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before the beginning of the added month; and that the saṅkrânti succeeding true Adhika Âśvina took place when 287 parts of the natural month Nija Âśvina had elapsed, or when 287 parts had elapsed after the end of the added month Adhika Âśvina.

85. The moments of the saṅkrântis are further given in tithis and decimals in cols. 10, 12, 10a and 12a. Thus, in the above example we find that the preceding saṅkrânti took place when 29.850 tithis of the preceding month Bhâdrapada had elapsed, i.e., when (30 − 29.850 =) 0.150 tithis had still to elapse before the end of Bhâdrapada; and that the succeeding saṅkrânti took place when 0.861 of a tithi of the succeeding month, Âśvina, had passed.

To turn these figures into time is rendered easy by Table X. We learn from it that the preceding saṅkrânti took place (50 lunation parts or 0.150 tithi parts) about 3 h. 33 m. before the beginning of Adhika Âśvina; and that the succeeding saṅkrânti took place (287 lunation parts, or .861 tithi parts) about 20 h. 20 m. after the end of Adhika Âśvina. This time is approximate. For exact time see Arts. 82 and 90.

The tithi-indices here shew (see Art. 88) that there is no probability of a different month being intercalated if the calculation be made according to a different authority.

86. To constitute an expunged month we have shewn that two saṅkrântis must occur in one lunar month, one shortly after the beginning and the other shortly before the end of the month; and in cols. 9 and 10 the moment of the first saṅkrânti, and in cols. 11 and 12 that of the second saṅkrânti, is given. For example see the entries against Kali 3506 in Table I. As already stated, there can never be an expunged month by the mean system

87. In the case of an added month the moon must be waning at the time of the preceding, and waxing at the time of the succeeding saṅkrânti, and therefore the figure of the tithi-index must be approaching 10,000 at the preceding, and over 10,000, or beginning a new term of 10,000, at the succeeding, saṅkrânti. In the case of expunged months the case is reversed, and the moon must be waxing at the first, and waning at the second saṅkrânti; and therefore the tithi-index must be near the beginning of a period of 10,000 at the first, and approaching 10,000 at the second, saṅkrânti.

88. When by the Sûrya-Siddhânta a new moon (the end of the amâvâsyâ) takes place within about 6 ghaṭikâs, or 33 lunation-parts, of the saṅkrânti, or beginning and end of a solar month, there may be a difference in the added or suppressed month if the calculation be made according to another Siddhânta. Hence when, in the case of an added month, the figure in col. 9 or 9a is more than (10,000 − 33 =) 9967, or when that in col. 11 or 11a is less than 33; and in the case of an expunged month when the figure in col. 9 is less than 33, or when that in col. 11 is more than 9967, it is possible that calculation by another Siddhânta will yield a different month as intercalated or expunged; or possibly there will be no expunction of a month at all. In such cases fresh calculations should be made by Prof. Jacobi's Special Tables (Epig. Ind., Vol. II.) or direct from the Siddhânta in question. In all other cases it may be regarded as certain that our months are correct for all Siddhântas. The limit of 33 lunation-parts here given is generally sufficient, but it must not be forgotten that where Siddhântas are used with a bîja correction the difference may amount to as much as 20 ghaṭikâs, or 113 lunation-parts (See above, note to Art. 49).

In the case of the Sûrya-Siddhânta it may be noted that the added and suppressed months are the same in almost all cases, whether the bîja is applied or not.

89. We have spared no pains to secure accuracy in the calculation of the figures entered in cols. 9 to 12 and 9a to 12a, and we believe that they may be accepted as finally correct,