NavList:

Well done Robin.  You’re way ahead of me mathematically; I was still sketching globes on flat planes waiting for inspiration.  Let me see if I follow your argument.  In getting Cot C in equation 1.1, you’re looking at the actual curves as drawn on the flat chart, so you’re using simple trigonometry.  That would work at the Equator for a gnomonic chart.  I’m not sure it’d work further North unless you’re envisaging some sort of imaginary ‘ideal’ chart.  On the chart in question, I’m not sure you’d still be dealing with a right angled triangle.
Your next equation comes from spherical trig using the ‘four parts rule’.
Your third equation is your second equation inverted and massaged to put it into the same form as equation 1.1.
Your fourth equation should give you the curved lines, one or a pair for each of the chlongs stated.
The fifth equation has x=0, so it must be the straight line up the centre.  
When you say you’re now able to replicate the curves, does that mean you’ve tried putting equations four and five into a function drawing programme to see what comes out?  DaveP
http://fer3.com/arc/m2.aspx/Great-Circle-Sailing-Chart-Stuart-jan-2016-g34260