the equation we find the condition

/ + ^(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

The least root of this equation is

z = 3.83171.

If h is the height to which the liquid will rise in a capillary tube
of unit radius, then the diameter of the largest orifice is
____ ___
2a = 3.83171 \/(2h) = 5.4188 \/(h).

Duprez found from his experiments
___
2a = 5.485 \/(h).

[The above theory may be well illustrated by a lecture experiment. A
thin-walled glass tube of internal diameter equal to 14-1/2 mm. is
ground true at the lower end. The upper end is contracted and is fitted
with a rubber tube under the control of a pinch-cock. Water is sucked up
from a vessel of moderate size, the rubber is nipped, and by a quick
m