Q:  Why do we skip over the FIRST following
      eclipse in order to measure the period? 


          Let's call the two stars A and B.
          Here's what we see as they orbit around
          each other:

          First,

              A passes in front of B   ---    primary eclipse

          then

              B passes in front of A   ---    secondary eclipse

          and finally

              A passes in front of B   ---    primary eclipse again


          The period is the time between the primary eclipse
          and the next primary eclipse.  We must skip over
          the secondary eclipse, which occurs after half a period.