of floating Bodies , and the Stability of Ships. 93 
fluent of KlT x 2 = 3?r r 4 ; and because PQ = In, and the area 
of the circle AIBHSA is nr, the volume of the part immersed 
V is = 7 rr 1 In; moreover GP = -j ; and OP == ; wherefore 
l In j v • fluent of KH x« 
GO = = d: and since the — — — 
2 12 V 
3 it r 4 
127 rr^ln ’ 
making fluen ^ °[ 2 y— = d, in order to obtain the limit or li- 
mits which separate the cases of permanent and unstable equi- 
librium, we obtain the equation = 1 ~ ^ x - or -^r = 72 
— w 2 ; ra 2 — « = ; or if 2r is put = b = the diameter of 
the base, n — n — and n = — =±= 
If therefore the diameter of the base bears to the axis a 
greater proportion than that of s/ 2 to 1, no value can be given 
to the solid's specific gravity, which will cause it to float in a 
state of insensible equilibrium ; or in other words, there is no 
specific gravity separating the cases in which the cylinder will 
float permanently, from those in which it will overset when the 
— Fluent of r x — z % z 2, i : 
arc H S 
z 1 /v r — z 2 2z 4 /r^ — z 1 - , r 4 
q- x V — 7^ T" x ^ *■ 
This quantity ought to be — o, when z — o-> wherefore the entire fluent of 
y/r z z 2 2 z 4 / r L _ z 2 r 4 
“v -7 x “V — ~ + T 
arc HS wr 4 HS 7 r 
X 7-, because the arc ~ — when z — o, or SH — SB ; when z ~ 
r 2 
16 
r, this fluent, that is, the fluent of r z — z i Y z while z increases from o to r is ~ 
