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A Treatise on Astronomy Theoretical and Practical by Robert

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A Treatise on Astronomy Theoretical and Practical by Robert NAPOLI Digitized by Google Digitized Google A TREATISE ov ASTRONOMY THEORETICAL and PRACTICAL. BY ROBERT WOODHOUSE, A M. F.R.S. FELLOW OF GONVILLK AND CAIUS COLLEGE, AND PLUMIAN PROFESSOR OF ASTRONOMY IN THE UNIVERSITY OF CAMBRIDGE. Part II. Vol. I. CONTAINING THE THEORIES OF THE SUN, PLANETS, AND MOON. CAMBRIDGE: PRINTED BY 3 . SMITH, PRINTER TO THE UNIVERSITY ; FOR J. DEIGHTON & SONS, AND O. & W. B. WHITTAKER, LONDON. 1823 Digitized by Google Digitized by Google : ; —; CHAP. XVII. ON THE SOLAR THEORY Inequable Motions of the Sun in Right Ascension and Longi- tude.— The Obliquity of the Ecliptic determined from Ob- servations made near to the Solstices . — The Reduction of Zenith Distances near to the Solstices, to the Solstitial Zenith Distance.—Formula of such Reduction.—Its Application . Investigation of the Form of the Solar Orbit. — Kepler’s Discoveries.— The Computation of the relative Values of the Sun’s Distances and of the Angles described round the Earth.— The Solar Orbit an Ellipse. — The Objects of the Elliptical Theory. In giving a denomination to the preceding part of this Volume, we have stated it to contain the Theories of the fixed Stars such theories are, indeed, its essential subjects ; but they are not exclusively so. In several parts we have been obliged to encroach on, or to borrow from, the Solar Theory and, in so doing, have been obliged to establish certain points in that theory, or to act as if they had been established. To go no farther than the terms Right Ascension, Latitude, and Longitude. The right ascension of a star is measured from the first point of Aries, which is the technical denomination of the intersection of the equator and ecliptic, the latter term de- signating the plane of the Sun’s orbit the latitude of a star is its angular distance from the last mentioned plane ; and the longitude of a star is its distance from the first point of Aries measured along the ecliptic. The fact then is plain, that the theories of the fixed stars have not been laid down independently of other theories : and it is scarcely worth the while to consider whether or not, for the 3 i Digitized by Google 430 sake of a purer arrangement, it would have been better to have postponed certain parts of their theories till the theory of the Sun’s orbit, and of his motion therein, should have been esta- lished. According to our present plan, indeed, (a plan almost always adopted by Astronomical writers) we shall be obliged to go over ground already trodden on. But we shall go over it more care- fully and particularly. In those parts of the solar theory which it was necessary to introduce, either for the convenient or the perspicuous treating of the sidereal, we went little beyond ap- proximate results and the description of general methods. For instance, in pages 137, 138, it is directed, and rightly, to find the obliquity of the ecliptic from the greatest northern and southern declinations of the Sun. But the practical method of finding such extreme declinations was not there laid down ; and on that, as on other occasions, much detail, essentially necessary indeed, but which would then have embarrassed the investigation, was, for the time, suppressed. Such detail is now to be given together with other methods, that belong to the solar theory. But it may be right, pre- viously to enumerate some of the results already arrived at. In Chapter VI, which was on the Sun’s Motion and its Path, it was shewn that the Sun possessed a peculiar motion tending, in its general description, from the west towards the east, almost always oblique to the equator, and inequable in its quantity. These results followed, almost immediately, from certain meridional observations made with the transit instrument and mural circle. By such observations two motions or changes of the Sun's in the direction of the meridian, place are determined ; one the other in a direction perpendicular to the meridian. The oblique motion of the Sun, therefore, is, in strictness, merely an inference from the two former motions : or, if we suppose die real to be an oblique motion, its two resolved parts will be those which the transit instrument and mural circle discover to (see is an equable one. us ; neither of which motions p. 126.) Digitized by fcoogle 431 But although the two resolved motions are inequable, it docs not at once follow that the oblique or compounded motion must be inequable. For, if it were equable, the resolved parts, namely, the motion in right ascension, and the motion in decli- nation, would be inequable. Some computation, therefore, is necessary to settle this point, and a very slight one is sufficient. Thus, by observations made in 1817, h 8' July 1, O ’s JR. 6 40" T.7 Decl. 23° 44" N. 2, 6 44 9-7 . 23 4 35 Jan. 1, 18 17 2.2 23 1 20 S. 2 18 51 27.0 2256 14 a • Compute the longitudes of the Sun by means of this formula. 1 X sin. O ’s long. = cos. O 's dec. X cos. O ’s .41, and we have Difference. ( July 1, O’s long 3" 9° ll 39") .0° 57' 1 1", Jan. 1, 9 10 48 38 ) .1 1 10 . 2, 9 11 49 48 I The oblique daily motions then, instead of being equal, are to one another as 3433 to 3670. Besides the results relating to the Sun’s path and motion, there were obtained, in Chapter VI, other results, such as the obliquity of the ecliptic, and the times of the Sun entering the equator and arriving at the solstitial points. But the methods by which the results were obtained require revision, or rather we should say, that these methods having answered their end, namely, that of forwarding us in the investigation which we were theg pur- suing, may now be dismissed, and make way for real practical methods. We will turn our attention, in the first place, to the determi- nation of the obliquity of the ecliptic. If at an Observatory, the Sun arrived at the solstice exactly when he was on the meridian, the observed declination would Digitized by Google : 432 be the measure of the obliquity. But it is highly improbable that such a case should happen : nor is it, indeed, on the grounds of astronomical utility, much to be desired. A solitary obser- vation, under the above-mentioned predicament, would not be sufficient to establish satisfactorily so important an element as that of the obliquity. It would be necessary to combine with it other observations of the Sun’s declination, made on several days before and after the day of the greatest declination, to reduce, by computation, such less declinations to the greatest, and then to take their mean to represent the value of the obliquity. In such a procedure, it is clearly of little or no consequence, whether the middle declination be itself exactly the greatest, or whether, like the declinations on each side of it, it requires to be similarly reduced to the greatest. The reduction of declinations to the greatest, which is the solstitial declination, is an operation of the same nature, and founded on the same principles, as the reduction of zenith distances observed out of the meridian to the meridional zenith distance : the formula* of which latter reduction, together with their demonstration, were given in pages 418, &c. It is con- venient, however, on the present occasion, to modify the result of that demonstration, or to express it by a different formula which we will now proceed to do. Let then, d(=St) be the Sun’s declination, d! ( — Xi ) the solstitial, X ' O ( = <v S), O ( = 90°) the corresponding longitudes, w, the obliquity of the ecliptic, Digitized by Google ; 433 then, by No per’* Rules, we have sin. d = sin. © . sin. w, sin. d' = sin. sin. © w ; consequently, ct — sin. 90° sin. d = sin. w (siu. — sin. © ), or (see Trigonometry, pp. 32, 42), — w d w + d . « sin. = sin. , sin. - i< 90° . cos. w , if — — © 2 2 2 Let w (= d!) = d + 2, then. ^ • _ - . I — - sin. sin. - sm. cos. w 1 = w , 2 V 2/ 2 o f 2 . 6) - -4- - and, sin. - vcos. w cos. sin. w sin. -> = sin. w . sin. 2 l '' 2j 2 * • * . 2 2 2 „ . 2 2 — Substitute, instead of sin. - cos. - and , and 1 2 , 2 , 2 48 8 3 1 respectively, (suppressing for the present sin. l", sin. l", sin. l", by which 2, 2’, 2*, ought, respectively, to be multiplied), and we shall have this approximate expression. ,2 * 2l - f , < cos. to sm. w - [ ^ cos. w — —H \2 48/ 1 8 2J - tan. w . sin. 2 2® 2 whence — = 2 48 1 t- - tan. w : 8 2 or, nearly, 2 2o i 2 . ,U( ...2 . ..2 2 I - . sin. - 1 -.- - = tan. w in. < tan. «> + —— . tan .’u>} 2 2 ( 2 222 22 i 2 2 2 6 2 * 2 * 2 ’ ; . 2 . ; from which expression, approximate values of - of sufficient * , Digitized by Google exactness may be obtained : for instance, to obtain a first approximation, neglect the terms on the right hand side of the equation, that involve S, and 5 . M (1st value) - = tan. to . sin. - . Again, retain the terms involving 8 and neglect those involving 4 S , and u \ . - . to ? (2nd value) tan. to . sin. tan. to sm. tan. 2 J U . « . , - 3 - = tan. to . sin. — tan. w . sin. 2 2 Again, substitute this new value, and neglect those terms that u involve higher dimensions of sin.
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