By putting these values equal to each other, and dividing by the radius, we obtain ZM)- COS OM cos (zs to zm) - cos sm sine zOX sine ZM sine zs X sine zm Hence, cos sm-cos (zs zm)=cos OM-cos (ZO ZM) X sine zs x sine zm sine ZO X sine ZM Now zs zm and likewise zO Zм will always be acute, being each less than 1°; but Oм and sm, being each of the same species, may be either acute or obtuse, therefore when the observed distance is more than 90°, Cossm=cos OM+COS (ZO ZM) X (zs zm). sine zs X sine zm sine zOX sine ZM And when the observed distance is less than 90°, Cos sm cos (zs o zm) cos OMX sine zs x sine zm sine zOX sine ZM (zs zm). Hence the following GENERAL RULE†. To the natural cosine of the difference between the apparent altitudes, 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. EXAMPLE 1. The apparent distance of the moon's centre from the star Regulus was 63°.35'.13", when the apparent altitude of the moon's centre was 24°29′44′′, the apparent altitude of the star 45°.9'.12′′; the moon's correction, or difference between the refraction and parallax in altitude 48'.1"; the star's correction, or refraction 57"; required the true distance? ↑ 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. Apparent Q Apparent distance 63°.35′.13" Nat. cos. 44485 Diff. app. altitudes 20°.39′.28′′ 93570 Note. The moon's correction is always to be added to her apparent altitude, to get the corrected altitude. But the sun or star's correction must always be subtracted. EXAMPLE II. The apparent distance of the moon's centre from the sun's was 106°.46′.44", when the apparent altitude of the sun's centre was 45°.32'.30', and the moon's 19.43.22"; the moon's correction 50.3", and the sun's 50"; required the true distance of their centres? Apparent distance - 106°.46'.44" Nat. cos. 28868 25°.49'. 8" Nat. cos. 90018 118886 5'07513 sec. 10:02625 Moon's app. altitude Natural number Diff. of true alts. 24°.58.15" Nat. cos. 90652 True distance 106°. 2'.19" Nat. cos. 27629 *Instead of the logarithmical secants of the apparent altitudes, you may take the logarithmical cosines, add them together and take the sum from 20, the remainder will be the same as the sum of the logarithmical secants without the indices. EXAMPLE required the true latitude, without making use of the supposed one? By this Answer. The latitude by the rule is 1°.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. method the latitude is 0°.39′ South. The former is nearest the latitude by account. (H) 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. (I) 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 zм 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 represent the observed zenith distance of the sun or of a star; zs the true zenith distance, Os 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 GENERAL PRINCIPLES. I. Find the segments of the base Ov and vм, (by the rule M. 200). With Oz and Ov find the angle zOv, which will be equal to the angle TOS. (N. 131.) With Os, and the angle 10s, find OT, which will be equal to SP. The triangle TOS, being indefinitely small, may be considered as a plane triangle. Again, with VM and zм find the angle zмv, with мm and the angle ZMV find the base RM, considering the right-angled triangle RM 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 мO+SP+RM=SM. II. Or, with ZO, ZM, and OM find the vertical angle Ozм, (G. 227.), and with zs, zm, and the angle Ozм, find the true distance sm. (E. 225.) 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, both for correctness and ease. 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. (K) 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 zм 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 H. 214, we have, in the triangle Ozм, 2 sinez rad. cos (ZO rad2 ZM)-rad. cos. O M ; and, likewise in sine zOX sine ZM the triangle szm, by the same formula, rad2 sine zs X sine zm By if they be decreasing; to the sum or difference add the augmentation*, and you will have the moon's truesemi-diameter at reduced time. Again, As 12 hours are to the second difference, so is the reduced time to a fourth number, which must be added to or subtracted from the horizontal parallax at the nearest noon, or midnight, preceding the reduced time, according as the tables are increasing or decreasing, and it will give the horizontal parallax at reduced time. III. Clear the observed altitude of the moon of dip + and semi-diameter ‡, and you have the apparent altitude of her centre: to the cosine of the moon's apparent altitude, add the logarithm of the horizontal parallax at reduced time in seconds, the sum rejecting 10 from the index, will be the logarithm of the moon's parallax in altitude in seconds §, from which take the refraction of the moon in altitude, the remainder will be the moon's correction. IV. Aditional Preparation for the Sun and Moon. Clear the observed altitude of the sun of dip and semi-diameter, and you have the apparent altitude of his centre. From the refraction of the sun's altitude take his parallax** in altitude, and you have the correction of the sun's altitude. V. To the observed distance of the sun and moon's nearest limbs, add their semi-diameters at reduced time, and the sum will be the apparent distance of their centres. IV. Additional Preparation for the Moon and a Star. From the star's observed altitude take the dip of the horizon, the remainder will be its apparent altitude. The refraction of a star is the correction of its altitude. V. To the observed distance of the moon from a star, add the moon's semi-diameter at reduced time, the sum will be the apparent distance; if the farthest limb was observed subtract the semi-diameter. VI. To find the distance.. With the apparent altitudes, their corrections, and the apparent distance, find the true distance by the general rule. (K. 298.) Viz. take their difference and add it to the observed altitude of the moon's lower limb; or take their sum and subtract it from the observed altitude of the moon's upper limb, according as the lower or upper limb has been observed. || Table IV. § (T. 90.). Viz. take their difference, and add it to the observed altitude of the O's lower limb; or take their sum, and subtract it from the observed altitude of the O's upper limb, according as the lower or upper limb has been observed. ** Table VI. Note. |