"prayatna - effort (of life which is breathing)
saithilya - smooth (make it smooth)
samapattibhyam - focusing on it
By making the breath smooth (and long), and by concentration or focussing the mind on the breath, the perfection of the posture is obtained.
Note: Krishnamacharya interprets this sutra differently than other teachers. he gives the correct technical meaning (in this context) fromn prayatna or Jivana prayatna, or effort of life which is breath. he says that it is the breath that should be made smooth and effortless, not the posture. it is not physical; it is the breathing" p55
*The translation and treatment of the sutra is from Patanjali's Yoga Sutras Based on the teaching of Srivatsa Ramaswami by Pamela Hoxsey and taught on the Vinyasa Krama teacher training course that I attended in 2010. This is relevant because Ramaswami spent over thirty years, from the 1950's to the 1980's, as Krishnamacharya's student.
"Going back to my notes on Yoga Sâtra classes with my guru, I found a very interesting interpretation of the sâtra, Prayatna-ùaithilya anantasamápattibhyám. The word prayatna, very commonly used in India, basically means “effort.” úaithilya indicates “softness.” So Prayatna- ùaithilya could mean “mild effort”; hence you find that many writers on the Yoga Sâtras declare that the way to achieve perfection in a yoga posture is to “ease into the posture effortlessly.” This is easier said than done. There are hundreds of practitioners who cannot relax enough to be able to easily get into a posture like the Lotus, for example. So we have to investigate the meaning of the word prayatna as used by the darùanakáras in those days. Prayatna according to Nyáya, a sibling philosophy to yoga, is a bit involved. Nyáya explains prayatna of three kinds (prayatnaê trividhaê proktam). Two of them are the effort put
in for happiness (pravätti) and the effort to remove unhappiness (nivätti). Every being does this all the time. One set of our efforts is always directed toward achieving happiness and the other toward eradicating unhappiness. But the third type of effort relevant here is the effort of life (jàvana-prayatna). What is effort of life? It is the breath or breathing. Now we can say that prayatna-ùaithilya is to make the breath smooth. Thus in ásana practice according to Vinyása Krama, the breath should be smooth and by implication long (dàrgha).
The other part of the sâtra refers to samápatti, or mental focus. Where or on what should the mental focus be? It is to be on ananta (ananta-samápatti). Now we have to investigate the contextual meaning of the word ananta, translated as “endless” or “limitless,” which many writers equate with infinity. So some schools tend to say that while practicing ásanas, one should focus the attention on infinity, which is inappropriate— and impossible, at least for the vast majority of yogàs. Ananta also refers to the serpent, Ädiùeüa, whose incarnation Patañjali is believed to be. So some schools suggest that one should focus on a mental image of Ädiùeüa or Patañjali. It may be possible, but it is uncomfortable to think that Patañjali would write that one should focus on his form for the success of ásana practice. So what might ananta symbolically signify? The word ananta can be considered to be derived from the root, “ana”—to breathe (ana ùváse). We are all familiar with the group of words práóa, apána, vyána, etc., names of the five práóas derived from the root “ana.” So in the sâtra, ananta could mean “breath”; ananta-samápatti is then translated as “focusing the mind on the breath.” In fact Ananta, or the serpent king, is associated with air. Mythologically the cobra is associated with air; there is a common mythological belief that cobras live on air. If you look at the icon of Naôarája (the dancing úiva), you will find all five elements of the universe (earth, water, air, fire, and space) represented symbolically in úiva. The matted red hair represents fire, the Gaïgá in his tresses, the water element; the air element is said to be represented by the snake around the lord’s neck. So ananta-samápatti would mean focusing the attention on the breath or práóa.
Thus this sâtra means that while practicing ásana, one should do smooth inhalations and exhalations and focus the attention on the breath. Since Vinyása Krama involves several aesthetic movements into and within yoga postures, to achieve the coordination of movement, breath, and mind, one should synchronize the breath with the movement with the help of the focused mind. By such practice, slowly but surely, the union of mind and body takes place, with the breath acting as the harness.
But why don’t other texts talk about it? There is a saying, “Anuktam anyato gráhyam.” If some details are missing from one text, they should be gathered from other complementary texts. Haôha-yoga-pradàpiká explains a number of ásanas but does not mention breath synchronization and other basic parameters. But Haôha-yoga-pradàpiká proclaims that its instructions are like a prerequisite for the Rája Yoga practice of Patañjali. These two texts are therefore compatible. Thus we can conclude that Patañjali gives the basic parameters of ásana practice (and also of the other aïgas like Práóáyáma), but for details we have to refer to compatible texts like Haôha-yoga-pradàpiká, Yoga- Yájñavalkya and others".
See here for the full article.
The video below is from 2010, the year I was there, mainly an excuse to put a slide show of some of the photo's from the asana element of the course to a backing of Ramaswami chanting. The first chant is a bit quiet, the second is the one I still listen to in savasana every morning.