eclipses, one solar and the other lunar, in this eclipse-season. And clearly the reasoning is precisely the same when the Moon is at M2 in the middle of an eclipse-season.

The same holds when the Moon is near to M, or M in the middle of an eclipse-season.

In these cases the two eclipses would not be so important generally as an eclipse occurring in the middle of an eclipseseason. If the Moon were exactly at M2 or M4 the solar eclipse would be central and the lunar eclipse partial;* but even the

It is easy to extend the calculation made above, as to the average duration of eclipse-seasons, to determine how long the opening of the Moon's orbit, as seen from the Sun, continues small enough for a particular order of eclipses to take place. In the following inquiry it will be understood that only mean values are considered. Farther on, the effect of the eccentricity of the lunar and terrestrial orbits, and other circumstances, in modifying the limits of eclipse-seasons, will be dealt with; but only in a general way.

For a central solar eclipse to take place—that is, either an annular or total solar eclipse-it is necessary that the opening of the Moon's orbit, as seen from the Sun's centre, should be so small when new Moon occurs as to intersect the Earth's disc. For clearly in this case a line from the Sun's centre to the Moon's, at the time of new Moon, will fall on the Earth's disc, and where this line meets the Earth there must needs be a central solar eclipse; whereas, if the line joining the centres of the Sun and Moon did not meet the Earth there could be no central eclipse. We have, then, the greatest opening of the Moon's orbit as before, about ths of the long diameter of the orbit, while the Earth's diameter is, as we know, but about a sixtieth part of this long diameter (or ths of the greatest opening). So that the time required to open the orbit to this extent bears to 867 days (half of 173·3, that is) the ratio which an angle whose sine is bears to a right angle. But the sine of an angle of 103° is about, and such an angle is about ths of a right angle. Hence the required time is about 10 days; and, counting it on either side of the middle of the eclipse-season, we get 20 days as the duration of what may be called the central-solareclipse-season.

Now the relation here considered has no counterpart among recognised orders of lunar eclipses, since the fact that a line through the centres of the Sun and Moon at the time of lunar eclipse crossed the Earth would correspond only to the fact that the Sun's centre was concealed from the centre of the Moon's disc, a relation not requiring special consideration.

solar eclipse, though central, would be less important (regarding the whole Earth, than a solar eclipse occurring during the

Let us, however, inquire what are the average limits of the two orders of lunar eclipses which are dealt with by astronomers-viz., total and partial lunar eclipses.

Total eclipses of the Moon are determined by the consideration that all parts of the Sun are concealed from the whole of the Moon's dise (always setting on one side the effects of atmospheric refraction), and, therefore, the opening of the lunar orbit must be less than the Earth's apparent diameter, as seen from the Sun, by the sum of the two quantities, which, in the inquiry in the text, were added to that diameter. We must, therefore, diminish the Earth's apparent diameter as seen from the Sun by about one-half; so that, instead of getting the angle whose sine is about, as before, we get the angle whose sine is but one-half of this, or an angle of about 5 degrees; and the corresponding proportion of 867 days (or about 5 days) is one-half of the total-lunar-eclipse-season, whose full length is therefore about 104 days.

Lastly, for the occurrence of lunar eclipses generally, we must have the opening of the lunar orbit such that from some part of the Moon's dise the whole of the Sun is concealed, and therefore, on the assumption hitherto made (which is not far from the truth), that the average value of the Sun's apparent diameter is equal to the Moon's, we need neither increase nor diminish the Earth's apparent diameter, as seen from the Sun. We therefore get the same results as when we were considering central solar eclipses-namely, 10 days for one-half of the lunareclipse-season, whose full average length is therefore about 20 days.

Now these results enable us to determine the general conditions under which various orders of eclipses will occur during the eclipse

season.

That there may be a central solar eclipse, the middle of the eclipseseason must occur when the Moon is somewhere within the arc m, M1 ~1, such that m, M, and n, M, are each equal to the space traversed by the Moon in about 10 days. That there may be a total lunar eclipse, the middle of the eclipse-season must occur when the Moon is somewhere within the are ng My my, such that n, M, and M, m, are each equal to the space traversed by the Moon in about 5 days. That a partial lunar eclipse may occur, the middle of the eclipse-season must take place when the Moon is somewhere on the arc ng M3 m3, such that ng Mg and M3 m3 are each equal to the space traversed by the Moon in about 10 days. And, lastly, it will be gathered from the inquiry in the text that for two solar eclipses to occur during the eclipse-season, the middle of this season must take place when the Moon is somewhere on

middle of the eclipse-season. It is easy to see in what respect it would be less important. When a solar eclipse occurs in the middle of the eclipse-season, the Moon's shadow traverses the centre of the Earth's disc as seen from the Sun. It therefore has a longer course on the Earth, and if total is rendered more remarkable by traversing that part of the Earth which is nearest to the Moon at the time. It is worthy of notice, however, that an annular eclipse, if its importance is measured

the arc n ̧ M3 m, such that n м, and м ̧ ̧ are each equal to the space traversed by the Moon in the excess of half an eclipse-season over half a lunation.

Now the places of the points m, m2, &c., will vary slightly, according to the length of the lunar month, the position of the Moon at new or full with reference to her perigee and apogee, and so on; and in particular it is to be noted that the limiting positions of m1 and m2, as of n, and n, are such that m, may be between m2 and M4, n, between n and M. But taking them as at present placed, and proceeding round the orbit from M, towards M2, &c., we have the following relations:

If at the middle of the eclipse-season the Moon is between M, and 23, there will be one central solar eclipse during the season; if between Nz and N21 there will be one central solar eclipse and one partial lunar eclipse; if between n, and n1, there will be one central solar eclipse and one total lunar eclipse; if between n1 and n, there will be one partial solar eclipse and one total lunar one; if between n and m1, there will be two partial solar eclipses and one total lunar one; if between m and m1, there will be one partial solar eclipse and one total lunar one; if between m1 and m2, there will be one central solar eclipse and one total lunar one; if between me and m, there will be one central solar eclipse and one partial lunar one; and, lastly, if the Moon is between m, and M1, there will be during the eclipse-season one central solar eclipse.

Thus there will be one solar eclipse if the Moon is on the arc m, Nz at the middle of the eclipse-season, a solar and lunar eclipse if the Moon is on either arc m ̧ m ̧ or n ̧n, and two solar eclipses and one lunar one if the Moon is on the arc n, m,. The dimensions of these arcs indicate the probability that an eclipse-season will include one, two, or three eclipses. Only when the Moon falls on either of the arcs m, m, and n, n, can there be a central solar eclipse and a total lunar one; this combination is, therefore, very infrequent. Still more infrequent is the occurrence of a partial solar and a partial lunar eclipse in the same eclipse-season; for this can only happen when m, is for the time between m, and M,, n, between n, and M.

by the breadth of the ring when the Sun is centrally eclipsed, is affected in a contrary manner when the shadow of the Moon falls near the centre of the Earth's disc; for that point being nearest to the Moon, the Moon appears relatively larger there, and the annulus therefore relatively narrower. It is true this part of the Earth is nearest also to the Sun, but his apparent magnitude is little affected, whereas the Moon's (owing to her relative proximity) is appreciably enlarged.

We may thus sum up the general characteristics and relations of our eclipse-seasons, the note on the preceding paragraph supplying the details on which the results are founded:

The most common of all orders of eclipse-seasons are those in which two eclipses take place. Of these one of course is sclar, the other lunar, and most commonly the solar eclipse is central, the lunar one partial, but in a considerable proportion of cases the solar eclipse is partial and the lunar one total. Very seldom does a total lunar eclipse accompany a central solar one, and yet more seldom are both partial. Next in order of frequency to the seasons of two eclipses are the seasons of but one eclipse, always a central solar one. Lastly come the seasons in which there are three eclipses, which are always-in order—a partial solar eclipse, a total lunar eclipse, and again a partial solar eclipse.*

From the preceding note it follows that the average frequency of the several orders of eclipses-omitting the case of two partial eclipses as of such infrequent occurrence-are fairly presented in the following table, in which the letters refer to fig. 104:

Now, with regard to the succession of these eclipse-seasons, it needs only to be noted that three seasons in which there are three eclipses never occur in succession.

We can now easily determine the greatest and least number of eclipses which may occur in any single year. The average interval between successive eclipse-seasons is 173-3 days. Two such intervals amount together to 346-6 days, or fall short of a year by about 19 days. Hence there cannot be three eclipse-seasons in a year. For each eclipse-season lasts on the average 33 days. Now suppose an eclipse-season to begin with the beginning of a year of 366 days. The middle of the season occurs at about midday on January 17; the middle of the next eclipse-season 173-3 days later, or on the evening of July 8; and the middle of the third occurs yet 173-3 days later, or on December 29, early in the forenoon; so that nearly the whole of the remaining half belongs to the following year. Now this is clearly a favourable case for the occurrence of as many eclipses as possible during the year. If all three seasons could be of the class containing three eclipses, there would be eight eclipses in the year, because the second eclipse of the third season would occur in the middle of that season. This, however, can never happen. But there may be two seasons, each containing three eclipses, followed by a season containing two eclipses, only one of which can occur in the fragment of the eclipse-season falling within the same year. In this case there would be seven eclipses in the year. So also there would be seven if in the first season there were three, in the second two, and in the third three, for then the fragment of the third falling within the year, being rather more than onehalf, would comprise two eclipses. So also if the three successive seasons comprise severally two, three, and three eclipses. The same would clearly happen if the year closed with the close of an eclipse-season.

There may then be as many as seven eclipses in a year, in which case at least four eclipses will be solar, and at least three of these partial, while of the lunar eclipses two at least will be total.