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103. Note that whenever a number of separately stated values for increase of ${\displaystyle a}$, ${\displaystyle b}$, ${\displaystyle c}$ have to be added together there is always a danger of slight error in the total, because, fractions below half being omitted, each separate increase may be a fraction too high or too low. But as a general N.B. rule one error rectifies another. In the case of mean intercalations six separately stated values of ${\displaystyle a}$, ${\displaystyle b}$, ${\displaystyle c}$ have to be added together, and the possibility of error should be remembered when the resulting totals are very near 10,000 or 0 respectively. For these reasons I find that my results for mean intercalations often differ by one or two units from those obtained by Sh. B. Dikshit.^[1] He wrote to me at the end of 1893 that he had invented a method by which the working was very easy (it was probably the same, or nearly the same, as Prof. Jacobi's in Epig. Ind. I., 418, § 35), the figures being obtained direct in tithis and decimals, and converted into the tithi-index (${\displaystyle t}$) by division by 30. I cannot say which method is least liable to error; possibly his, but I have tested a number of cases and find the difference between us very slight, so that my system may, I am satisfied, be depended upon as sufficiently accurate for all practical purposes. Close cases must always be carefully examined by Prof. Jacobi's "Special Tables."

104. In this publication I am dealing principally with the First Ārya and Present Sūrya Siddhānta. It would be impossible to give rules for the conversion of results by one authority into the same by another according to the "true" system, unless we knew A. The true or
apparent system. the exact lengths of true or apparent months as fixed by each; and this we do know for the two Siddhāntas mentioned, but not, I believe, for any other. Results by the mean system could, however, be easily converted, each mean month being exactly one-twelfth of the fixed year-length in each case. Amongst the Tables below will be found a list of these year-lengths, so far as I have been able to gather them. (Table XXXVII., col. 3.)

105. The causes of difference between results arrived at by using the First Ārya or the Present Sūrya Siddhānta respectively are, so far as the "true" system is concerned in which calculations are made by the true or apparent motions of the sun and moon, (1) the different lengths of the sidereal year ("Correction I." § 46 above), and (2) the different lengths of the true solar months as fixed by each ("Correction II." § 55). The figures to be applied are given in Tables XVII. and XVIII.C.

106. It will however be noticed that, whereas these correction-Tables can always be applied direct when a result for which conversion is desired stands expressed in time or in terms of ${\displaystyle a}$, ${\displaystyle b}$, ${\displaystyle c}$, this is not quite the case when we wish to convert a result expressed in terms of ${\displaystyle t}$, as in cols. 9 and 11 of Table I. where the times of true samkrantis are given according to the Stirya Siddhania. For such cases we still use Table XVIII.C, and apply to the given ${\displaystyle t}$ the value of ${\displaystyle a}$ contained in it, ignoring the values of ${\displaystyle b}$ and ${\displaystyle c}$; but the result is only approximate. The reason for this is as follows.

107. In every calculation on the ${\displaystyle a}$, ${\displaystyle b}$, ${\displaystyle c}$ system the desired result ${\displaystyle t}$ is obtained by adding to the given ${\displaystyle a}$, not the given ${\displaystyle b}$ and the given ${\displaystyle c}$, but the equation of ${\displaystyle b}$ and the equation of ${\displaystyle c}$. Now in Table XVII., for Correction I., the highest value of ${\displaystyle b}$, even for a year removed 1400 years from the base-year, is only 4; and ${\displaystyle c}$ never suffers any change. In Table XVIII.C, for Correction II., the value of ${\displaystyle c}$ again suffers no change; and therefore both under Tables XVII. and XVIII.C, if preferred, ${\displaystyle c}$ and its equation may be altogether ignored. As regards ${\displaystyle b}$ in Table XVIII.C the greatest difference is 3, for the Karka, Siṁha, and Kanyā saṁkrāntis. It is 2 for the Tulā saṁkrānti, 1 for five others, and 0 for three. Taking the greatest difference, 3, and turning to Table VI., Ind. Cal., it will be observed that equation ${\displaystyle b}$ (which alone affects the ultimate result) for the value ${\displaystyle b}$ = 3, though it may be itself 3 in certain positions of

1. ↑ The figures given in cols. 9a to 12a of Table I. in the Indian Calendar are those fixed by Mr. Sh. B. Dikshit.