According to the internal revenue service, [federal estimated tax payments are due on this schedule](http://www.irs.gov/publications/p505/ch02.html#d0e7559):

 - April 15
 - June 15
 - September 15
 - January 15 (of the next year)

June is only two months after April. I would expect estimated tax payments to be evenly spaced three months apart.  Why are estimated taxes for the second and third quarter due a month earlier than I would expect?  (I would expect July and October.)

I usually bill my clients at the end of the month.  In June, taxes are due for the 3rd quarter, but I haven't even received the June payments yet.  Do I owe taxes at that time on the money that I expect to receive that quarter, but which hasn't actually come in yet?