One lonely night I decided to crank

One lonely night I decided to crank up my computer and
grind out just what was happening. Since the balloons we fly are
practically spherical, I started with a perfectly spherical shape
(the spherical shape makes the math ALOT easier!).
I also assumed a linear increase in pressure from the throat to the crown.
To begin with, I calculated the sums of the vertical components of force at each level of the envelope. This was accomplished by
breaking the sphere up into many narrow bands similar to the 2
shown in figure 8 and using a little trigonometry