70 Mr. Atwood’s Propositions determining the Positions 
From the equation GZ = — -L 2 L£ — C JL it i s inferred, 
A 12 acn 2 
that when the specific gravity of the solid is of very small 
value in respect to that of the fluid, because — a s must in this 
1 7 12 acn 
case be necessarily greater than — — the solid will float 
permanently with the line SL vertical, that is, with the flat 
surface EF parallel to the horizon. Secondly, the specific 
gravity .21183 causing the solid to float in the insensible equi- 
librium, is the limit at which the solid ceases to float with 
stability ; if therefore the specific gravity is increased beyond 
.21183, and the solid is placed on the fluid with the flat sur- 
face upward, the equilibrium thus formed will be that of in- 
stability, from which the solid will be deflected into some 
other position in which the equilibrium is permanent. While 
the specific gravity is augmented from .211 to .788, the insta- 
bility increases at first, but admits of a maximum, which is 
found by putting the least increment of the quantity ~~~ — 
5 x c — cn _ ^ considering n as a variable quantity, and mak- 
ing a == c ; in which case n appears to be equal to ”7==. If the 
value of the specific gravity is increased beyond the 
instability becomes less, and at last vanishes when the spe- 
cific gravity is at its second limit = .78868 : whatever va- 
lue is given to the specific gravity between .78868 and 1, the 
solid will float permanently with the line SL vertical, or with 
its flat surface horizontal. 
These cases arise from assuming the height of the parallelo- 
pined SL, in a greater proportion to its base CD than that of 
