Return to main pageWe already know from a previous calculation that, in the range of the series whose data were used for the analysis, the distance between two flat sections is 1.40 gamma units. As 1.40 is quite a bit less than the gamma limit for an eclipse to happen at all [1.57], as long as the graph is reasonably symmetric about the gamma = 0 line it will be easily possible to fit in 3 flat regions but only 2 steep regions. A series with the curve in this position will have a larger number of eclipses than usual because the extra numbers added by the flat regions will exceed those subtracted by the steep sections.
As the curve shifts, either the top or bottom plateau will move towards the +/-1.57 limits. If we start with a perfectly central curve, the leeway before a plateau crosses a limit is clearly (1.57-1.40). The total leeway between a plateau just coming within one limit and the other just moving outside the second limit will thus be twice this amount which, at 0.074 gamma units per shift (also calculated on the previous maths page), will take 2 x (1.57-1.40)/0.074 = 4.6 shifts i.e. there will be an average of 4 or 5 consecutive Saros series with a larger number of eclipses. In practice, this is something of an overstatement because, while in the six millennia from 3000BC to 3000AD the initial value of gamma for a series varied from 1.4571 to 1.5674, only 11% of the values exceeded 1.55. With this value the maximum number of consecutive "long" series will be 4, as shown by the actual data.
At the point a plateau crosses a 1.57 limit there will again be just two plateaux and two steep sections on the graph and so the total number of eclipses will decrease. This decrease will be quite abrupt, due to the flatness of the plateau, and so the total number will quickly fall from a high value to a lower one, with few intermediate values (as observed). However, with continuing shifts, another plateau will appear within the other limit and so the total number will jump up again. As previously calculated, for series numbered around 10 to 55 this will take 1.40/0.074 = 19 shifts.
Due to the magnitude of the aphelion/perihelion effect, the number of eclipses on a plateau is about 12 and so this will be the difference between the minimum and maximum numbers in a phase of a series. The latitude allowed between a plateau appearing at one end and another moving off the other end accounts for the width of the maximum part of the "total number of eclipses" graph and also the horizontal offset of the "sawtooth" at the top and bottom edges of the Saros-Inex Panorama.