9%, Mr. Atwood's Propositions deter?nining the Positions 
the fluid being — 1 ; let the centre of gravity be G ; the 
centre of gravity of the immersed part = O ; GO = d ; let 
AIBHSA represent a circular section of the cylinder coincident 
with the fluid's surface ; draw any diameter IS ; and a diame- 
ter AB perpendicular to IS ; let the axis passing through the 
centre of gravity round which the cylinder is moveable be pa- 
rallel to IS ; through any point W of the diameter IS draw the 
ordinate KW perpendicular to IS, and produce KW till it in- 
tersects the circle in the point H ; make QW = z ; NP = l ; 
or = 3.141 59. It appears from page 66 that the solid will 
float permanently in the given position of equilibrium 
with the axis vertical, when the fluent of - KH - - X ? - is 
12.V 
greater than d, the letter V signifying the volume im- 
mersed under the fluid's surface ; it is also shewn in page 
66, that if d is greater than fluent H x g , the equilibrium 
will be unstable ; when the fluent of = d, the equi- 
librium will be the limit separating the cases in which the 
solid floats with stability from those in which it is momentary 
and unstable. To ascertain the limit in the present case it is 
necessary to find the fluent of — 77^— • Since QS = r, and 
OW = 38, WH = s/p—z, KH = 2 X vV — 38% and 
K H z = 8x r z — ■ x z ; the fluent of which quantity, while % 
increases from c to r is — and for bbth semicircles, the 
3 — — — . <§ 
* Fluent of r 1 k — fluent of r z x r % — z x z — fluent of r 2, — z z z 1 x. 
Fluent of r 2 x ~r 2 "— z* x a = r z x the area QBHW. (fig. 23.) 
