A history of Greek mathematics - Wilbourhall.org

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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
« has values lying between five pairs of ratios respectively.
Only in the case where s is not less than (h — ^pf/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 AG = 2 CM, K is a point on GA
such that AM-.CK =15:4. CO is measured along GA such
that GO = %p, and R is a point on AM such that MR = §C0.
A 2
is the point in which the perpendicular to AM from K
meets AB, and A 3
is the middle point of AB. BA 2
B 2
, BA
Z
M
are parabolic segments on A 2
M 2
, A 3
M 3
(parallel to AM) as axes
and similar to the original segment. (The parabola BA.,B 2
is proved to pass through G by using the above relation
AM: GK =15:4 and applying Prop. 4 of the Quadrature of
the Parabola.) The perpendicular to AM from meets the
straight lines
parabola BA 2
B 2
in two points P 2
, Q 2
,
and
through these points parallel to AM meet the other parabolas
in P-p Q 1
and P 3
, Q are
3
respectively. P X
T and Q 1
U
tangents to the original parabola meeting the axis MA produced
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