/ 3}_w.
The last period of twenty-five centuries terminated with 1800; therefore,
in any succeeding year, if c be the number of the century, we shall have x
= c - 18 and x + 1 = c - 17. Let ((c - 17) / 25)_w = a, then for all years
after 1800 the value of M will be given by the formula ((c - 18 - a) /
3)_w; therefore, counting from the beginning of the calendar in 1582,

M={(c - 15 - a) / 3}_w.

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 3121/2,
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 t