96 
ARCHIMEDES 
where k is the axis of the segment of the paraboloid cut off by 
the surface of the fluid*) 
V. Prop. 10 investigates the positions of stability in the cases 
where h/\p> 15/4, the base is entirely above the surface, and 
s has values lying between five pairs of ratios respectively. 
Only in the case where s is not less than (h — f p) 2 /h 2 is the 
position of stability that in which the axis is vertical. 
BAB 1 is a section of the paraboloid through the axis AM. 
G is a point on AM such that AC — 2 CM, K is a point on CA 
such that AM-.CK — 15:4. CO is measured along CA such 
that CO = \p, and B is a point on AM such that AIR = | CO. 
A 2 is the point in which the perpendicular to AM from K 
meets AB, and is the middle point of AB. BA 2 B 2 , BA^AI 
are parabolic segments on A 2 M. 2 , A a M :i (parallel to AM) as axes 
and similar to the original segment. (The parabola BA.,B tl 
is proved to pass through G by using the above relation 
AAI:GK =15:4 and applying Prop. 4 of the Quadrature of 
the Parabola) The perpendicular to AM from 0 meets the 
parabola BA 2 B 2 in two points P 2 , Q 2 , and straight lines 
through these points parallel to AAI meet the other para 
bolas inP l; Q 1 and P 3 , Q 3 respectively. P X T and Q, U are 
tangents to the original parabola meeting the axis MA pro 
duced in T, U. Then 
(i) if s is not less than AR 2 : AM 2 or (h-%p) 2 :h 2 , there is 
stable equilibrium when AM is vertical; 
(ii) if s< 
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(iv) if s > 
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(v) if s< 
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(I) 
(II) 
1523.2