Time at Ship

14

Difference is long. in time 10 59 36=164° 54′ E. from Greenwich,

68° 43′ 49′′

EXAMPLE IV.

Suppose that on the 31st October, 1820, sea account, at about 1h. P. M. in the longitude of 750 W. by account, the following observations of the sun and moon were taken. Required the true longitude?

Time per watch.

Oh. 58' 5

This Corr. Corr. Tab. XIX. 10' 12'--Corr. Tab. A 40"+Corr. Tab. B 0-10′ 5

Suppose that on the 5th May, 1820, sea account, at about 4h. 4' P. M. in the latitude of 50° 1' S. and in the longitude of 10 E. by account, the following observations of the sun and moon were taken. Required the true longitude?

Preparation.

Suppose that on the 8th of February, 1820, sea account, at about Sh. 36′ A. M. in the longitude of 21° W. from Greenwich by account, six distances of the sun and moon's nearest limbs were observed by a circle of reflection to be 464° 10′ 12′′ the corresponding times and altitudes being as in the following Table. Required the true Iongitude?

*The correct altitude is found by subtracting the refraction 3 from the apparent altitude 15o 50'. The Potar Distance is found by adding the Declination 16° 4' N. (corresponding to the reduced time)

This logLog. Table XIX. 2016+Log. Tab. € 9=2025.

This Corr-Corr. Table XIX. 3.' 7''+Corr. Tab. A *13" Corr. Tab. B. Q′′ (52′ 20 /

Second method of finding the true distance of the Moon from the Sun or a Star.

From the sun's refraction (Table XII.) take his parallax in altitude (Table XIV.) the remainder will be the correction of the sun's altitude.

The star's refraction (Table XII.) is the correction of its altitude.

From the proportional logarithm of the moon's Horizontal Parallax* increasing the index by 10, take the sine of the moon's apparent zenith distance (Table XXVII.) the remainder will be the prop. log. of the parallax in altitude, which must be found in Table XXII. and the moon's refraction (Table XII.) subtracted therefrom, the remainder will be the correction of the moon's altitude.†

Add together the apparent distance of the sun and moon (or star and moon) and their apparent zenith distances (or complement of their apparent altitudes) and note the half sum of these numbers: the difference between the half sum and the moon's apparent zenith distance call the first remainder; and the difference between the half sum and the sun's (or star's) apparent zenith distance call the second remainder.

To the constant log. 9.6990 add the co-secant of the half sum and the sine of the apparent distance (both taken from Table XXVII.) the sum, rejecting 20 from the index, will be a reserved logarithm.

To the reserved logarithm add the sine of the sun's (or star's) apparent zenith distance, the co-secant of the first remainder (both taken from Table XXVII.) and the prop, log. of the correction of the sun's (or star's) altitude (Table XXII.) the sum, rejecting 30 from the index, will be the prop. log. of the first correction to be found in Table XXII.

To the reserved logarithm add the sine of the moon's apparent zenith distance the co-secant of the second remainder (Table XXVII.) and the prop. log. of the correction of the moon's altitude (Table XXII.) the sum, rejecting 30 from the index, will be the prop. log. of the second correction, to be found in Table XXII.

Then to the apparent distance add the correction of the moon's altitude, and the first correction, and subtract the sum of the second correction and the correction of the sun's altitude, the remainder will be the corrected distance.

Add 60' to the correction of the moon's altitude, and 60' to the difference between the oorrection of the moon's altitude and the second correction; find both these sums in the side column of Table XX. and in either of the vertical columns, under the corrected distance, find the seconds corresponding, § the difference of these two numbers will be a number of seconds to be added to the corrected distance when less than 90°, but subtracted when above 90°, the sum or difference will be the true distance.

→ Instead of finding the moon's horizontal parallax from the Nautical Almanac, we may find the proportional logarithm thereof in the same page of the month of that work. Thus if we would work Example III. preceding, by this rule, we might take out the logs. 5198 and 5187, instead of the Hor. Par. 54′ 25′′ and 54′ 31', and obtain by means of Table XI. the sought log. 5195 without referring to Table XXII.

All these corrections may be found by means of Tables XVII. XVIII. and XIX. Thus the correc tion of Table XVII. subtracted from 60 minutes, will give the correction of the star's altitude. The cor rection of Table XVIII. subtracted from 60 minutes, will give the correction of the sun's altitude. The correction of Table XIX. subtracted from 59′ 42′′ will give the correction of the moon's altitude. Perhaps the use of these Tables in this and in the following method, would not be inconvenient.

This logarithm was found before, in calculating the correction of the moon's altitude. Observing to take both numbers from the same vertical column It may be observed that the num bers in one of the columns of Table XX. of this collection, are the same as those of Table XIX. of edi tion 1, and the numbers in the other column differ 18 from the former; but the numbers in the side column of Table XIX. differ 60 from those i: Table XX. so that in using Table XIX. edition 1st. it is unnecessary to add 60' to the correction of the moon's altitude and the first correction; this renders that Table rather more convenient in this second method, than Table XX. of this collection.

EXAMPLE (the same as Example I. preceding.)

Suppose the apparent distance of the centre of the moon from the star Aldebaran was 47° 33′ 48′′, the apparent altitude of the star 50° 31', the apparent altitude of the moon's centre 70 47, and the proportional logarithm of the moon's horizontal parallax 5199. Required the true distance of the moon from the star?

We shall now give a third method of correcting the apparent distance, being an improvement on Witchell's method, which was published in the former edition of this work. This improvement was made in consequence of a suggestion from a gentleman eminently distinguished for his mathematical acquirements,* that, by a small variation in the calculation, the number of cases might be lessened: and, upon examination, it was found that by making other alterations, the number of cases might be farther decreased, and the manner of applying the corrections rendered more simple. The method thus improved is as follows.

Third method of finding the true distance of the Moon from the Sun or Star. From the sun's refraction (Table XII.) take his parallax in altitude (Table XIV.) the remainder will be the correction of the sun's altitude.

The star's refraction is the correction of its altitude.

From the proportional logarithm of the moon's horizontal parallax, increasing the index by 10, take the co-sine of the moon's apparent altitude (Table XXVII.) the remainder will be the proportional logarithm of the moon's parallax in altitude; from which, subtracting the moon's refraction (Table XII.) the remainder will be the correction of the moon's altitude.†

1. Add together the apparent altitudes of the moon and sun (or star) and take the half sum; subtract the lesser altitude from the greater, and take the half difference; then add together

The tangent of the half sum,

The co-tangent of the half difference,

The tangent of half the apparent distance.

The sum, rejecting 20 in the index, will be the tangent of the angle A, which must be sought for in Table XXVII. and taken out less than 90° when the sun's altitude is less than the moon's, otherwise greater than 90°. The difference of the angle A,

• The late Chief Justice Parsons.

These corrections may be found by Tables XVII. XVIII. XIX. as was shown in the note to the second method, page 174.

Every co-tangent in Table XXVII. corresponds to two angles, one greater than 90°, the other less: