It's a Flyer...or is it a Flier?

> I'm working on an invoice for a client. I created 2 email flyers (or is it fliers?) Seems like it's one of the more inconsistently-spelled words I come across. I've got a client whose been in advertising and marketing for more years than I've been in design, and he spells it "flier". Others I have worked with insist it's spelled  "flyer". 

> To me, it seems like "flyer" is better. Why? Well, if you fly a lot, you are a  frequent "flier"—right? So if "flier" is the correct spelling in that instance, then the spelling "flyer" seems more appropriate for a very different use of the word—such as a single page design which communicates an event, service, product, or activity. Without using any style guide or dictionary, that's just what seems right to me.

> Hoever logical my own reasoning is (you agree, right?), apparently both are technically correct. I ran across this article that gives credence to both. What do you prefer?