balloon at any point z may be expressed as:
$$
d _ {z} = 2 \left[ \left(\frac {D}{2}\right) ^ {2} - \left(\Delta z - \frac {D}{2}\right) ^ {2} \right] ^ {1 / 2}
$$
$$
= 2 \left[ D \Delta z - \Delta z ^ {2} \right] ^ {1 / 2}
$$
Therefore, maximum allowable balloon pressure at any plane z-z will be:
$$
\Delta p _ {z} = \frac {4 S _ {f} t}{2 \left(D \Delta z - \Delta z ^ {2}\right) ^ {1 / 2}} \mathrm {p s i}
$$
Equation (5) may be plotted in terms of bursting pressure and $ \Delta z $ for any given diameter balloon. A straight line through the origin and tangent to the plot of Equation (5) will indicate the maximum allowable $ (dp/dz)_{air} $ (1-B) for any given diameter balloon. Comparing the maximum allowable $ (dp/dz)_{air} $ with a chart of altitude vs. pressure in the atmosphere will indicate the minimum altitude at which the balloon can be allowed to be full. From an altitude-buoyancy table for any given diameter balloon, the maximum allowable buoyancy, or maximum allowable gas inflation can be obtained.
Figure 26 is a plot of equations (3) and (5) for .001" polyethylene （ $ S_{f}=\frac{900}{2} $ psi ）balloons of 20'，30' and 70' diameters.
Fig. 25. Relationship d/ $ \Delta Z $ , for balloon.
Fig. 26.
Graph of equations (3) and (5).