BackI noted in "part 1" that after 31st December 2009 the next eclipsed Blue Moons would be on 31st January 2018, 31st December 2028 & 31st January 2037. Note the absolutely exact 19yr interval between each December date and between each January date - this is another example of the Metonic cycle noted previously. The cycle only works exactly if there are 4 leap years in the 19yr interval, however - there would have been a further eclipsed Blue Moon on 31st December 2047 but, due to there being only 3 leap years in this 19yr interval rather than 4, things get 1day out and so the (total) eclipse is actually on 1st January 2048. Also note the intervals of 8yrs 1mth (Dec to Jan) and 10yrs 11mths (Jan to Dec) which add up to one Metonic cycle. These intervals happen because each of them is equal to an exact number of lunations: 8yrs 1mth is 100 lunations, 10yrs 11mths is 135 lunations. As with the full Metonic cycle, this means that the Moon will be at the same phase after these intervals. Not only that, but each interval is also (nearly!) equal to an exact number of "node to node" lunar cycles, plus a half: 108.52 and 146.52 respectively. As explained in my articles on eclipses, the nodes are the [two] points at which the Moon's orbit crosses the plane of the Earth's orbit - more critically, they are the points near to which all eclipses must occur. An extra half orbit will just move the point of interest from one node to the other, so isn't really important, and the value does not have to be exactly 0.5 (or zero, of course) for an eclipse to happen - a slight error is acceptable.
Taking these two facts together, after each of these intervals not only will the Moon be full on the same calendar date plus or minus one month, which will make it very likely to be a Blue Moon again, but if it was close to a node (i.e. eclipsed) the first time it will be reasonably close to the other node the second time and thus will probably also be eclipsed again. This explains the periodic nature of eclipsed Blue Moons. The long gap is caused by the fact that because the "extra half" is actually quite a bit larger than 0.5 the Moon quickly gets too far from a node for an eclipse to happen. There is then quite a time to wait before the alignments correct themselves again - in fact, during this time there are several instances of the first Full Moon in a Blue-Moon month being the one that is eclipsed.
The difference between the 235 lunations and the 255 "node-to-node" cycles in a full Metonic period is 0.57 days. In this time the Moon will travel 7.57deg away from the node which, due to the tilt of its orbit, will shift it 0.67deg higher or lower than the eclipse in the previous Metonic cycle. Given that the width of the Earth's (umbral) shadow is about 1.45deg, that means it only takes fractionally more than 2 shifts for the eclipse position to move right through the shadow i.e. there will be at most 3 eclipses in a Metonic-period sequence of eclipsed Blue Moons, as found above. The overall sequence can be doubled by taking account of the 8yr and 11yr cycles of course, as these are just formed from interlinked Metonic cycles, one at each node. Each set of 3 eclipses will either be small partial - total - total (as at present) or large partial - total - large partial (as on, for example, 31st March 2238, 30th March 2257 & 30th March 2276) so the "relative rarity" of partially-eclipsed Blue Moons will change as the centuries go by.
The Metonic period has one more quirk, and that relates to the comment made on the main page that the first January Full Moon of the Jan/Mar pair in 2018 was very close to perigee. Looking forward 19yrs to 2037 we find another Jan/Mar pair (as might be expected), when the first March Full Moon is also very close to perigee. The actual differences are 4hrs 29min in 2018 and 4hrs 40min in 2037 (both after perigee). This is, of course, not a coincidence! In the same way that the Metonic period is 235 lunations and 255 "node-to-node" cycles it is also 251.85 perigee-to-perigee cycles. While this is not a whole number, add 59 days to the Metonic period and we do get a whole number - exactly 254 perigee-to-perigee cycles. And what is 59 days? Why - it's just the minimum interval between Blue Moon pairs! So, if either of the January Full Moons of a Jan/Mar pair is at perigee, the corresponding March Full Moon in the pair one Metonic period forward will also be at perigee, just as we found [The conclusion holds for any orbital distance of course, not just perigee, but only perigee and apogee are particularly "interesting" distances].
Unfortunately, because the 59 day interval does not divide evenly into 365, once an "interesting" pairing has happened it will not happen again for a long time. To find out how long, I did a manual search for interesting pairings from 2018 onwards. There were a number of "near misses" on the way, but the next genuine case will not be until 2390/2409! This is an exact duplicate of the 2018/2037 pairing, but Full Moon will be before perigee by 8hrs 25mins rather than after it by [an average of] 4hrs 35mins. The near-misses were 2124/2143, when because 2124 is a leap year the second pair of Full Moons will fall on 29th February and 30th March rather than 1st and 31st March; 2200/2219, when in 2219 the second Full Moon of the second pair falls on 1st April rather than 31st March, but by a mere 2 minutes (!), and 2257/2276, when in 2276 the second Full Moon of the first pair falls on 1st February rather than 31st January, but only by 21 minutes. The 2200/2219 pair was the only case I found when Full Moon was close to apogee rather than perigee (16hrs 10mins [average] after). It was also the only case where the Full Moon in question was the second of the pair (i.e. the Blue Moon itself).
Note that the difference between both 2124 & 2257 and 2257 & 2390 is 133yrs: this is not only 7 Metonic periods (1645 lunations) but also 1763 perigee-to-perigee cycles, explaining the repetition of the [almost] "interesting" perigee date pairs above. There is not an interesting pairing in 1991 or 2523 (133yrs before or after) though, because the difference between 1645 lunations and 1763 perigee cycles is 0.85 of a day, enough to upset the lunation/perigee correspondence after only a couple of repetitions. Also, although 133yrs is equal to 7 Metonic periods, the period itself is actually 2hrs longer than 19yrs and so lunations "gain" more than half a day relative to the calendar per repetition. This means that cycles of Metonic periods do not themselves run for ever, explaining the fact that the difference between 2018 and 2124 is not an exact number of Metonic periods. The two genuine interesting pairings I found are thus not related but belong to different Metonic period series, in the same way as eclipses belong to different Saros series. However, as in the case of Saros series, calculating how many Metonic series are running at the same time is not an easy task, so I think I shall leave my Blue Moon analysis there.
Fascinating stuff though!