Ml 



EQUATORIAL INSTRUMENT. 



EQUATORIAL INSTRUMENT. 



939 



Correcting for the index error, we have 27 Iff 52* + 8' 9" = 2P 20 1 1", 

 for the inxtrumental N.P.D., whereas the Nautical Alnmnao gives the 

 true N.P.D. of this star = 27 21' 42'. The difference is 1' 41", which 

 is the quantity by which the pole of the instrument is to the west of 

 the pole of the heavens. We have therefore determined the error of 

 the polar axis and the index error of the declination circle, which may 

 be corrected, if necessary, by altering the screws. 



8 Aquilie is very near the equator, and therefore proper for deter- 

 mining the error of collimation in K.A. Tho sidereal time between the 

 observations in 11 29"5, and the difference between the readings of the 

 hour circle is 11 S3 1 ' 2, bunco the error of collimation in the equator is 

 3*7 

 -g- or 1"85, which is to be added to the hour angle of the observation, 



last. W., and subtracted, Inst E. For stars out of the equator this 

 Correction is to be multiplied by the secant of declination. Now the 

 secant of declination for ft Urea; Minoris= 3.82, hence the effect of 

 colliinati'iu for this star = 3'82x ! 86 = 7''1 nearly, ami subtracting 

 this from the hour angle of the first or E. observation, and adding 

 it to the second or W. observation, we have E. 0* 2* 49''5 and 

 W. 0* 14 23 >- 9 for the hour-circle readings corrected on account of 

 collimation. The difference between these is 11 m 3 4*. 4, while the 

 sidereal time elapsed is 11 39'3, and half the discordance between 



4-9 

 these two results -g- or 2" 45 is the error due to the inclination of the 



declination uxii. As this error varies as the tangent of declination, 



2-45 

 which in ft Ursa) Minoris = 3 - 69, the error for any other star = j^gg 



x tan. S = 07 x tan ! nearly. It is evident that this correction is to 

 be subtracted from the instrumental hour angle, Inst. E., and to be 

 added, lust. West. The sign is to be changed if the correction is to be 

 applied to the sidereal time of the observation, that is, if the observer 

 wishes to adjust his instrument (when it is E., for instance), he must 

 make the time of passage later than it is, which is done by lowering 

 the west end of the declination axis. In this case the quantity is 

 0"7orlO"-5. 



No considerable error arises from omitting the effect of inclination 

 upon S Aquilse in the above example ; but it in more satisfactory to 

 deduce both the coefficients of collimation and inclination at once. 



Let e be the constant of the correction for the collimation and i for 

 the inclination, both + when lust. W. and when the correction is to be 

 applied to the hour angle ; then substituting the numerical values of 

 the secants and tangents of the two stars, we shall have for the cor- 

 rected hour angles. 



Ursuo Jlinoris E. 2 56-fi-3'82<r-3-69; 

 W. 14 



Difference ... 11 20-2+7-64c + 7'38i = l 



SAquils, W. Ot 215'-2 + l-OOc + 0-05i 

 K. 13 48-4-l'OOc-0-05i 



39' 



Difference . . 11 33'2 2'OOc 0'10j=llk 29'5, 

 which give for the determination of c and * 

 7-64e + 7'38i'=19-l 

 0t'= 37 



from which t in found= 071 c=l'81 nearly as before. 



In this way, by forming an equation for each star, and combining all 

 the observations in which i has a small coefficient into one equation (when 

 stars have south declination, or are sub polo, the coefficient is negative), 

 and those where the coefficient of i is large into another, the value of i 

 and consequently of c, may be determined with great accuracy. It is not 

 absolutely necessary that the observations for deducing these corrections 

 should be near the meridian, but it is desirable that as little time as 

 possible should elapse between each of the pair of observations, on 

 account of the variation of refraction and of the effect of polar 

 error, if that be considerable. As a general rule, it would be well to 

 keep within a few minutes of the meridian, for in the above example 

 the variation of refraction iu 3 Ursic Miuoris is 0'5, while the variation 

 of the effect of polar error is no less than 1''3. Besides, the instrument 

 is always moat perfect near the meridian, and is to be used there when 

 possible. 



If the errors be corrected by adjustment, the index error of the hour- 

 circle is simply the difference between the observed hour angle and the 

 true hour angle. Or supposing the true sidereal time unknown, the 

 index error must be determined by a level, as we have described above, 

 after placing the declination taxis horizontal. But if instead of actually 

 adjusting the instrument, the errors are noted and corrections applied, 

 we have yet to compute the effect of polar deviation upon the obser- 

 vations in R. A. before the index error can be correctly obtained. In 

 order to do this, we must consider the polar deviation more minutely ; 

 and as we have reason to think that from a want of skill in detecting the 

 polar error of an equatorial, or in applying the corrections which 

 depend upon this error to observatioiiB, especially in H. A., observers 

 have been led to impeach too hastily the character of instruments, we 

 shall explain thU part of the subject very fully. 



Let P b the place of the pole of the heavens, and p that of the pole 



of the instrument when prolonged indefinitely, M seen on the sphnv 

 of the heavens by a spectator outside ; K P z the meridian of the place, 





which, when produced towards z, passes through the zenith and the 

 south point of the horizon ; EPW the 6-hour meridian, which panes 

 through the east and west points of the horizon. Let fall p m, and p n 

 perpendicular on z jj and E w, and let p m or P n = *" and j> n or P m 

 =y". It is presumed that the index error of the declination circle has 

 been obtained by reversed observations, which is indeed the universal 

 rule, and therefore that the observed N.P. distances, corrected for 

 index error and refraction, express the actual distances of the stars 

 from p. 



Let a tolerably distant star be observed, the place of which is in the 

 direction of p t when continued ; draw p k perpendicular to p s, and 

 also p 9 from the true pole towards the star, hence z p s is the true hour 

 augle of the star. 



Then, since the instrument is nearly in adjustment and the star not 

 very near the pole, p s and p s are nearly parallel, and p i- is perpen- 

 dicular to both, hence the distances of the star from p and k are equal, 

 and the effect of polar error on the N.P.D. of the star is to moke its 

 instrumental polar distance too large by p k. Drawing in > perpen- 

 dicular, and p IB parallel to p t, 



=p m x cos. mp r+P m x cos. m r f ; 



or, since mp v is the complement of up r, which is = z r s, aud m r i 

 = zrs, 



j) k = x" x sin. east hour angle + y" x cos. east hour angle. 



From this formula, if x and y be known in quantity and direction by 

 observations in the meridian and at six hours from it, Ihe value of p k 

 may be computed aud applied as a correction to the observed N.P.D. 

 of any star, and the observed hour angle will serve for the compu- 

 tation. 



In practice we have found it sufficiently accurate to draw p n p on a 

 scale where seconds are visible, that is, equal to about ^ inch, and 

 then making the angle nps equal to the observed hour angle, and 

 letting fall tk perpendicular on p s to ascertain the value of ji I, -by 

 compass and scale. It is not possible to commit an error of more than 

 2" or 3", which is generally of little importance ; and besides the supe- 

 rior rapidity of the operation, there is no danger of confounding the 

 tii/n of either part of the correction, whether the star be above or 

 below pole, which even careful and experienced computers can scarcely 

 at times avoid. In any case the graphical process will afford 

 useful check. 



The effect of the displacement of the polar axis upon the readings 

 of the hour circle may easily be gathered from the same figure. 

 Suppose the two lines PS and pi to be continued till they meet at 

 the star, and to be produced, if necessary, until they cut the equator 

 in 3 and a. The'reader may imagine or draw the figure. If the star 

 be north of the equinoctial, the lines p 8 2 and p s a cross at the star, 

 and the hour angle corresponding to p is to the east, and greater than 

 the hour angle corresponding to the true pole r. Hi-ncc the reading 

 of the hour circle is too small in the case represented in the figure, or 

 2 (r is to be added to the reading of the hour circle. Also, since 

 a 2 : P k : : sin. S : cos. 8 .*. a S = p i x tan. S. Again, 



r t = m v m w 



pmxsia. mpv-rntx sin. oitw 



= x" x cos. east hour augle y" x sin, east hour angle. 



