hi!
you have the equation.  let's ignore the third (gravitational) term.

i hope this diagram clarifies ( " * " outlines a 3-section tube, " + " outlines the pistons.):
[code]
  *  *  *  *  *  *  *  *  *
                 +                 *
                 +                     *
                 +                         *  *  *  *  *  *  *  *  *  *  *  *  *
                 +                                                    +
                 +       gas enclosed                            +
                 +                                                    +
                 +                         *  *  *  *  *  *  *  *  *  *  *  *  *
                 +                     *
                 +                 *
  *  *  *  *  *  *  *  *  *
[/code]
basically, we're changing the shape of an ideal gas, keeping the volume the same.  but we're looking at the process while it is happening, not a "before" and "after" situation.
it's not clear that we can ignore the temperature.
nor is it clear that the pressure on one piston is the same as the pressure on the wall of its corresponding straight tube section.
nor is it clear that the density in the straight tube sections is the same.

i thought a simulation might help see what's happening.  i have no idea if anyone has done this.

thanks again.