254 P A R P A R A glance at the table is sufficient to show that neither ap parent magnitude nor apparent motion affords a criterion of the parallax of any fixed star. Similar researches must, in fact, be carried out on a much more extended scale before any definite conclusions can be drawn. At present we can only conclude that different stars really differ greatly in absolute brightness and absolute motion. The following are the formula} which will be found most useful in computing the corrections for parallax: For the Sun, Moon, and Planets. Put 7i =the equatorial horizontal parallax ; A =the distance of the object from the earth ; l and = the geocentric and apparent zenith distances respec tively; A and A = the geocentric and apparent azimuths respectively; . <f> and <p = the geographical and geocentric latitudes respec tively; p = the earth s radius corresponding to 4> ; a and a =the geocentric and apparent right ascensions of the object respectively; 5 and 5 =the geocentric and apparent declinations of the object; t =the hour angle of the object (reckoned + when west of meridian). 1. To find the parallax of the moon in zenith distance and azi muth, from the observed (or apparent) zenith distance and azimuth. Put 7= (<-< ) cos A . Then sin (( - ()-=p sin TT sin ( - 7); _ p sin TT sin (<p - <p ) sin A - 4. To find the parallax of the sun, planets, or comets in right as. -ciision or declination. 4 ir cos sin t The corresponding quantities are found with all desirable precision for the sun and planets by the formulae - =pjrsin ( - 7); or approximately = ir sin ; A - A = pir sin (<j> - <j> ) sin A cos ; the latter quantity may generally be neglected. 2. To find the parallax of the moon in right ascension and decli nation from the true (or geocentric) right ascension and declination, psin TT cos cos t cos 8 tan (a - a ) = tan 6 tan (45 + J0) tan t . tan (f) cos ^(a - a ) Put then Put fl /_psiuirsin0 cos(7- 8) sin 7 then tan (8 - 8 ) = tan ff tan (45 + *0 ) tan (7 - 8). 3. To find the parallax of the moon in right ascension and decli nation from the observed (or apparent) right ascension and decli nation. 3 ,. p sin IT cos a> sin t sm(a-a) = - ; cos 8 tan 7 = )si(a- a ). C0s[< - |(a - a )] Bin (8 - 8 ; - sin 7 bourg, ser. vii. vol. i. (0" 1470" 009 ?); Brunnow, Dunsink Obser vations, vol. i. (0" 2120"-012), vol. ii. (0" 1880" 033) ; Hall, Washington Observations, 1879, Appendix I. (0"-1800" 005). 9. Krueger (heliometer), Ast. Nach., 1403 (0"-1620" 007). 10. Johnson (heliometer), Radcliffe Obs., vol. xvi. p. xxiii (0" 138 0""052). 11. Wichmann (heliometer), Ast. Nach., No. 841 (0" 087 0" 02) ; Brunnow, Dunsink Obs., vol. ii. (0"-0890" 0l7). 12. Brunnow, Ibid. (0" 0700" 014). 13. Brunnow, Ibid. (0" 054 0" 019). 14. Gill and Elkin, Mem. R. A. 8., vol. xlviii. p. 40 (0"7470"D13), p. 51 (0"76 021), p. 71 (0"780" 028), p. 82 (0" 68 027), independent investigations. 15. Gill and Elkin, Ibid., p. 97 (0"-870"-009), p. 115 (0" 39 023), independent investiga tions. 16. Gill, Ibuk, p. 154 (0" 285 0" 02). 17. Gill and Elkin, Ibid., p. 130 (0"-270" 02), p. 138 (0"-1700" 03). 18. Gill, Ibid., p. 160 (0" 166 0" "01 8). (0" 140"-02). 20. Elkin, Ibid., p. Elkin, Ibid., p. 184 (0" 030" 03). (-0" 0180"-019). 1 In the case of the sun, planets, and comets this distance is generally expressed in terms of the earth s mean distance from the sun, that distance being reckoned unity. 2 Here must first be found by subtracting the value of - from the observed value of . 3 In preliminary computation of (a - a ) employ 8 for 8. With this value compute 7 and 8 - 8 . Finally, with resulting value of 8, correct preliminary computation of a - a . 19. Elkin, Ibid., p. 174 (0" 060" 02). 22. Gill, Ibid., p. 180 21. 167 a - a = cos 8 tan <b tan 7= -7-; cos f sin (7 - 8 ) fnip 0and r o and 8 sin 7 When the distance of the object from the earth (A) is given (the earth s mean distance from the sun being reckoned unity), as is usually the case in ephemerides of minor planets and comets, we have mean solar parallax 8" 78 7T = . . A A The reader will find the proof of these formulae in Chauvenet s Spherical and Practical Astronomy, vol. i. pp. 104-127. For the Parallax of the Fixed Stars. = the maximum angle subtended by the mean distance of the earth from the sun at the distance of the star, = the star s annual parallax; = the obliquity of the ecliptic; = the sun s longitude and radius vector; and a ) _the star s true and apparent right ascensions and and 8 ) ~ declinations respectively. 1. To find the heliocentric parallax of a star in right ascension and declination, its annual parallax (^j) being known. o - a = -pr sec S(cos sin a - sin cos e cos a) ; 8 - 8 = -pr sin 0^cos e sin 8 sin a - sin e cos 8) -pr cos cos a. 2. To find the effect of parallax on the distance (s) and position angle 5 (P) of two stars, one of which has sensible annual parallax. 6 As=prm cos (0 - M); AP=prm cos (0 - M ); where m sin M = ( - cos a sin P + sin 8 sin a cos P) cos e - cos 8 cos P sin e ; m cos M = sin o sin P + sin 8 cos a cos P ; If -i m sin M = - (cosacosP + sin8sinasinP)cose + cos8sinPsin e ; s L J ?;i cosM = sin a cos P - sin S cos asm P . PARALLELS, THEORY OF. The fundamental princi ples of mathematics have not in general received from mathematicians the attention which they deserve. Mathe matical science might in fact be compared to a building far advanced in construction. As to the firmness of its foundations there can be no doubt, to judge by the weighty superstructure which they carry ; but the aspect of the building is not a little marred by the quantity of irrelevant rubbish which lies around those foundations, concealing their real strength and security. The question of the parallel axiom in Euclid s geometry is to some extent an exception. There have been endless discussions concerning it. The difficulty is well known, and will be found succinctly stated in the article GEOMETRY (vol. x. p. 378). Those who have treated the subject have devoted themselves either to criticizing the form of Euclid s axiom, suggesting modifications or substitutes (sometimes with undoubted advantage, e.g., Playfair), or to questioning its necessity, offering either to demonstrate the axiom or to dispense with it altogether. It would serve no useful purpose to attempt a complete account of the literature of the subject ; we may refer the reader who is curious in such matters to the various editions of Perronet Thomson s Geometry without Axioms. It will be sufficient to mention Legendre s views, which, although by no means reaching to the root of the matter, may be held as indicating the dawn of the true theory. 7 The delicacy of the question 4 t and S may be used instead of t and 8 in these formulae without sensible error. 5 The position angle is to be reckoned from north through east, the star which has sensible parallax being taken for origin. 6 Obviously, also, P may here express the relative parallax of the two stars. 7 For some interesting controversy on this subject see Leslie s Geometry, 3d edition, p. 292 ; and Legendre, Elements de Qeometrie. 12th edition, p. 277.
