Diminish the latitude by the equation in Table B, and the result will be the geocentric, or reduced latitude: to the logarithmic radius corresponding to this in Table C, add the log. sine of the moon's reduced horizontal parallax; the sum, abating 10 in the index, will be the log. sine of the moon's horizontal parallax adapted to the oblate spheroidal figure of the earth.

Take the star's right ascension and declination from the Nautical Almanac under the head "Occultations," between pages 455 and 465; and let these elements be considered as the apparent right ascension, and the apparent declination of that point of the moon's limb which is in contact with a fixed star or planet at the exact instant of an immersion or an emersion.

To the mean time of observation add the mean sun's corrected right ascension; and the sum will be the right ascension of the meridian : the difference between which and the apparent right ascension will be the apparent horary distance of the moon's limb from the meridian. Convert this into motion by Problem II., page 342, and it will express the value of the apparent hour angle. Now,

To the sine of the apparent hour angle add the co-sine of the reduced latitude, and the sine of the adapted horizontal parallax ; call the sum, abating 20 in the index, the reserved logarithm. To this add the secant, less radius, of the apparent declination, and the constant logarithm 9.698970;* the sum, abating 10 in the index, will be the sine of the parallax in the apparent hour angle;† which being subtracted therefrom, the result will be the moon's corrected hour angle. -See Article 13, page 544.

To the co-sine of the moon's corrected hour angle, add the co-sine of the reduced latitude, the sine of the apparent declination and the sine of the adapted horizontal parallax; the sum, abating 30 in the index, will be the sine of the minor part of the parallax in the moon's declination ; which is to have the same name as the apparent declination when the corrected hour angle is more than 90 degrees; but, a different name, if it be less than 90 degrees.- See Article 14, page 544. To the sine of the reduced latitude, add the co-sine of the apparent declination, and the sine of the adapted horizontal parallax; the sum, abating 20 in the index, will be the sine of the major part of the parallax in the moon's declination;† which is to have the same name as

The logarithmic versed sine or co-sine of 60 degrees.

The arc is to be taken which corresponds to the tabular log. sine that comes nearest to the computed log. sine, whether the tabular one be the greatest or the least.

the apparent declination when this and the latitude are both north, or both south; but, a contrary name, if one be north and the other south. -See Article 14, page 544.

Now the sum or the difference of the minor and major parts of parallax, according as they may be of the same or of different denominations, will be the effects of parallax in the moon's declination, with the name of the major part: then, this being applied to the apparent declination by addition, when the names are alike, or by subtraction if unlike, the result will be the correct declination of the moon's centre.

To the secant, less radius, of the correct declination of the moon's centre, thus found, add the reserved logarithm; and the sum will be the sine of the parallax in right ascension ;* to which prefix the sign +, when the moon is west, or the sign, when she is east, of the meridian. The reader would do well to turn back and re-peruse the whole of Article 15, page 544.

Find the difference between the correct declination of the moon's centre and her approximate declination; then, to the sines of the sum and the difference of this and the moon's true semidiameter, add the secants, less radius, of those declinations: half the sum of these four terms will be the sine of the semi-chord of the occultation in right ascension; to which prefix the sign +, at an emersion; but the sign —, at an immersion.-See Article 16, page 545; and the paragraph above Article 11, page 543.-Now, if the moon's parallax in right ascension, and the semi-chord of the segment of occultation in right ascension have signs alike, take their sum; but, if unlike, their difference, with the sign of the greater term. Multiply this sum, or difference, as the case may be, by 4; and it will be converted into time; observing that seconds of a degree produce thirds of time, and minutes of a degree, seconds of time. Annex a cipher to the thirds; then divide by 6, and they will be reduced to two places of decimals, or to hundredths of a second. Apply the minutes and seconds, thus found, to the apparent right ascension by addition or subtraction, according as its sign may be either affirmative or negative; and the result will be the correct right ascension of the moon's centre.

Enter the Nautical Almanac and find, under the given day, the difference between the right ascension of the moon's centre, determined as above, and her next less right ascension; and find, also, the difference between the next less and the next greater right ascension; which dif

* The arc is to be taken which corresponds to the tabular log. sine that comes nearest to the computed log. sine, whether the tabular one be the greatest or the least.

ferences convert into seconds. Then, from the sum of the logarithm of the first difference and the constant logarithm 3. 556303,* subtract the logarithm of the last difference; and the remainder will be the logarithm of a portion of time in seconds: raise this to minutes &c., to which prefix the hour corresponding to the next less right ascension, and it will express the mean time of the immersion, or emersion, at Greenwich. Now, the difference between the mean, time at Greenwich, thus found, and the mean time of observation, being converted into motion, will be the correct longitude of the ship or place of observation; which will be east, if the mean time at ship be the greatest; otherwise, it will be west.

Remark. The above General Rule is adapted to the second volume of this work; in which the sines and the secants of all arcs are given to every second in the semicircle: and as it departs from the customary modes of computation, by determining the different values of the moon's horizontal parallax in the sines of arcs instead of in their lineal measures; I think it right to observe, that this is done for the purposes of simplifying and expediting the numerical calculations:—and I shall now show, that in thus studying the accomplishments of these points, I have not sensibly departed from the strict line of mathematical correctness. The greatest value of the moon's horizontal parallax is 61:32; beyond this point it can never pass, so long as the ordinary laws of nature remain unchanged. Now, since the Trigonometrical Tables are adapted to a circle whose radius is unity, or 1; and that it is not necessary, at least for practical purposes, to extend the numbers in those Tables beyond six places of decimals: and since the absolute length of the arc of 61:32" is 0.017608; and that the sine of the same arc is also 0. 017608, as may be readily proved by calculation; therefore, the one may be substituted for the other, and so may all other small arcs and their sines that correspond to six places of decimals. But, since the different values of the moon's horizontal parallax will always be considerably below 60 minutes; therefore, it is clearly manifest, that for the important purpose of facilitating the calculations, the measures of the sines may be safely substituted for the lengths of the arcs. See the latter part of Article 16, page 545.

This is the logarithm of 3600', or of one hour reduced to seconds.

20. The following Example will show the practical application of the above general Rule. But, as the size of the page will not admit of all the terms that enter the calculations being inserted at length, it becomes indispensably necessary to make use of certain contractions.

Let R. A.
Ap. R. A.
Ap. dec.

Apx. d.

Ap. h. a.

Par. in h. a.

Cor. h. a.

Red. lat.

Log. s. h. p.

As thus:

= or express the right ascension.

= the apparent right ascension.

=

=

=

=

=

the apparent declination.

the approximate declination.
the apparent hour angle.

parallax in the apparent hour angle.
the corrected hour angle.

the latitude reduced to the oblate figure of
the earth.

the sine of the D's hor. par. adapted to the oblate figure of the earth.

Log. adapted h.p. Log. of 's horizontal parallax adapted to ditto.

the minor part of the parallax in declination.

the major part of the parallax in declination.

= the effects of parallax in declination.

=

the moon's correct declination.

the moon's semidiameter, or semidiameter for calculation.

the parallax in right ascension.

= half the chord line of the occultation in right

ascension.

= the correct right ascension of the moon's

centre.

Example.

April 25th, 1836, in latitude 39:28:14" north, and longitude, by account, 5:22:45" west, the emersion of Leonis from behind the moon (west of the meridian) was observed at 8:58:58 mean time; required the true longitude of the place of observation?

Remark. Although the apparent horary distance, in time, is determined to hundredths of a second; yet, this is not absolutely necessary; for, in every instance, the nearest second will prove to be sufficiently exact,