The solution for this equation can be achieved analytically if
both the pressure effects, 
, and the gravitational
effects, 
, are small. In this event, equation
11 reveals that the velocity U is 1 and the force
balance equation (10 simplifies to
The solution presented here will now take a different course than the one that Taylor did and follow the work of Boussinesq to find the radius of the bell as a function of the distance away from the jet Z.
Define the radius of the bell as a function of the distance away
from the jet, 
. Some geometric facts that are very
useful:
Beginning with equation 15, taking the derivative with respect to Z yields:
Substituting equation 12 and using equations 13 and 14 results in a familiar final ordinary differential equation for R(Z)
with the boundary conditions that the initial slope be given by
and the initial radius be zero. The solution of the ODE
is a catenary of the form
Applying the boundary condition regarding the initial slope
yields that 
. Applying the zero
initial radius condition requires that 
such
that
The form of this equation is quite straight forward and easily verifiable through experiments. Taylor used a horizontal jet to verify his equation, which after inspection, is essentially the same as equation 19. The choice of a horizontal jet is quite interesting because it allows him to say that the gravity term is not that important because of the symmetry he obtains in his experiments. His selection of the orifice size made that quite true (proof by selection, so to speak).
