I have exactly the same problem. After a while it's difficult even to type. Even if I do only word processing with MS Word and some Internet with Safari. (No video!)
So I wonder, if the MBA ssd 2nd gen was a notebook with an e-paper screen, (that means no back-light at least, without talking about the minimum power requirements), Atom cpu, no battery built in, and also no fan, then it would be a machine with very low or even no heat emissions? (And no fan noise!)
And another question: What is the main heat responsible (internal) piece of the MBA? The battery? The power charger? The cpu? The graphics cpu? The back-light lamp?