/ + ^(ve) stable.
(p^2 + q^2)T - ([rho] - [sigma])g = < 0 neutral.
\ - ^(ve) unstable.
That the surface may coincide with the edge of the orifice, which is
a rectangle, whose sides are a and b, we must have
pa = m[pi], qb = n[pi],
when m and n are integral numbers. Also, if m and n are both unity,
the displacement will be entirely positive, and the volume of the
liquid will not be constant. That the volume may be constant, either n
or m must be an even number. We have, therefore, to consider the
conditions under which
/ m^2 n^2\
[pi]^2 ( --- + --- )T - ([rho] - [sigma])g
\ a^2 b^2/
cannot be made negative. Under these conditions the equilibrium is
stable for all small displacements of the surface. The smallest
admissible value of
m^2 n^2 4 1
--- + --- is --- + ---,
a^2 b^2 a^2 b^2
where a is the longer side of the rectangle. Hence the condition of
stability is that
/ 4 1 \
[pi]^2 ( --- + --- )T - ([rho] - [sigma])g
\a^2 + b^2/
is a positive quantity. When the breadth b is less than
__________________
/ [pi]^2 T
/ ------------------
\/ ([rho] - [sigma])g
the length a may be unlimited.
When the orifice is circular of radius a, the limiting value of a is
_______
/ T
/ ------- z, where z is the least root of the equation
\/ g [rho]
2 z^2 z^4 z^6
--- J1(z) = 1 - --- + ------- + ----------- + &c., = 0.
z 2.4 2.4^2.6 2.4^2.6^2.8