I was asked about this series of posts from a few years ago. I've repaired the links.
A series of posts exploring the 'Ashtanga Rishi Series' mentioned at the end of Nancy Gilgoff's Article (see link below) and outlined in a reply by David Willams on his forum below ( the headings in block capitals are mine.
I'll be starting each of these posts with this same introduction/reminder of the context.
'Originally there were five series: Primary, Intermediate, Advanced A, Advanced B, and the fifth was the “rishi” series.
Nancy Gilgoff 'Yoga as it was'
Ashtanga Rishi Approach
'...Doing a practice of 10 postures for up to 50 breaths is a method of preparing for "advanced series" after one has learned 1st and 2nd. It can be done once or twice a week. One does the "salutations" and then starts going thru the series, holding each posture for as long as comfortably possible. Notice which postures could be held for 50 breaths. The next time you practice this way, the postures which you could hold for 50 are omitted and new ones are added at the end. One gradually works thru the series, dropping and adding asanas, still doing 10 asanas per session. I have gone all the way thru 1st and 2nd this way several times over the years and have found it beneficial.
Ashtanga Rishi Series
'Then, once one has mastered all of the asanas, one can practice "the rishi series", the most advanced practice. One does the 10 postures that one intuits will be the most beneficial and appropriate for that day, holding each posture for up to 50 comfortable breaths'.
Ashtanga Rishi Blog post series
Ashtanga Rishi Approach, first day Paschimottanasana to Janu sirsasana A
Ashtanga Rishi Approach, second day Janu Sirsasana B to Navasana
Ashtanga Rishi Approach, third day Bhuja pindasana to badha konasana
Ashtanga Rishi Approach, fourth-day Upavishta konasana to Supta bandhasana
Ashtanga Rishi Approach, fifth day Pasasana to Kapotasana
Ashtanga Rishi Approach, sixth-day Supta vajrasana to Ardha Matsyendrasana
Ashtanga Rishi Approach, Seventh Day Eka pada sirsasana to Tittibhasana C
Ashtanga Rishi Approach, Eighth day to seven headstands
Rishi series made from asana named after Rishi An alternative take on the Rishi series
See also Srivatsa Ramaswami's December 2012 newsletter for more on the Rishis
UPDATE: 25/03/16 It strikes me that there is a flaw to this project, while exploring the longer stays of fifty breaths in primary and intermediate asana I've tended to stick the regular current rate of breath, a couple of seconds inhalation and the same for exhalation. However, Krishnamacharya in his 1934 Mysore book Yoga Makaranda (and also Pattabhi Jois in several interviews) stressed the importance of long slow inhalations and exhalations, 10,15 even 20 seconds for each. Krishnamacharya also stressed the employment of Kumbhaka (retaining the breath in or out). Fifty breaths at 10 seconds each for inhalation and exhalation plus a 5-second kumbhaka equals 25 seconds per breath, that's twenty minutes for one asana., three hours and twenty minutes for 10 asanas practiced in this manner.
However Pattabhi Jois is quoted as saying "up to 50 breaths", so while we may well choose to stay in one asana for twenty minutes taking fifty long slow breaths we might also choose to stay for twenty-five slow breaths bringing the practice of 10 asanas well under two hours.
