| π² ( | 4 | + | 1 | ) T − (ρ − σ)g |
| a² | b² |
is a positive quantity. When the breadth b is less than √[π²T/(ρ − σ)g] the length a may be unlimited.
When the orifice is circular of radius a, the limiting value of a is √[(T/gρ)z], where z is the least root of the equation
| 2 | J1(z) = 1 − | z2 | + | z4 | + | z6 | + &c., = 0. |
| z | 2·4 | 2·4²·6 | 2·4²·6²·8 |
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).