By the substitution of these values of J, S and M, the equation of the epact becomes
| E = | N + 10(N - 1) 30 | r - (c - 16) + | c - 16 4 | w + | c - 15 - a 3 | w. |
It may be remarked, that as a = ((c - 17) / 25)w, the value of a will be 0 till c - 17 = 25 or c = 42; therefore, till the year 4200, a may be neglected in the computation. Had the anticipation of the new moons been taken, as it ought to have been, at one day in 308 years instead of 312½, the lunar equation would have occurred only twelve times in 3700 years, or eleven times successively at the end of 300 years, and then at the end of 400. In strict accuracy, therefore, a ought to have no value till c - 17 = 37, or c = 54, that is to say, till the year 5400. The above formula for the epact is given by Delambre (Hist. de l'astronomie moderne, t. i. p. 9); it may be exhibited under a variety of forms, but the above is perhaps the best adapted for calculation. Another had previously been given by Gauss, but inaccurately, inasmuch as the correction depending on ''a'' was omitted.
Having determined the epact of the year, it only remains to find Easter Sunday from the conditions already laid down. Let
P = the number of days from the 21st of March to the 15th of the paschal moon, which is the first day on which Easter Sunday can fall;
p = the number of days from the 21st of March to Easter Sunday;
L = the number of the dominical letter of the year;
l = letter belonging to the day on which the 15th of the moon falls:
then, since Easter is the Sunday following the 14th of the moon, we have
p = P + (L - l),
which is commonly called the number of direction.
The value of L is always given by the formula for the dominical letter, and P and l are easily deduced from the epact, as will appear from the following considerations.
When P = 1 the full moon is on the 21st of March, and the new moon on the 8th (21 - 13 = 8), therefore the moon's age on the 1st of March (which is the same as on the 1st of January) is twenty-three days; the epact of the year is consequently twenty-three. When P = 2 the new moon falls on the ninth, and the epact is consequently twenty-two; and, in general, when P becomes 1 + x, E becomes 23 - x, therefore P + E = 1 + x + 23 - x = 24, and P = 24 - E. In like manner, when P = 1, l = D = 4; for D is the dominical letter of the calendar belonging to the 22nd of March. But it is evident that when l is increased by unity, that is to say, when the full moon falls a day later, the epact of the year is diminished by unity; therefore, in general, when l = 4 + x, E = 23 - x, whence, l + E = 27 and l = 27 - E. But P can never be less than 1 nor l less than 4, and in both cases E = 23. When, therefore, E is greater than 23, we must add 30 in order that P and l may have positive values in the formula P = 24 - E and l = 27 - E. Hence there are two cases.