1584, N = 8, J = 7 + 11 = 18;
1585, N = 9, J = 18 + 11 = 29;
1586, N = 10, J = 29 + 11 - 30 = 10;
and, therefore, in general J = ((26 + 11(N - 6)) / 30)r. But the numerator of this fraction becomes by reduction 11 N - 40 or 11 N - 10 (the 30 being rejected, as the remainder only is sought) = N + 10(N - 1); therefore, ultimately,
| J = | N + 10(N - 1) 30 | r. |
On account of the solar equation S, the epact J must be diminished by unity every centesimal year, excepting always the fourth. After x centuries, therefore, it must be diminished by x - (x/4)w. Now, as 1600 was a leap year, the first correction of the Julian intercalation took place in 1700; hence, taking c to denote the number of the century as before, the correction becomes (c - 16) - ((c - 16) / 4)w, which
must be deducted from J. We have therefore
| S = - (c - 16) + | c - 16 4 | w. |
With regard to the lunar equation M, we have already stated that in the Gregorian calendar the epacts are increased by unity at the end of every period of 300 years seven times successively, and then the increase takes place once at the end of 400 years. This gives eight to be added in a period of twenty-five centuries, and x/25 in x centuries. But 8x/25 = 1/3 (x - x/25). Now, from the manner in which the intercalation is directed to be made (namely, seven times successively at the end of 300 years, and once at the end of 400), it is evident that the fraction x/25 must amount to unity when the number of centuries amounts to twenty-four. In like manner, when the number of centuries is 24 + 25 = 49, we must have x/25 = 2; when the number of centuries is 24 + 2 × 25 = 74, then x/25 = 3; and, generally, when the number of centuries is 24 + n × 25, then x/25 = n + 1. Now this is a condition which will evidently be expressed in general by the formula n - ((n + 1) / 25)w. Hence the correction of the epact, or the number of days to be intercalated after x centuries reckoned from the commencement of one of the periods of twenty-five centuries, is {(x - ((x+1) / 25)w) / 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. |