PARALLAX.] 



ASTRONOMY. 



1029 



case of the moon, a correction will be necessary, which is 

 found thus 



_ . , p" 3 sin. 3 1* 

 sin. p = p sin. 1 * 77 nearly ; hence 



, ;>" 3 sin. 3 1" / , , ir" 3 sin. 3 1"\ . 

 p" sin. 1"- = Iv" sin. 1" ? sin. Z' 



6 V 6 I 



butp" 3 sin. 1 ' -a-" 3 sin. 3 1" sin. 3 Z', by neglecting powers 

 higher than the third ; hence, by substitution, 



/("sin. l" = 7r"sin. 1" sin. Z ^ (sin. Z' sin. 3 Z') 



, . ., , 7r" 3 sin. 3 1* . , ,, 



= TT sin. 1 sin. Z sin. Z cos. Z 



6 



Consequently the correction is 



r-^~ n. z ' cos. 2 Z' . . . (5) 



This correction is applied in the Greenwich Lunar Re- 

 ductions from 1750 to 1830. 



Again, since sin. ir = =-, for any other distance D', 

 we shall have sin. *' =- ; therefore, 



sin ir sin ir' -1 - 



D ' D ' or D~ D" 



If D' be expressed in parts of D taken as unit, we have T 

 = ~: ; therefore, if n- be the sun's horizontal parallax at 



th. mean distance from the earth, and D' the distance of 



any planet from the centre of the earth, expressed in 



7t of the sun's mean distance, the planet's horizontal 



1 

 parallax will be represented by JT X *y and for a planet 



we shall have 



p = x -g sin. Z 1 ; . . . (6) 



It is by this method the parallax of the planets has 

 been computed in the Greenwich Planetary Reductions 

 from 1750 to 1830. 



Equation (1) gives the means of finding the parallax 

 when the true zenith distance is given ; thus 



sin. p = sin. sin. (Z + P\ *"* v (sin- Z cos. p + coe. 



Z sin. p) 

 tan.p = sin. v sin. Z + tan. p cos. Z sin. ir 



sin. TT sin. Z 



tan. t) = : ?7 



1 1 BUI. IT cos. Z 



(7) 



And by the usual method of development, 

 P 



sin. Train. Z sin. 1 ir sin. 2 Z sin. 3 a-sin. 3Z 



+ ~2 sin. 1" " t ~3 sin. 1" ' &c - ' 



sin. 1" 



We have seen that sin. tr = -= ; hence, if D be very 



great, IT will be very small ; and, therefore, bodies at an 

 extremely great distance, compared with the radius of 

 the earth, have no sensible parallax. 



Before proceeding any further, it may be well to re- 

 mark that the value of sin. p depends on lines drawn 

 from the centre of the earth, or from what is called the 

 geocentric zenith, which differs from the astronomical 

 zenith for this reason : All astronomical instruments 

 lifted for determining the north polar distance of an 

 object depend on a zenith point, or something equivalent 

 to one ; and this is nothing more than the instrumental 

 reading corresponding to a vertical position of the tele- 

 scope, and is determined by observing the same star by 

 direct vision and reflection. The zenith point is there- 

 fore the reading for a point perpendicular to the horizon ; 

 or, which is the same, a line drawn from the point thus 

 determined in the heavens to the place of observation, is 

 a normal to the curve by which the earth is generated, 

 supposing it to be a surface of revolution. The earth is 

 supposed to be a surface of revolution, generated by an 

 ellipse revolving round its shorter axis ; the ratio of the 

 axis beint; 300 : 299. Now, it is well known that nor- 

 mals to an ellipse do not pass through its centre ; hence 



the lines drawn from the centre of the earth to the place 

 of observation, which determines the geocentric zenith, 

 will make a small angle with the normal to the earth's 

 surface at the same place, which determines the astrono- 

 mical zenith ; and this angle is called the angle of the 

 vertical. The geocentric zenith is nearer to the equator 

 than the astronomical ZBiiith by the value of this angle ; 

 and as all the bodies (some comets excepted) for which wo 

 have to compute parallax, lie to the south of the zenith, 

 the angle of the vertical is subtracted from the astrono- 

 mical zenith distance, in order to have the geocentric 

 zenith distance. This angle is easily determined, for it 

 is the difference between the astronomical and geocentric 

 latitude of the place of observation, and is thus formed : 

 Let P (Fig. 218) be the pole of the earth ; E a point of 



Fig. 218. 



the equator ; O the place of observation ; Z' the geocentric 

 zenith ; Z the astronomical zenith ; then Z' C E is the geo- 

 centric latitude = L g, and Z G E is the astronomical 

 latitude L a, and Z'OZ= COGis the angle of the 

 vertical. Let o and 6 be the axis of the ellipse, the 

 e illation of the normal Z G is 



o* y' 

 V V'=&- ^ x ~ x= 6 



therefore, 



a 1 

 tan. La = =-. tan. ~Lg, or tan. L$r 



.... (9) ; 



300' 



hence, L g becomes known. For an ellipticity of 



and the astronomical latitude of Greenwich, viz., 61 28' 

 39', we find the geocentric latitude, 51 17' 29", and con- 

 sequently the angle of the vertical = 11' 12". 



The value of r 1 , by which the moon's horizontal paral- 

 lax must be multiplied, to have the horizontal parallax 

 at the place of observation, is thus computed (Fig. 219) : 

 Fig. 219. 

 P 



CO=r', OCE La; then by the property of the ellipse 

 ft 2 *" + a 2 y 2 = a 2 6", we easily find 



a 2 sin. s L<7-f-& 2 cos. 



aU" 



'a 2 (a 2 6 3 )co3. 2 L0 



6 



.'. r = 6 see. 



_._2r-cos>. Lg 



where sin. 2 cos. 1 L a. 



a' * 



