Mujitsu and Tairaku's Shakuhachi BBQ

As described in Christopher Blasdel's book 'The Shakuhachi: A manual for learning'
"Sometimes a small circle is written in place of the dot to indicate a downbeat which is held over, and a small triangle written where an upbeat is held over."
I find this definition to be slightly ambiguous, especially to the person who is questioning these marks in the first place.  I was taught that the circle in place of a beat mark indicates an 'empty beat' which means a note is not initiated on that beat.  The previous note may be slightly to completely carried over that empty beat depending on the following beats and notes.  As such, a certain sense is necessary and may best be imparted by your teacher.  The triangle is exactly the same, but falls on the upbeat or left side of the musical line.
Hope that helps.

I think the triangle means play through the beat plus one more half beat, but I am not sure how this affects the following notes.

I think in most instances the triangle increases the "time" of the note preceding it by one half and the next note is often a grace note followed by a note played according to how it is marked.  There are a couple of children songs, the names of which escape me, which have several of the "triangled" notes in succession and helped me get more familiar with this.  I am not much of a musician, so I hope anyone will correct me if I am incorrect.