54 Mr. Atwood's Propositions determining the Positions 
axis is placed vertically ; the cylinder, under these circum- 
stances, must always float permanently with its axis vertical. 
When the diameter of the base bears to the length a less 
proportion than that of \/e to 1, two values of the specific 
gravity may always be assigned, which will be the limits of 
the cases in which the solid floats with stability or oversets ; 
i. e.n — d-^ y/ _L — If the specific gravity should be 
given, the proportion of the cylinder's length to the diameter 
of the base may be defined which limits the cases of stability 
or instability of floating with the axis vertical ; for since n — 
tf = it follows that y = v/8 n — S?i z ; consequently n be- 
ing given, if the diameter of the base should be to the length 
of the axis in a greater proportion than that of v/ 8 n — 8 to 
l, the solid will float permanently with the axis upward ; but 
if the base should be to the length of the axis in a less pro- 
portion than that of \/ 8 n — %n to l, the solid will overset. 
Thus if v/ 8 n — 8 n — y / ' = 1.2247 J if therefore 
the diameter of the base should be in a greater proportion to 
the length of the axis than 1.2247 to 1 > ^ float perma- 
nently with the axis vertical, if in a less proportion, it will 
overset from that position. 
Suppose a parabolic conoid CEDK (fig. 24.) of given di- 
mensions and specific gravity, should be placed on the surface 
of a fluid with the vertex downward, and the axis vertical ; to 
ascertain the limits (depending on the length of the axis, the 
parameter of the parabola from which the conoid is formed, and 
the specific gravity,) which separate the cases in which the solid 
* Page 48. 
