g6 Mr. Atwood's Propositions determining the Positions 
i g - za ^ n x \ which being put = o, in order to obtain the 
limiting value required, we obtain ~ ^ v/ ” = 
3 — 4 g — 4 a v 7 >z __ anc | n== . — ~ 3 - ; consequently v/ : 1 
From this determination it appears, that if the axis should 
be to the parameter in a proportion less than that of 3 to 4, no 
specific gravity can be given to the solid which will make it 
float in the equilibrium, which is the limit between the stabi- 
lity and instability of floating ; secondly, if the specific gravity 
of the solid bears a greater proportion to that of the fluid 
than the proportion which the square of the difference be- 
tween the axis and ^ of the parameter bears to the square 
of the axis ; when the axis is placed vertical, the solid will 
float with stability in that position ; and thirdly, if the spe- 
cific gravity of the solid bears a less proportion to the specific 
gravity of the fluid than that which the square of the afore- 
said difference bears to the square of the axis, the solid will 
overset when placed on the fluid with the axis vertical, and 
will settle permanently with the axis inclined to the verti- 
cal line. These limits agree precisely with those which are 
demonstrated by Archimedes, in the second book of his tract, 
intituled De iis quce in humido vehuntur,* prop. iii. and prop. iv. 
* The demonstrations of Arch i me des, which relate to the parabolic conoid, are 
founded on a supposition that this solid is generated by the revolution of a rectangular 
parabola on its axis; that is, of a parabola which is the section of a rectangular cone ; 
in which case the line, called by the author (or rather by his translator, the original of 
this treatise being lost) “ ea qua; usque ad axem,” is half the principal parameter, be- 
ing equal to the perpendicular distance between the plane which touches the cone, and 
the plane parallel to it, which is coincident with the parabola. This solid is termed by 
Archimedes, “ conois rectangula,” but the limitation appears to be unnecessary, 
because the demonstrations of the author are equally applicable to a solid generated by 
