The effect was discovered by Max Wertheimer, one of the founders of Gestalt psychology, and first published as: Experimentelle Studien über das Sehen von Bewegung. Zeitschrift für Psychologie 1912, 61, 161-265, 1912. (see a list of Wertheimer's publications, and much more, in this excellent website).
Now, this simple effect as been hugely influential and useful for theories of movement perception, and vision and cognition in general. There are hundreds of variations that you can think of (and that have been most probably already studied, make no mistake), as well as thousands of papers on the phenomenon. It has become quite hard to do anything original in this field, at least since Shiffrar and Freyd's excellent study on "apparent motion of the human body" , which I plan to discuss someday in the context of phantom limbs. One of the questions raised by the Phi phenomenon is of course its mechanisms: is it primarily a bottom-up thing, based on mindless "motion detectors", or is it modulated by top-down factors, like beliefs, knowledge, expectancies, etc.?
Well, I now see that the journal Perception has a short and intriguing piece  that might bring something new to the topic. Whereas Shiffrar and Freyd focused on the perception of apparent motion for other person's body parts, Ekroll and Scherzer, the authors of this new paper, propose a demonstration of the Phi illusion for one's own body. Here's their abstract:
We describe how illusory apparent motion of one’s own fingers and other body parts can easily be experienced in spite of grossly conflicting proprioceptive signals. This simple illusion may be used to shed light on the crossmodal integration of visual and proprioceptive signals. Our preliminary observations suggest that the visual and proprioceptive signals do not merge into a common crossmodal percept. Instead, mutually inconsistent visual and bodily sensations are experienced simultaneously.So, how do you induce apparent motion using your own body? Here's what the authors say:
To experience the phenomenon, hold one of your hands with the fingers pointingTaken from Perception's website, the figures below of a hairy hand illustrates the idea. From my experience, you have to do this rather fast, and it's not so easy to be consistent (it is very important that you always see the same "amount" of finger sequentially, otherwise it seems to me that any effect there might be is destroyed).
away from you (...). Only the index finger should be stretched out and visible from your point of view. Then lower the index finger and lift the middle one instead. Keep the hand steady while alternating swiftly and cyclically between these two finger positions, ie move the two fingers up and down in counterphase. When this is done swiftly enough, you will have the visual impression of a single finger tip moving horizontally instead of two fingers moving up and down. (...) One may initially find it difficult to move the fingers swiftly enough while preserving the appropriate coordination of the two fingers. (...) The effect seems to be more easily experienced
when the fingers are viewed monocularly with a half-closed eye. (...) Furthermore, viewing the fingers in peripheral vision, say, by looking at the wrist instead of directly at the fingers, also seems to enhance the effect.
The animation can be seen here. It is really a matter of subjective opinion whether there is a strong feeling of apparent horizontal motion or not while looking at this (I mean, looking at the animation, i.e. not doing it yourself). The authors also recommend trying this with the feet, as in the follwing figures:
The animation can be seen here. This works even less, I think, but try it anyway.
Ok, good. Neither these effects work so well for me, but I accept the general idea (and maybe I should practive a little bit more, while trying I did have brief glimpses of a quite effective phi, but this was very transient). Theoretically, when this works, it should induce the perception of intermediate phantom fingers in between the real fingers, just like phantom dots are created by the brain in between two flickering dots. I'll keep trying.
Well ok, this is all fun, and makes for a very cool classroom demonstration. But one might ask, from a theoretical point of view, what's the point? Well, the authors valiantly present three theoretical perspectives arising from this funny experience:
1) "proprioceptive signals from the hand could have dominated over visual motion signals, making the subject perceive the veridical vertical finger motion, although a second passive observer perceives illusory horizontal motion"Well that's a good idea. The problem is that it will be hard to gather good data from this illusion, as subjects have to do it themselves, and therefore it can hardly be controlled, plus the feeling is by definition subjective, and so forth. What is more, if you want a "second passive observer", then this guy would have to see the fingers from someone else's point of view, hardly a practical experimental setting. Another approach to test hypothesis 1) would be to induce in the fingers some kind of proprioceptive disturbance, to see if this enhances the perception of phi. But I have no idea how to do this without disturbing motor processes. It seems we reached an experimental dead-end. But proprioception is not the only thing getting in the way of vision here. As I said, it is very difficult to produce consistent, identical and repetitive finger movements while looking at them from a specific angle, so the visual information itself is quite unstable, which might be enough to reduce any phi effect.
2) vision may dominate over proprioception, and the active subject may have theSure. I mean, this is sort of trivial, for if the illusion that is described works at all, then it implies that vision does indeed, somehow, dominate over proprioception.
bodily sensation of a horizontally moving finger.
3) "proprioceptive and visual information may fail to merge into a coherent crossmodal percept. In our experience, the latter seems to be the case."I'm not sure what this last hypothesis means. In the author's words: "one experiences a visual illusion of horizontal motion although the fingers are felt to move vertically". What would it feel like if a "coherent crossmodal percept" between proprioception and vision was achieved? How is hypothesis 3) different from hypothesis 2)? If vision dominates over proprioception, well then they have not "merged" appropriately, right? On the other hand, it might be a sort of figure-ground phenomena, akin to what happens during binocular rivalry or the Necker cube: there are times, when doing this illusion, where you perceive either proprioception, or vision, but not both at the same time (and when vision is in the foreground, then phi is experienced). But again, it's close to impossible to actually test this thorougly. It gets also quite confused in the conclusion, where the authors write:
"the proprioceptive and visual signals, which in many other cases of proprioceptive/visual conflict merge into a common percept, fail to do so in the present case. A possible explanation for this would be that the horizontal apparent motion is so discrepant from the vertical motion that would be expected on the basis of the efferent motor commands that the visual percept is more plausibly attributed to an object not belonging to the observer's own body."This they link to the literature on experimental disturbances of agency (of the Jeannerod type). But obviously this goes a step too far. I'm of course speaking on my own experience (or lack thereof) of this illusion, but it certainly does not feel like the moving fingers, whether they seem to move vertically or horizontally, are someone else's! I don't think there's any disturbance of agency or ownership in this illusion, as far as I'm concerned I think this is the wrong approach.
But then I don't really have anything better to propose, and I think overall that the publication is very cool and intriguing.
 Shiffrar M & Freyd JJ (1990). Apparent motion of the human body. Psychological Science, 1 (4): 257-264.
 Ekroll V, Scherzer T R, 2009, "Apparent visual motion of the observer’s own limbs" Perception 38(5) 778 – 780