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EXAMPLE X.

In North latitude when the sun's declination was 230.29' North, at 8h.54' in the forenoon the sun's corrected altitude was 48°.42', and at 96.46' the altitude was 55°.48'; required the true latitude ? British Palladium 1773, page 72.

Answer. The true latitude is 499.49%.28" North.

EXAMPLE XI.

In a supposed latitude of 69.50' North, at 76.30' in the forenoon the true altitude of the sun's centre was 220.30', and at 106.36'.40" his altitude was 63o.40', the sun's declination being 22°.48' North; required the true latitude, without using the supposed one?

Here 106.36'.40" — 75.30'=35.6'.40" the elapsed time, hence half the elapsed time=1b.33'.20"=23°.20'. I.

II. Rad, sine of

10.00000 Cot į elap. time 230.20% 10.36516 : sine pol. dist. 670.12 9.96467 : rad, sine of

1000000 :: sine į elap. time 230.30 9.59778 : : cos pol. dist. 670.12' 9.58829 : sine į arc 1st 21°.25 9.56245 : cot arc 2d 80°.S0.37, 9.22313

Arc Ist=420.50'.
III. Arc first

42°.50 co-secant •16758
Greatest zenith dist. 67o.30'
Least zenith dist. 26°.20 co-secant •35302

90°

900

O

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IV. Sine polar distance 670.12

9.96467 Sine least zenith dist. 26°.20

9.64698 { fourth arc

370.31'.21" sine x2=19:56933

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This latitude being so very different from the latitude by account, it will be necessary to examine it by the notes to the rule.

There can be no error, except in the fourth arc, and this is examined with very little additional trouble, there being only three numbers to take out of the tables :

Arc the second= 809.30.37" its supplement=999.29.23" Arc the third = 1550.33'.20" its supplement=240.26'.40"

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One of the two latitudes found above is certainly the true one, the latter, being the nearest to the latitude by account, may

be taken as the proper one.

EXAMPLE XII.

In latitude 10.50 North, by account at 10h.24' in the forenoon the true altitude of the sun's centre was 640.59', and at 11. 20' it was 780.57', the sun's declination being 09.30' North;

required the true latitude, without making use of the supposed one?

Answer. The latitude by the rule is 19.36'.24" North, but the place being so near to the equator, it will be proper to examine arc the fourth as directed in the notes to the rule. By this method the latitude is 09.39' South. The former is nearest the latitude by account.

(R) In all the preceding examples the two observations of the sun's altitude are supposed to have been made at the same place, and the latitude determined by the solution agrees to that place. Although the two altitudes are generally taken at the same place at land, yet at sea that is seldom the case. An allowance, therefore, ought to be made for the run of the ship during the elapsed time; thus, find the angle contained between the ship's course and the sun, if it be eight points no correction is necessary, but if less or more than eight points the correction must be applied to the first altitude, by addition or subtraction. Consider the angle contained between the ship's course and the bearing of the sun as a course, the distance made good during the elapsed time as a distance; with these find a difference of latitude and apply it as above.* The result will reduce the first altitude to what it would have been if taken at the same place where the second was taken. The latitude must be found with these altitudes, thus corrected, in the same manner as before: this will be the latitude of the place where the second altitude was taken.

'The difference of longitude during the elapsed time may likewise be taken into consideration, though in general it is of little or no consequence. The change of the sun's declination during the elapsed time might likewise be considered, but this, like the longitude, will cause no sensible error; particularly, if the declination answering to the middle time between the observations be used.

(S) PROBLEM XVII. (Plate III. Fig. 12.) Given the apparent distance of the moon from the sun, or a star, and their apparent zenith distances, to find their true distance, as seen from the earth's centre.

Let zo be the observed zenith distance of the moon, zm the true zenith distance; mm being the difference between the moon's refraction and her parallax in altitude. And let z o represent the observed zenith distance of the sun or of a star; zs the true zenith distance, O s being the difference between the sun's refraction and his parallax, or the refraction of a star.

* Vide Mr. Cotes's De estimatione errorum in mixtâ Mathesi.

GENERAL PRINCIPLES. I. Find the segments of the base Ov and vm, (by the rule W.231.). With Oz and Ov find the angle zov, which will be equal to the angle TOs. (N. 135.) With Os, and the angle TOs, find or, which will be equal to sp. The triangle TOS, being indefinitely small, may be considered as a plane triangle.

Again, with vm and zu find the angle zmv, with mm and the angle zmv find the base RM, considering the right-angled triangle MRM as a plane triangle.

Lastly, MO +SP-RM=sm the true distance in all cases except where the angle at the zenith is acute, and the angle at the moon obtuse, then MO +Sp+RM=sm.

II. Or, with zo,zm, and om find the vertical angle Ozm, (P. 248.), and with zs, zm, and the angle OzM, find the true distance sm. (N. 245.)

The various methods which have hitherto been made use of for determining the distance between the moon and the sun, or a star, are derived from one or other of the above principles. Those methods which are derived from the latter, are generally preferable to those derived from the former, as being more correct and simple. When the observed distance is small, or the moon's parallax great, and the star's refraction considerable, two other corrections are necessary in order to render the first of these principles generally correct. We have considered the little triangles as right-angled, but the fact is NPO and NRM are each of them isosceles triangles, and, therefore, OP and Rm are not strictly perpendicular to sm and OM; these corrections, therefore, consist in determining how much sp and RM deviate from the bases of right-angled triangles. A true method of determining these four corrections may be seen in the Edinburgh Transactions, Article VII., Physical Class. (T) INVESTIGATION OF A GENERAL RULE FOR DETERMINING

THE TRUE DISTANCE OF THE MOON FROM THE SUN, OR FROM A FIXED STAR. (Plate III. Fig. 13.)

Let om be the observed distance, and sm the true distance. Also, let zo be the observed zenith distance of the moon and zm the true zenith distance; zo the observed zenith distance of the sun or of a star, and zs the true zenith distance, as above. Then, by one of the formulæ F. 184, we have, in the triangle Ozm, 2 sine? } z rad.cos (20 ~ZM)-rad. cos OM

; and, likewise rad

sine zo x sine ZM in the triangle szm, by the same formula, 2 sine z

rad . cos (zs ~ Zm)— rad. cos sm

COS

By putting these values equal to each other, and dividing by the radius, we obtain cos (20 ~ZM)-cos OM cos (zs ~ zm) - cos sm

sine zo x sine zM sine zs x sine zm

Hence, cos sm-cos(zs ~ zm)=cos OM-cos(z0-~ZM) sine zs x sine zm sinez o x sine zm®

Now zs~zm and likewise zo~ZM will always be acute, being each less than 1°; but Om and sm, being each of the same species, may be either acute or obtuse, therefore when the observed distance is more than 90°,

sine zs x sine zm Cossm=cOSOM+cos(20~2M)

sine zo x sine zm (zs - zm). And when the observed distance is less than 90°,

sine zs x sine zm Cos sm=cos (zs ~zm)- cos OMX

sine zo x sine ZM (zs - zm.) Hence the following

GENERAL RULE. * To the natural cosine of the difference between the apparent altitudest, add the natural cosine of the apparent distance if more than 90°, or subtract it if less than 90°, and find the common logarithm of the remainder; to which add the logarithm secants of the apparent altitudes, the logarithm cosines of the true altitudes, and reject the tens from the index. The difference between the natural number answering to this sum in the table of logarithms, and the natural cosine of the difference between the true altitudes, will give the natural cosine of the true distance.

NOTE. The moon's correction, which is the difference between the refraction and the parallax in altitudes, must always be added to the apparent altitude to obtain the true altitude;

COS

* Several other rules, differing in the form of expression, may be deduced from the foregoing demonstration, but this has been preferred on account of its shortness, and the ease with which it may be applied : requiring no other tables in its application than those which are common and well known.

A collection of short rules, without demonstration, may be seen in Nicholson's Philosophical Journal, for November 1806, vol. XV. page 254.

Dr. Mackay's 1st Method (page 150, vol. I. 3d edition) of his valuable treatise on the longitude, is the simplest I have ever met with, when his tables are used.

Mendoza's method is likewise very short and easy, but his tables are large and expensive.

+ The apparent altitudes, are the observed altitudes corrected for dip and semidiameter, and the apparent distance, is the observed distance corrected for seini. diameters.

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