Is there an intuitive explanation for the tax incidence formula from elasticity? You're probably familiar with the formula for tax incidence (in a standard Principles framework) from elasticity. Specifically, that the consumer share is

$$\frac{\varepsilon_S}{\varepsilon_S+|\varepsilon_D|}$$
and the producer share is
$$\frac{|\varepsilon_D|}{\varepsilon_S+|\varepsilon_D|}$$

I had always assumed that this was just a byproduct of linear supply/demand and therefore not of much interest, but just now have proven to myself that it's not with a few nonlinear supply/demand forms that also produce this result. 

Is there an intuitive reason why the tax incidence takes this particular functional form, or is it just a common mathematical happenstance? This isn't the only functional form that satisfies what I think of as the intuitive tenets of the tax incidence solution (consumer + producer shares add to 1, increase in your side's elasticity increases your side's burden). Another form that satisfies this, but is not in fact a calculation of incidence share, is $$\frac{(|\varepsilon_D|/\varepsilon_S)}{(|\varepsilon_D|/\varepsilon_S)+(\varepsilon_S/|\varepsilon_D|)}$$

I suppose an alternative explanation is that the counterexamples I happened to test this with satisfy some condition necessary to have those be the shares, and it's not actually universal.
Ah, never mind, got there.

Both supply and demand share the same change in quantity as a result of the tax, so

$$\Delta Q_S = \Delta Q_D$$

The amount by which the quantities changed is the derivative of that quantity wrt price, times the change in price that side experienced

$$\frac{\partial Q_S}{\partial P_S}\Delta P_S = \frac{\partial Q_D}{\partial P_D}\Delta P_D$$

Multiply both sides by the equilibrium P/Q

$$\frac{\partial Q_S}{\partial P_S}(\frac{P^*}{Q^*})\Delta P_S = \frac{\partial Q_D}{\partial P_D}(\frac{P^*}{Q^*})\Delta P_D$$

which of course is the formula for elasticity at the equilibrium

$$\varepsilon_S \Delta P_S = \varepsilon_D \Delta P_D$$
$$\frac{\varepsilon_S}{\varepsilon_D}\Delta P_S = \Delta P_D$$

So then the producer's share is 

$$Incidence_S=\frac{P^*-P_S}{T}=\frac{|\Delta P_S|}{|\Delta P_S|+|\Delta P_D|} = \frac{|\Delta P_S|}{|\Delta P_S|+|\frac{\varepsilon_S}{\varepsilon_D}\Delta P_S|}$$

the price change cancels out and you get producer's share as

$$Incidence_S=\frac{1}{1+|\frac{\varepsilon_S}{\varepsilon_D}|}=\frac{|\varepsilon_D|}{|\varepsilon_D|+\varepsilon_S}$$

So it all comes from the fact that this is the unique formula that produces the price-change ratios that generate identical quantity changes in both sides of the market. Probably not any intuition that can come down to a principles course level but makes a lot more sense than it did before!