of floating Bodies , and the Stability of Ships. 
103 
*= = 'jp+jii±^ 57^-.°^, . consequent . 
1 y, restoring the value of m = 1 — n, 1 — s/ n 
= llL±lil^f and therefore v/« = 
5 c a 
2 6a — 15 p ± 6 ‘S 2. a x ^ 8 a — 15/) 
50a 
Various inferences follow from this determination. In the 
first place, although the object of the preceding investigation 
was, to find a single value only of the specific gravity, which 
would cause the solid to float permanently with the extremity 
of the base coincident with the fluid's surface, yet by the re- 
sult it appears, that there are two values of the specific gravity 
which will answer this condition under a certain limitation, 
which is also discovered by the solution ; this is, that the axis 
( a ) shall be to the parameter ( p ) in a proportion greater than 
that of 15 to 8 ; for if that proportion should be less, 8 a will 
be less than i$p ; in which case the quantity \/ 8 a — i$p 
becomes impossible. From which circumstance it may be in- 
ferred, that whenever the axis is to the parameter in a less 
proportion than of 15 to 8, the solid will float permanently on 
the fluid with the whole of the base extant above the fluid's 
surface, whatever may be the specific gravity of the solid. 
This limit is precisely the same with that which is demon- 
strated by Archimedes, in the second book of his tract, 
intituled de us quce in humido vehuntur, prop. vi. When the 
axis bears a greater proportion to the parameter than that 
of 15 : 8, the solid will float either with the base entirely out 
of the fluid, or partly immersed under it, according to the 
specific gravity. Having given the axis a in a greater pro- 
portion to the parameter p than 13 to 8, by making the 
