Sorry, real work interfered. Something about working full time and taking 20 credits to finally finish school... I left all my work at home, too. I got pretty close to an expression, but it includes an expression for the angle of wrap.

Cicadashell: Your expression for alpha doesn't seem to account for the fact that the radius that touches the "end" of the chain as it leaves the top/bottom of the chainring/cog isn't strictly at the pi/2 and 3*pi/2 position. Specifically, I expect the denominator in that expression to be strictly less than d (or equal to d if the rings are the same size...)

I suggest you draw a diagram with drastically different sized cogs very close together, that way you can see where exactly (in general...) the tangent lines touch the two circles. That was how the problem began to unravel for me, until I had to get back to real engineering work.