Page:Sewell Dikshit The Indian Calendar (1896) proc.djvu/264

${\displaystyle \textstyle t=(\lambda +\mu )+{\frac {15}{n}}\times 0{,}9966\sin {\phi _{1}}\cos {k}-{\frac {15}{n}}\cos {\phi _{1}}\sin {k}\cos {(K+t)}+{\frac {15}{n}}m\cos {M}}$.

The angle ${\displaystyle M}$ being, at the moment of greatest phase, always sufficiently near 90° or 270", ${\displaystyle \textstyle {\frac {15}{n}}m\cos {M}}$ can be neglected; and, introducing for ${\displaystyle \textstyle {\frac {15}{n}}}$ its mean value 27,544, and identifying ${\displaystyle \phi _{1}}$, with ${\displaystyle \phi }$, the value of ${\displaystyle t_{0}}$ can simply be determined by the expression ${\displaystyle t=(\lambda +\mu )+27{,}447\sin {\phi }\cos {k}-27{,}544\cos {\phi }\sin {k}\cos {(K+t)}}$ instead of determining it by the whole of the above formulæ. Now in this last expression ${\displaystyle k}$ and ${\displaystyle K}$ are mere dependents on ${\displaystyle L'}$, and therefore the values of ${\displaystyle t}$ can be tabulated for each value of ${\displaystyle L'}$ with the two arguments ${\displaystyle (\lambda +\mu )}$ and ${\displaystyle \phi }$. Table D is constructed on this formula, only instead of counting ${\displaystyle t}$ in degrees and from true noon it is counted, for Indian purposes, in ghaṭikâs and their tenths from true sunrise.

The value of ${\displaystyle t}$ for the instant of the greatest phase at the given place being found, it can be introduced into the formula ${\displaystyle m\sin {M}=\gamma -0{,}9966\cos {g}\sin {\phi _{1}}+\cos {\phi _{1}}\sin {g}\sin {(G+t)}}$.

As ${\displaystyle M}$ is always near 90° or 270°, ${\displaystyle \sin M}$ can be considered equal to ±1, so we have ${\displaystyle \pm m=\gamma -0{,}9966\cos {g}\sin {\phi _{1}}+\cos {\phi _{1}}\sin {g}\sin {(G+t)}}$ where the sign ± is to be selected so that the value of ${\displaystyle m}$ may always be positive.

The second part of the above expression ${\displaystyle -0{,}9966\cos {g}\sin {\phi _{1}}+\cos {\phi _{1}}\sin {g}\sin {(G+t)}}$ (which, for the sake of brevity, may be called by the letter ${\displaystyle \Gamma '}$) contains only values which directly depend on ${\displaystyle L'}$, such as ${\displaystyle \cos g}$, ${\displaystyle \sin g}$, ${\displaystyle G}$, or which, for a given value of ${\displaystyle L'}$, depend only on ${\displaystyle (\lambda +\mu )}$ and ${\displaystyle \phi }$, and therefore the values of ${\displaystyle \Gamma '}$ can be tabulated for each value of ${\displaystyle L'}$ with the two arguments ${\displaystyle (\lambda +\mu )}$ and ${\displaystyle \phi }$. This has been done in the Table B which follows, but instead of ${\displaystyle \Gamma '}$ the value ${\displaystyle 1+\Gamma '=\Gamma }$ has been tabulated to avoid negative numbers. The value of ${\displaystyle m}$ can then be found from ${\displaystyle m=\pm (\gamma +\Gamma ')}$.

Both Tables B and D ought to consist of two separate tables, one containing the values of ${\displaystyle L'}$ from 0° to 360° in the case of ${\displaystyle P}$ being near 0°, the other containing the values of ${\displaystyle L'}$ from 0° to 360° for the case of ${\displaystyle P}$ being near 180°. To avoid this division into two tables, and the trouble of having always to remember whether ${\displaystyle P}$ is near 0° or 180°, the two tables are combined into one single one; but, whilst in the case of ${\displaystyle P}$ being near 0° ${\displaystyle L'}$ is given as argument, in the case of ${\displaystyle P}$ being near 180° the table contains, instead of ${\displaystyle L'}$, ${\displaystyle L'+400^{\circ }}$ as argument. We need therefore no longer care whether the moon is in the ascending or descending node, but simply take the argument as given in the first table. With the value of ${\displaystyle m}$, found by ${\displaystyle m=\pm (\gamma +\Gamma ')}$, we can find the magnitude of the greatest phase in digits ${\displaystyle \textstyle =6{\frac {u'_{a}-m}{u'_{a}-0{,}2736}}}$, which formula can also be tabulated with the arguments ${\displaystyle u'_{a}}$, and ${\displaystyle m}$, or with ${\displaystyle u'_{a}}$ and ${\displaystyle (\gamma +\Gamma )}$. This has been done in Table C. As ${\displaystyle u'_{a}}$ when abbreviated to two places of decimals has only the six values 0.53, 0.54, 0.55, 0.56, 0.57 and 0.58, every column of this Table is calculated for another value of ${\displaystyle u'_{a}}$, whilst to ${\displaystyle \gamma }$ the constant 5 has been added so that all values in the first Table may be positive. Instead of giving ${\displaystyle u'_{a}}$, directly, its last cipher is given as tenths to the value of ${\displaystyle (\gamma +\Gamma )}$ so that there is no need for ascertaining the value of ${\displaystyle u'_{a}}$.

Of all elements, then, given by the Canon we want only the following ones;—