comprehend between them a small portion of the 
solid ; and since according to the conditions of the case, the 
angle of inclination KGS, or NXB, is evanescent, the sine of 
this angle (which has been denoted by the letter s ) will also 
become evanescent ; and since the space or volume immersed 
in consequence of the inclination, that is NXP, is equal to the 
volume elevated above the surface IXW, and the angles NXP, 
IXW, are vertical ; the point of intersection of the lines IN 
and AB, that is, the point X will bisect the line AB, and the 
* Page 49, and page 50. 
