fi 1260-1263.

l'he prominence, which iR due in many astronoiiiicsl re- ortlcr to restore this uniforiiiify , which is otiviously of the searches to the long and excellent series of the Greenwich iiiost esseritial iiriporhiicbe, 1 have rel'errctl all the otiser- mericliorial oliservations, gives to any changes of the instrri- vatioiis to the Circle reatlirigs. which corresporrtl to the Naclir nients, by means of which these ohservatioiis are procurecl, observations of the wire. It iiiight have heen tlcsirahle to a higher and more general importance, than they woulcl other- get rid, as much as pnssilile, of' pere~nalecpitions in the wise possess. Hence the interest, with which astrononicrs reading of iiiicroscope - iiiicronieters etc. hy iisiclg for each are wont to regRrtl the construction and cfhiency of any new observer his own Zenithpoints. A closer inspec:tiori sjiotv,, instrunient of superior pretensions, is greatly enhanced in however, that, owing to several ciiwes, this (:nurse is for the case of the powerful Greenwich Transit Circle, and the the past observations inipracticable. I have 'coiiserperitly asiral question, concerning the degree of correctness, which considered it best to adopt the same periods of uri;iltered the results of a new apparatus have attained, acquires addi- Zenithpoints, as have Iieerl used in the Greenwiclr rechictioris. tional claims to be answered. As I an1 not aware, that a The values of the corrections, which it was accordingly seces- strict determination of this point has yet been attempted, 1 sary to apply to the single ohservations, fluctuate I)ct\veeri shall here niake it the suhject of inquiry with reepect to -fO"45 and TO"71. Polar Distances, so far as it may be safely decided from The correction for fl ex u re, which has heen eriiployed the data, which arc furnished by the printed obsevations of' in the Greenwich reductions, is incorporated in the tiible for the first four years. division errors. The amoimt of the correction is not NtatecI and the appendix to the volume for 1852, to tvbich tbe § 1. reader ie referred, does not supply the deficiericy. Prorii The stars, which 1 bave selected for the purpose of the tables it appears however, that the adopted correction irr reducing all their available observations to some conimon 1851-1852 is +0"73 sinz and in 1853-1854 +O'50 sittz, epoch and of comparing the single results with each other, where Z denotes the southern Zenith Distance. The latter are those of the enlarged catalogue of the Nautical Almanac coefficient is the mean of the three deterniinafions of the (so far as it concerns Greenwich and with the exception of horizontal flexure, taken Iiy help of the colliniators arid nien- the star vorioois, of which there are no observations), the tinned in the appendix. But I caniiot discover the origin of clock-stars given in the Greenwicb volume fur 1854 and, the other coefficient, unless it he assunied that iri coiiibiiiing besides a few others of special iatereet, all those, which the two earlier deterniiriatioris cloulile weight has heen allowed have been observed at least ten time8 during the four years. to the second. In order to restore uniforniity, the ohser- That the observationsl of these stars niay allow of a strict vations of t.851 - 1852 have been corrected, c1o that they comparison with each other, it is first neceesary to Clear the contain the same flexiire correction +O"jO sfit Z, which is single results, which are given in the Greenwich volumes, *) employed in succeeding ,years. of the systematic errors introduced by the w 9 n t of u n i f o r - The reduction, which the rcflexion - observalioris require ni i ty in the deduction of the adopted Z e n ith p IJ i n t s. In 011 eccount of the divergency of the norilia18 at the $ace of the quickoilver ourface arid at the Circle's centre, *) In the aection ,,Mean North Puler Dietansce of atare deduced has also been applied, a reduction, which, so far as I know, frum each day's observation." has always heen neglected in British oliriervatories. In the

f) The prcnent payer W86 originally writtdd with a view to its bein& printed in England, HI the yrolmlile length aeemed tii t'orliid ite ineertiun in tht Aetron. 'Nwhr. The c*lrcumhnL!r, which yretented it frum bdng pulilidhed iii the wily first intended, render any essential changes undihitakde. The rluthor huyee therefire, thht thid will be bnme in mind, if miwe poitit8 nrc found ru be treated in rather P dilfercnt mitniier Proin thnt, which he would otherwide had deemed most ruit;rlilo 53r Hd. I2 179 Nr. 1260. construction of the present Greenwich Transit Circle the quick- Alm. and clock-stars excepted) and if only the exact means silver is carried by an apparatus, which keeps it nearly at of the dates of observations were given. an equal distance of about fronr the centre; the correction 96". The observations have then further beeri reduced for amnunts consequently to sh +0"16 2. precession and proper motion to 1855.0, which epoch has In order that the Greenwich annual results may be heen chosen with a view of preparing for the future addition hrought up properly to some comnion epoch, they require, of the results of following years. For the same reasou the besides the reduction for the intervening years, two minor correction of the colatitude has also been employed, which corrections. The first of these is owing to the circum- has been deduced in the volonie for 1854, so that the reduc- stance, that the observations are, or profess to be, reduced tions contain the eame colatitude J8°31'22u00, which is to Jan. 1 Greenwich mean noon and that consequently there adopted at Greenwich from the beginning of 1856. is every fourth year a break of continuity in the progress of the epochs. For tlie future this source of needless in- After applying the different quantities just mentioned, accuracy is happily stopped, since in the Naut. Alm. for and excluding those observations, which have been found 1857 Bessel's reckoning is adopted, according to which the too discrepant and which will he spoken of presently, I have sidereal year begins at the moment, when the mean AR. of taken the means and, after comparing with them the single the Sun is = 280'. Front 1857 to 1860 therefore this cnr- results, have deduced the probable errors of the latter. rection will only be needed by those star-places, which are There seems to be no fixed rule laid down at Greenwich reduced by means of the Greenwich day numbers. - The tor the exclusion of discrepant observations; at least in some second of these minor corrections has its origin in the cases observations are without any apparent special reason neglect of proper motion in making the reductions to the he- rejected, which are less discrepant than others, which are ginning of the year for the stars, which are not contained retained. Since niistakes have obvinusly soinetinres happened in the Naut. Ah. This neglect does no harm in the annual and since therefore sonre assumption in drawing the line catalogues, as the fraction of the year is there given, which hetween less accurate observations and mistakes cannot be corresponds to the mean date of the ohservations. But it is avoided, 4 have considered 4" to be a fair liitrit for stars otherwise in the Greeiiwich general catalogues; for , since of lcss than 70" or 73" %. D., so that, in their cases, every the annual variations, which have been iieed in their con- resiilt has heen rejected, which differs niore than 4" from struction, include proper motion and yet the deficiencies for tlie riieari of all the results, the discordant one included. lo the respective fractions of the different years are not supplied, conformity with this rule, ten observations have been excluded the accuracy of the nrean places is by this oversight sonie- tiesides those already thrown nut in the Greenwich volumes; times very sensihly affected, as may he seen from a few but of the latter on the other hand four have heeii retained. instances in the last Greenwich general catalogue (Cat. of The results for all the stars selected have then been 1576 stars for 1850), iu which the follnwiiig places require arranged in the following table in the order of increasing on this account the corrections North Polar Distances, D and R denoting the results of the in AH. inSPD. - - direcl and reflexion-observations, n their nunilier und 8 their No. 46 VCassiopeae -0' 13 -0"15 prohable errnr as fnund hy comparing the single obser- 47 B. F. 75 -0903 -It00 vstions with their means. In a few cases of large proper 159 Piazzi 11. 123 -0909 +I902 niotion and in all eases, where the number of observations 770 1830 Groonibridge -0711 -1982 etc. is less than 10, these prohahle errors are omitted. - The Of course under the circumstances it would be niuch resultiiig North Polar Distaiices Sub Pola are nf course to better, if proper motions were disregarded entirely (the Naut. he considered negative.

NO. AR. Star D 1855,O 11 6 R n 6 -- - -7 - Lc --- 1 20b 9" A Ulnae niin. 1" 7' 27"79 ?I O"548 27"60 'LO O"543 s. P. 27982 31 570 28 t 04 11 525 2 17 a Ursae nrin. 1 27 48r5Q 259 607 48-93 28 784 s. P. 48 t 96 306 622 48 * 36 19 916 3 6 31 51 Hev. Cephei 2 44 50976 31 578 52t61 2 s. P. 51.76 42 618 50 9 42 3 4 18 I9 d Ursoe min. 3 31 tt26 87 474 I ~88 9 s. P. 1968 56 174 . 181 Nr. 1260. 182

No. AR. Star D I E R n E - -h/l ------50 5 10h 8" B. A. C. 3495 0' 58"lI 3 59"77 2 s. P. 579 14 2 6 17 I E Ursae niin. 7 43 53987 27 O"439 53934 15 O"394 s. P. 54922 3 7 15 49 JUrsae min. 11 45 42726 19 531 41 954 10 40 1 8 23 33 y Ceph~i 13 10 36998 24 487 36 9 73 14 731 s. P. 36957 4 9 14 28 5Ursae min. 13 39 33766 3 339 1 I 1 s. P. 34 9 69 2 10 14 51 ,B Ursrle min. 15 15 7919 42 565 7979 20 480 s. P. 7934 2 11 12 27 x Draconis 19 24 43905 15 533 43 7 55 12 359 12 21 27 p Cephei 20 4 31941 64 374 31 957 30 496 s. P. 30974 4 13 20 2 p Dreconis 22 32 23702 17 472 22969 15 513 14 4 40 9 Carnelopardali 23 54 39910 I8 532 40913 15 79s 15 9 20 hUreae mai. 26 18 28935 16 566 28941 15 726 16 10 55 m Ursae niaj. 27 28 2942 40 695 2763 31 687 s. P. 2306 10 633 17 21 15 a Cephei 28 I 39999 50 487 40920 16 488 s. P. 39 9 99 4 18 16 22 n Draconis 28 9 24709 22 41 5 24979 6 19 0 32 b Cassiopae 34 15 31925 17 37 I 30959 I s. P. 31 997 P 20 11 46 7 Ursae miij. 35 29 57930 17 784 s. P. 56931 6 21 17 27 B Draconis 37 35 23908 16 604 22 9 23 8Ursae maj. 37 39 53724 14 09613 s. P. 52 7 67 10 17246 23 17 53 y Draconis 38 29 32953 37 0,481 s. P. 34944 3 24 13 42 ~Ursaemaj. 39 57 41,115 17 494 s. P. 41937 8 25 3 14 a Persei 40 39 32987 19 634 s. P. 30925 I 26 8 49 1 Ursae maj. 41 23 33912 21 669 s. P. 33 9 07 8 27 56 a Aotigae 44 9 18913 20 580 t9912 3 28 20 36 1y Cygni 45 14 9957 49 555 9986 2 s. P. 11 990 3 29 12 49 I 22 Canum 50 53 51954 15 800 51 937 3 30 11 45 B.A.C. 4010 51 $14 28969 13 31 11 51 Lal. 22547 18 59965 14 (531) 32 18 32 J Lyrae 20 55986, 94 630 56 9 30 4 33 21 0 61' Cygni 57 40980 46 528 41958 2 34 21 0 61"Cygni 57 46903 28 522 46 9 06 i 35 0 49 ,u Andromedae 52 17 17942 15 472 16963 2 36 10 19 31 Leonis min. 52 33 5-15 I1 567 5790 3 37 12 p Andromedae 55 8 57990 9 38 18 45 ,61 Lyrae 56 48 10969 104 486 10786 8 39 18 45 p2 Lyrae 48 49-91 26 559 51 935 3 40 4 48 1 Aorlgae 57 4 5994 36 613 6940 7 41 7 25 a' Geniiriorum 47 53351 62 612 52999 2 42 16 36 Herculis 58 7 54999 47 612 56 $06 20 475 43 2 35 o Perbei 58 10 29989 13 660 30943 7 44 16 55 E Herculis 6t 26937 15 886 27 9 00 8 45 14 26 p Bootis 59 24728 30 453 24 9 47 9 46 21 7 II CYWi 60 21 56-86 69 482 56937 4 47 0 31 c Andronredae' 61 98 35900 31 09645 34 7 65 13 07772 12' 1 83 Nr. 1260.

ho. An. Star D 11 6 R n 6 - - '- -' )rrr - --cy-r -- 48 jll 17"' p Tauri 61'31' lI"51 64 O"622 1 l"89 9 49 7 36 13 Geminorum 37 39965 88 54 1 38967 2 50 01 p! Antlroniedae 42 36979 56 656 37 9 69 3 51 7 55 6 Cancri 18 11929 31 559 11966 6 52 17 41 + Herculis 62 I1 30934 26 397 309 11 7 53 14 39 6 Bootis 18 41905 71 648 54 14 58 + Bnotis 29 4951 24 858 2998 1 55 20 48 32 Vulpeculae 29 30924 40 520 29 9 76 11 O"677 47 40985 SQ 15 29 a Coronae 99 514 41 9 47 2 57 9 45 p Lennis 63 18 44987 36 459 44 9 56 11 816 58 6 35 E Geminorurn 64 43 47965 10 918 59 21 16 16 Pegasi 45 20908 38 663 19955 21 699 60 9 38 E Leonis 65 33 38901 74 623 37984 2 61 19 23 a Vulpeculae 37 34910 16 652 31962 3 62 22 43 p Pegasi 66 9 47986 27 813 47934 5 63 3 36 17 Tauri 20 47949 10 643 47 9 26 1 64 3 39 7 Tauri 20 19995 50 738 48954 2 65 1 59 x Arietis 67 13 32901 75 587 32941 2 66 6 14 p Geniinorum 25 0918 42 570 0993 2 67 66 7 Geniinorum 27 21923 31 715 209 79 5 68 7 11 6 Geminorurn 45 18936 56 735 18998 4 69 8 35 y Cancri 68 0 48989 29 620 70 11 6 d Leonis 40 57985 46 717 57 9 70 6 71 R 24 7 Cancri 69 4 11947 38 546 10995 12 229 72 6 56 3 Geririnorurir 13 17909 17 73 1 17950 3 73 10 12 y Leouis 25 36993 29 662 379 13 5 71 1 17 ,El Arietis 51 10967 27 447 9984 1 7- I3 14 9 a Bnotis 70 3 39900 125 561 389 64 6 76 16 16 y tler(*uli~ 30 12938 25 750 12999 4 77 3d 4 .4rietis 49 30953 13 512 30919 1 78 13 In 7 Bootih 52 25943 88 580 259 17 6 79 4 20 E Tauri 71 8 43974 14 616 419 07 4 80 9 11 83 Caricri 40 58926 38 868 57934 11 685 81 13 40 T Rootis 49 8925 37 616 8903 17 62 1 n2 ti 29 y Geniirioruni 73 28 53907 12 654 83 11 7 0 Leonis 46 43970 12 73 1 t+1 4 28 x Tauri 47 11909 b4 642 65 15 50 y Serperitis 51 41933 18 662 44 9 52 1 46 Y 50 26 Lennis 74 5 21955 12 4 80 67 10 I4 42 Lennis 17 42929 14 77 I nn 'Lo 33 x lrelphini 35 18979 21 679 49921 1 n9 11 i2 ~3 Leoriis 37 3974 70 616 2996 2 90 4 1'2 y Tauri 43 35963 15 440 36 9 70 4 91 10 11 R. A. c. 3579 55 2906 14 419 I- 92 10 39 k Leonis 13 2 27352 12 762 30956 2 93 I8 53 E Aquilae 7 31969 34 561 31975 10 446 91 I 21 Pisciuni 24 11993 32 524 11988 6 95 17 H a Herculis 26 28905 62 699 96 22 58 a Pegasi 75 34 27910 61 68 1 25 9 82 1 97 06 y I'egitsi 37 22972 57 630 98 11 34 < Sootir 38 50920 16 533 99 20 31 8 Delphini 54 24908 15 611 23 9 86 3 100 10 4 34 Leonis 55 52934 11 568 101 18 59 ( Aqnilae 76 20 569 11 84 633 I02 10 1 a Leonis 77 19 33982 92 645 31 980 2 103 17 28 JC Ophiuchi 19 51953 105 565 52915 3 104 8 51 a Cancri 35 1998 17 818 2931 1 105 3 33 A Teuri 55 23900 11 524 23 9 72 1 I06 I9 If o Aquilae 78 39 46971 56 09752 46916 10 0 I 490 185 firl 1260. 1.86

KO. -4R. Stiw D PI E R n- E \-ru ------I - 20"04 107 Ob42m 1 Leorris 78"41' 36 O"869 20"35 7 108 90 x Cnacri 45 4937 32 625 109 20 26 E Delphirii 79 11 12990 36 665 139 10 110 9 33 o Leonis 2i 1974 35 664 1976 ill 19 39 y Aquilae 41 13939 78 560 12 9 82 112 22 34 { Pegasi 55 28928 64 665 28 9 03 113 10 25 p Leouis 56 56907 42 878 54974 114 89 fl Ca 11 c ri 80 22 16920 16 472 115 16 51 x Ophiuchi 23 46949 30 63 1 46923 1 116 18 0 72 Ophiuchi 27 12946 25 539 2 117 11 58 o Virginis 27 41958 16 956 1 I I8 3 19 $ Tauri 46 34927 16 671 119 21 37 Pegasi 47 16952 62 691 15976 2 120 9 53 TLeorris 81 15 13999 34 747 44958 I 121 7 19 fl Canis ah. 25 20986 21 651 122 3 17 o Tauri 29 4~99 24 661 123 19 44 a Aquilae 30 41941 88 713 124 1 38 o Piscium 34 26913 16 528 125 10 58 x Leonis 52 52936 27 785 126 2 20 ,$' Ceti 82 11 33925 32 694 127 11 53 a Virginis 34 37998 31 638 128 5 17 OL Orioriis 37 27917 77 739 129 0 55 F. Pisciurii 53 30918 17 640 130 19 27 ,u Aquilae 55 32968 36 694 131 8 39 E Hydrae 83 3 8923 71 655 132 I5 37 OL Serpentis 6 54939 77 733 133 11 14 aLeonis 10 36916 10 538 134 23 58 w Piwiuni 56 22-66 26 196 135 19 48 ,8 Aqailae 57 8737 62 622 136 7 32 J Caeia niiu. 84 21 24385 79 745 137 22 3 8 I'egijsi 30 50908 11 503 138 15 14 E Serperitis gj 4 57979 14 833 139 23 32 I Pisciuin 9 33769 39 662 140 I 34 Y Pisciuni 14 52909 31 676 141 21 9 a Equnlei 20 57718 22 558 142 17 36 ,8 01) hi ir c*hi 22 6922 36 700 143 10 53 d Lennis 36 in.38 22 770 144 10 35 34 Sextaritis 39 38741 38 681 115 17 19 u Ophiuchi 43 47973 29 597 146 12 A$ BVirgini~ 48 ~,n5 33 668 147 11 20 r LeoriiK 86 20 41959 30 778 148 2 55 aCeti 28 55921 55 658 149 19 18 d Aquilae 87 10 15-07 55 573 150 17 41 y Ophiuchi 14 4710 17 581 151 2 36 y Ceti 22 40922 53 700 152 I1 43 .a Virgiiiits 25 5738 a 153 23 10 y Pisciuni 30 31901 33 778 154 13 54 r Virginis 45 6959 25 180 155 23 19 x Pisciuni 89 32 15951 31 952 I56 13 27 < Virgjnis 51 10767 36 680 157 12 12 7 Virginis 51 38-17 36 647 158 11 30 u Leonis 90 1 21986 22 760 159 5 25 d Orionis 24 37907 30 09747 I60 12 34 y' Virginis 39 10992 11 19214 161 12 34 y2 Virginis 39 14909 11 0 9 889 162 22 28 7 Aquarii 51 48968 33 826 163 21 58 OL Aquarii 91 1 20967 55 815 164 20 4 B Aquilae 14 54916 32 599 165 5 29 .s Orionie 17 54963 31 09810 187 Nr, 1260. 188

No. AR. Stnr D n 6 Nn. AR. Star D n 6 w .-- u *- - -,- - - - '-' - -- 166 22h 14" y Aquarii 92" 6' 59"05 34 O"563 213 5b26"' aleporis 1 07 5 5' 4 6"2 3 15 O"741 167 18 14 ?Serpentis 55 57184 28 727 214 20 21 p Capricorni 108 17 22~34 I 168 16 7 d Ophiuchi 93 15 2171 55 557 21 5 0 36 ,BCeti 37 0921 45 804 169 16 11 6 Opliiuchi 94 20 7163 13 644 216 15 57 p Scorpii 109 24 16-55 28 683 170 0 23 12 Ceti 45 33100 31 693 217 15 59 w2 Scorpii 110 28 22179 10 547 171 13 2 B Virginia 45 49171 18 80 1 218 18 5 ,u Sagittarii 111 5 31170 70 697 172 21 24 p Aquarii 96 12 23193 38 750 219 12 3 B Corvi 48 47727 18 09746 173 23 58 33 Pisciuni 31 7159 13 79 1 220 15 52 d Scorpii 112 12 18188 8 174 23 7 Q Aquarii 49 48110 15 518 22 1 4 59 6 Leporis 34 9907 18 1,005 175 2 10 67 Ceti 97 5 33917 22 590 222 12 27 ,B Corvi 35 39147 63 0,742 * 176 43 01 Eridarii 13 7188 223 81 15 Argus 113 53 20720 34 07660 177 49 o2 Eridani 52 53148 11 224 7 43 Argus 114 29 55157 18 0,889 178 9 20 a Hydrae 98 I 56185 46 624 225 17 13 B Ophiuchi 50 59396 18 11383 179 22 45 h Aquarii 21 0963 14 934 226 19 28 h2 Sagittarii 115 11 57119 2 180 58 ,B Orionis 22 22948 60 652 227 18 19 A Sagittarii 29 48983 12 07717 I81 21 30 Aquarii 30 8108 22 612 228 19 7 JI Sagittarii 30 7976 14 09679 182 22 9 B Aquarii 30 12185 30 800 229 16 21 LY Scorpii 116 6 20,84 38 0,965 183 15 9 fi Lib rae 50 41106 73 796 230 72 dCanis maj. 9 57124 13 17241 184 1 17 8 Ceti 55 58934 54 603 23 1 18 46 aSagittarii 28 19900 6 185 14 5 %Virginis 99 35 47170 23 774 232 18 37 QSagittarii 117 8 9988 16 0,790 186 0 12 1 Ceti 37 40183 10 431 233 19 48 b Sagittarii 33 2141 10 1,174 187 3 26 SEridani 57 6980 13 634 234 19 54 c Sagittarii 116 6 32-34 37 0,847 188 20 40 8 Aquarii 100 1 24925 24 934 235 6 53 B Canis maj. 46 38791 25 19010 189 16 29 < Ophiuchi 16 10109 18 594 236 23 41 d Sculptoris 55 54703 16 1,287 190 13 18 aVirginis 24 10175 129 71 I 237 22 50 or Piscis austr. 120 23 23105 46 1,501 191 14 49 g2 Librae 49 17194 14 863 238 22 23 p Piscis austr. 123 5 16909 11 1,645 192 6 47 8 Canis maj. 101 51 38905 17 YO0 239 5 34 or Columbae 124 9 15969 12 29159 193 20 10 LY~Capricorni 102 57 10110 12 829 194 20 10 a2 Capricorni 59 27125 38 659 § 2. 195 3 51 y Eri d a ni 103 55 26992 37 746 196 11 12 d Crateris 59 39988 45 586 The foregoing mean Polar Distance8 require to be cor- 197 15 15 o2 Librae 104 36 46799 13 0,857 recfed for the constant error of the adopted Zenithpoints, for 198 15 36 qLibrae 105 12 25144 12 19043 the error of the assunied flexure correction and for that of 199 20 13 9Capricorni 14 8199 21 01716 the colatitude and of refraction. But it will be desirable 200 14 43 a1 Librae 23 28113 6 previously to inquire into the law, which the probable errors 201 6 57 cani is maj. 25 20165 7 follow, in order that proper weight niay be given to obser- 202 14 43 a2Librae 26 10159 46 837 vations in different Zenith Distances, and the causes, whicb 203 17 2 ~Ophiuchi 32 27159 19 739 have affected the accnracy of the observations, may also be 204 12 22 6 Corvi 42 27191 23 740 opportunely considered. 205 15 20 (1 Librae 106 12 27985 12 924 For the purpose of deducing the law of errors for the 206 15 46 BLibrae 17 58193 12 664 d i re c t observations, their probable errcw, as they ore given 207 19 34 e' Sagittarii 27 35152 14 842 in the foregoing table, have heeo divided into gwps and 208 6 39 aCanis niaj. 31 12147 75 713 those in each group properly combined. The resuhg mean 209 22 47 d Aquarii 35 26187 13 990 probable errors 6 are found in the following table, io which 210 21 39 6 Capricorni 46 59100 22 698 z denotes the iitean Zenith Distance, n the uuniber if obser- 21 1 21 14 1 Capricorni 107 26 57188 19 885 vations and tn the numbsr of stars in each .group. The last 212 20 58 B Capricorni 48 22108 30 0,710 columns e' and 6'' will be explained hereener. 4 89 Nr. 1260. 190

Group Stare L n m 6 8' bU - / ------I B Ursae maj. S. P. -76' 7' 10 I l"246 01'9 1 1 0"884 2 u Ursae maj. S. P. -65 57 10 I 0,633 721 674 3 d Ursae niin. S. P. to A Ursae min. s. P. -40 20 435 4 5990 6335 5595 4 AUrsae min. s dUrsae min. -36 35 421 4 5714 6298 5555 5 ~Ursaeniin. 5 bursae min. -26 7 112 4 5121 6230 5526 6 x Draconis 5 VDraconis -13 50 242 8 5243 6191 5615

7 a Cassipeae I aCygni T 3 9 227 10 5684 6178 5849 (The mean North Polar Distance of this group is 40°20'.)

8 t 2 Canuni to 31 Leonis inin. +13 9 209 6 5907 6189 6053 9 13 Lyrae 5 pBootis 19 1 333 8 5730 6204 6159 10 {CYPi s VTauri 24 8 860 19 5996 6222 6249 11 CL Arietis = booti is 30 55 760 17 6265 6296 6363 12 y Geminorum s ATauri 37 7 884 24 6276 6303 6465 13 w Aquilae rc EPegasi 41 12 544 14 6819 6345 6532 14 ?r Leonis = aLeonis 43 48 561 14 6961 6378 6576 15 w Pisciuni rc TVirginis 47 16 7 10 20 6619 6432 6639 16 x Piscium I tiorionis 51 57 306 9 7610 6529 6737 17 y .4quarii rc dOphiuchi 54 20 117 3 5993 6594 6797 18 E Ophiuchi rc BVirginis 56 8 62 3 7142 6652 6819 19 P Aquarii 5 67 Ceti 58 2 88 4 6769 6723 6913 20 o Hydrae I BCeti 60 t 299 7 6991 681 I 6990 21 %Virginis : laLibran 61 49 231 7 7244 6907 7075 22 B Canis maj. 63 18 17 1 900 700 716 23 o* and u2Capricorni 61 25 50 2 699 708 723 24 y Erid a 11 i to 02Librae 65 29 95 3 685 71 7 730 25 7 Librae I dCorvi 66 53 121 5 802 729 742 26 (1 Librae rc d Capricorni 67 57 148 6 759 741 752 27 I Capricorni = aLeporis 69 10 64 3 769 756 766 28 /3 Ceti 70 14 45 1 804 771 780 29 ,O Scorpii 70 50 28 1 683 781 789 30 w2Scorpii 71 54 10 I 547 799 806 31 p Sagittarii 72 31 70 I 697 812 818 32 6 Corvi 73 14 18 1 746 827 833 33 ~Leporisand PCorvi 74 0 A1 2 799 846 850 34 15 Argus 75 18 34 1 Ot660 883 886 35 l Argus and d Opbiuchi 76 5 36 2 1, 136 910 912 36 h and +Sagittarii 76 55 26 2 0,696 943 944 37 uscorpii and dCanie uraj. 77 31 61 2 1,034 0,970 0,971 38 @ Sagittarii 78 32 16 I 0,790 1,024 1,023 39 bSagittarii 78 57 10 1 1,174 1,049 11048 40 c Segittarii 79 30 37 1 0,847 1,086 1,084 41 6 Canis maj. and dSculptoris 80 14 41 2 1, 102 19143 1, 140 42 u Piscis austrini 81 46 46 I 1,501 19300 1,296 43 PPiscis austrini 84 25 It I 1,645 1 t 803 1,795 44 OL Columbae +85 27 12 I 2,159 2,170 2,161 The forniula, which 1 have finiilly adopted for the pro- g2 = O"6177" + 0a16552 tang *Z or bable error €, which helongs to an observation in Z. D. z and E = p,i9079] v(i + [8,85641] ton9 '=). N. P. D. w, is this The prohable errors, Hhirh are given by this forrtiiilit, are E = V(E,,'+ q2 tang 22 + 2p" sii, m2), tahul;itetl above as irt the last coluniri hut one. It ilppears in which the constants E,,, p, 4 are to he deteritiiitetl froiii that the forniula is iiisnflicieiit to represent fairly the data. - the foregoing values of E. I ant by no nieaiis satisfied, that Assunling this is the true form of the law; but ott the whole it seems 2) 6' = E~~ + 9' tony 'lz + 2 p2 sin 2w, it follows, that to represent the cliita as far as their rather large discrepan- E = [9,73376] v(1f [8,96703] tany'z + [9,59718] sill 'w) or cies fairly allow, so that I have preferred lint to introduce E'= O"5117' + 0"16492 /nny 'lz + 0'430362siti 'm. any terni, which could not lie inechatiically espliiined. 111 ,' 1 he sun1 of the sqii;ires of the remaining errors, after the calculalioiis however the niore genetal fnrni liein; reduced Ity the cowtant term froni 4489.2 to 1106.67 E = Y/(E,,?+ q2 tm9 22 + 2 p* sin '[w + r:l) atitl then to 165.79 11. the terni, which is proportional to has been assumed. refiaction, is further tliiiiitiislied by this additional term to If E has heen derived froni 71 obscrviitiotis of 71) stars 143.04. It is ery questiortahle, whether the cliffererire of and S denotes the sum of the 111 sunis of tlifferences. froni accwriiry between stars of slow and quick motion is really which the prottable errors for the single sfars have hen eal- wwh, that it may he represented by ii prohable error, which xs ih c.\;ictIy proportinrial to rin m. Indeed, the ohservatinns culated, we have E = -. , where H = 0,815317, arid in the ininiediatc rteighltnuihood of the pole do not seem to Vn (n-nr ) cwnforrn to this assuinptiort, its may he seen by comparing accordingly, hy putting the data with the prnhattle errors E" in the last column of n2x = Eo2 + p2 n2g = -y'l fox 2r the table above, which are calculated according to this for- x2q p2 sin 'L T, x*y = 10 42 = + riiula. But, owing to the want of a sufficient riuniher of the equations of condition, from which the constants E~,Q: observations below the pole, it caiinot well he decided, 0. r are to be deduced, become whether the deviations in the vicinity of the pole are to be attributed to insufficiency of the formula or to some cause, which has affected only the observations in this part of the -- to which may be given the weight Vn(n-m). - The real nievitlian. - Since I am not aware that practical conaide- value of E will then depend on the number of terms, which rations suggest any more general form of the law, I have are retained in the formula. Assuming not gone beyond adopting 1) €2 = E'* + 4"' tang '2 arid treating the equations, 3) the form E~ = E,,' + q2 tan9 '.z + 2p2 sin 2(w + z), which correspond to the 44 mean prohahle errors of the fore- The treatment of the equation of conditiori by the method of going table, by the method of least squares, it is found, that least squares leads then to the final equations: 0 = +8373.2x 4-2218.31~ -4421.3g 4-2097.1 q -5321.9 0 = +2218.3 +6980.8 -1409.5 -1195.1 -3862.5 0 = -4424.3 -1409.5 +5795.7 - 860.3 4-31 83.2 0 = +2097.1 -1195.1 - 860.3 +2579.2 - 578.0, the solution of which gives the constant term of which expression hecomes for stars near ga = 0/1j45g2 + 0*:623* tang 'z f 0"3591* .h 2(w - 14'30'). the pole = O"6455'. The sum of squares of the residual errors is thereby diiiiiiiished to 137.73. - Althollgh it can The introduction of the new term would reniove only the .scarcely be doubted, that some causes, as for instance in- 48" part of the siini of squares of the remaining errors, exact reduction to the nieridiari , bave sontetinies. increased which would he diminished to 140.95. - Supposing (he apparent error of' observations of stars near the pole, I 4) that the ohservations in the vicinity o! the pole (frorii am unable to specily ariy likely cause, which mould explain d Ursae niin. S. P. to 6Ursae min.) are affected by some purely fortuitous errors, the prohable value of which amounts additional error, the probable value of which 1 ani obliged to O"315. Therefore rejecting thie and also the foregoing to assume Constant, the equations, when altered accordirigly, hypothesis and abstaiiiing from farther investigation by nunieri- give for star6 in general cal calculation, I have had recourse 6% = 0"47j52 t O"1 6532 tang 2z + 0"4365'si,i '2w, 5) to graphical construction by drawing by hand the 1'33 Nr. 1261. I94 siniplest curvc, which scenis nearest to satisfy the 4iriginal Star D R R-D i(R+ D)observ, \yI -.-- prohatile errors, when they are laid down as urdin;ttes of - - - 17 E Antlroniedae l"013 O"820 l"575 O"688 14 their Zenith Distances. By constructiiig likewise the curve, 18 3? Vulpeculae 0,440 677 0,694 396 11 which represents the protiahle errors tletliiced from the forniiila 19 p Leonis 702 81 6 19177 161 11 E = \/(0"54172 + 0'j16192 tn7rg 2z t 0N30369sin 'GI), 20 16 Pegasi 510 716 1,102 352 18 it alipears, that both curves, althoirgh they are not identical, 21 7 Cancri 569 222 01612 337 11 ,igrre at least so far, that the prerwencr of the first tu the 22 83 Caneri 606 685 0,798 168 11 aec:nnd could Imlialily not well be justified. - An exact 23 r Bootis 675 615 1,141 242 16 w Aquilae 0,490 O~flO1 knowletlge of the limit6 of Zenith Disfaneea, within which 24 09705 0,170 10 it Iilils he assuiiied that accuracy is affected to the same The Zenith Distances of these stars range froni -40" extent by the discoinfort arising from the hoclily position of to +-to". If here, where only the relatire weight of the R the ohserver, niight have suggested a more precise form of and D observations is to be deterniined, the variations of the law of errors; in the absence 01' this lirrowledge I have the probable errors within these limits are neglected, the the less hesitation in adopting the ahove formula as a fair proper combinatioo of these 382 ol~servatioiis of 21 stars expression for the probable errors, as 1 feel far from certain gives the prohable errors of that the discrepancies of the observatioiiu are really due to D R A-D &(R+D) purely fortuitous errors only. 1 apprehend that herein lies - c-v- ChN- the great and perhaps uncntiquerable difliciilty of this in- O"5876 O"5546 O"9666 O"3783. vestigation. The reflexion observations appear therefore to lie a little In order to determine the prohable error of the reflexion more consistent with each other than the direct once. By observations, its variation has been asslimed to be propor- comparing the single results of the foregoing table, it is tional to the variation of that of the direct ones and there- clear, that this difference of accuracy cannot be warranted, fore the rclative weight only of both kinds of observations but it seems to prove, that the error, which niay be occa- is investigated. To make the results as pure as possible sioned by the trembling of the quicksilver, is nrore than and at the same time to gain some means of estimating the counterbalanced by the error, which has its origin in the fortuitous errors , by which the adopted Zenithpoint Correc- more inconvenient position and sometimes it niay he in the tions ilre affected, the corresponding observations of the stars, greater hurry, in which the direct observations are taken. which have Been observed at least ten times both directly Considering that this latter circumstance has probably more and by reflexion at the same transit, are alorie taken into affected these 382 Observations, than the direct observations account. In this way are foiind the probable errors of in general, it seems best to give equal weight to both direct Star D R R-D #R+D) observ. and reflexion observations and to adopt finally ------e2 = O"5417" + 0@3036%sill 2w + 0'1 64g2tun9 2z 1 zUrsae iirin. S. P. O"637 O"916 l"257 O"628 19 2 A LTIsae niin. S. P. 5.39 525 1,016 205 11 or the reciprocal of the weight 3 hl'rsae inin. 379 502 0,825 225 19 rC1 = 1 + [9,59718]sin 'IU + [8,96703] tang 2~ 4 Ursae niin. 521 784 .q ,082 428 28 the unit of weight beitig that, which an ohservation of ii 5 eUrsae mirr. 39i 494 0,802 196 15 star at the pole would possess, if there were no refraction. 6 (Lrrsae niin. 735 401 01761 135 I0 7 yCephei 521 731 0,913 175 14 § 3. 8 ,B Ursae min. 508 4i? 0,821 269 18 Before entering upon the consideration of the scvernl 9 xDraconis 583 359 0,903 216 12 sources of error, which have coinbinetl to produce those diu- 10 /3 Cephei 362 443 09728 307 28 erepancies aniougst the observations, of which F is the pro- 11 pDraconis 468 170 0,796 181 13 hable amount, it is proper to explain the reason, why in this 12 9 Canielopardali 56 1 795 1,273 306 15 investigation no attempt has been iuade to deduce the tem- 13 /i Ursae niaj. 583 726 09812 453 15 perature - coefficient of refraction froni the Greenwich obser- 11 a Ursae niaj. 623 645 1,064 413 28 vations. When it is borne in mind, that Uessers refraction- 15 aCephei 720 488 0,890 432 16 tables are based upon Observations extending over less than 16 (llerculir 0,841 388 1,030 132 19 two years and made with an instrument, which at the present 53r Hd. 1 3 I05 h r. l!!Cil. .I96 day woiild sc;trcelj* lie cnrisitlerecl one of the first order, it effect of the clitTereiit assuniptioris regarding the c:oiistitufiori is clear, t1i;it the inr~)rnveiiicntof these taliles, or at least of the atniosphere is but little sensible, arises, as 1 need an exaniiiiatioii of their trustworthy applicaBility should rer- scarcely remark, frniii the circuiiistance, that the amount of tainly fnriii an essential part of a tliscussion of iiiericlinnal aqueous vapour iii the atniosphere at the tiiiiea of obser- ohservatioiis which profess to airii at very great accuracy. vation is unkriowri and that its effect on refraction canrint be It appears riot urilikrly , that the Greenwich atniosliliere in cnlculatetl. In order however to niake some proper allowance observing weather is less tlry than that of Koeriigsherg, and for it, Besscl adopted, tvhiit appeared to he the only safe t.hat accordingly tlie Greenwich oliservatioiis niay possibly and open course in this difficulty, when he cleterniined the require a sinaller teiilper;iture-coefIicient than Besael's, which teniperature - coefficient directly froiii his observatiorls and has been clcrivutl I'roni observations at the latter place. The thus took into account the average amount of riioisture in Greenwich observations however (lo not allow an investigatiori fine weather. His result (09003614 for 1" C. with proh. err. on this point to lie made with confidence; for it seems, that 216 units of the last figure) fell about niiclmay hetrveen the the nieteorological iristrunients are open to corrections, without two limits, which were indicated by physical experimeote, the kriowletlge of which it is of course impossible to proceed namely Gay Lussac's coefficient for dry air (0,00375) and satisfactorily in this delicate iiiquiry. Whether the errors tlie coeflicient for air saturated with nioisture (0~003566), of tlie exterior theriiiometer have lieerr ascertained arid taken which Bessef hiinself tletlucetl so an to represent very nearly into ;~ccourit, i* iiot stated and the circiuiistance, that its the kiiotvn physical data. Since the time of this tleteriiiina- reatliiigs iil'e iii a few instalices higher than those of the tinn the higher of these liniits haw heeri diniinislietl, as is niaxiniuiit-llierir~oiiieteriti the iiieteorological oltservatorj, niay well kriorvri , by Rwlherg arid the united teatiiiinny of his possihly Iw cliic to real differences of teniperature. Biit there esperinierits and those 01' Rcpndt iirid 31nyntrs ;ippears to is soiiie t1out)t. with respect to the correctness of the liaro- leave very little tloiilit with respect to the correctness (11' the iiieter rcailiiigs. Tlie coidiidirig reniark of the statelticlit .iii niotlern coellicient for dry air 09003665. On tlte other harid the Greenwich introductions .,this barometer is not fit for it is to lie remarked, that in consequence of this same any delicate purpose, but it is sufficiently accurate for coni- alteration the lower limit has lreen raised s-), so that the putatinn of refraction", I am afraid, must he corisidered iiiore range of the two probable limits has been reduced to less than questionable. To decide the point, 1 have compared than one half of its foriiier extent. Bcssel's coenicient for a conaiderable number of maxim and riiiiiinia readirigs of aveiage nioisfure lies within this range, about one fourth of the baronieter in the meteorological observatory with nearly it from the upper limit, and although this coincidence must sinrultaiieous reacliirgs of the Circle - rooiii barometer, after be attributed partly to chance, it tends at least to dispel baving reduced the latter according to the indicatiotis of the exterior thernioiiieter, in conforniity with the practice followed *) Assuming the constant of refraction for 760""", 0" C. and in the Greenwich computations of refraction. Froni this coiii- d ry uir lo be 60"396 and ndiipting RrgaauL's deterininrlions parisoii it appears, thiit on the whole the readings of the of the elastic force and Gay Lusrac'e number for the relative deneity uf vayour, I find, that the refractions of air eatu- Circle-r~ioiii baronieter require a positive correction of perhaps rated with iiinistiire and of the temperature 7" C. imy be 0". 02 to wake thew agree' with those of the standard liaro- iiearest represented, Iietween the hits -10' and +30° C., meter, but that thc very variable reiriainrler of the differences 60 "4 07 hy the formuln or, if .Wide's density is ofleii large enough to atlect very nensilily the computed 31 + 0 3 0035824 7' 60"4 1 5 refractions iii lower attitudes. Any atteiupt to deduce the is adopted, by the formula The refrac- temperature-coefficieiit would uiider these circunistauces only I + 0,0035264r' lions calculated from these foriiiulne and their rcdidunl errors lead to clnuhtful results arid I have felt still less inclined to are ae follows enter into this hopeless investigation, sioce froiii a prelinii- temyernture Gay Lursac Mu& nary scrutiny of the observations of sonie stars iii lotv iiltitudes p-p.- -. it is rjufficieiitly apparent, that the reiiiaining errors do not -1OOC. = I4OF. 62"651 -0"021 62"623 *-0"040 show airy decided dependence oii temperature. It niny horv- 0 32 60,407 +O,Oll 60,415 +0,019 +lo 50 58,316 +0,027 58,363 +0,052 ever riot be aniiss tb state here sonie corisiderationr, which +20 68 56,368 +0,016 56,435 +0,028 tend to remove any undue distrust with respect to the eniploy- +30 86 54,545 -0,032 54,635 -0,054 ment of Sessers coefficierit for Greenwich. 1 do not know, whether modern experiinenta have cleared The chief difficulty. as regards the regular variations of up the difference existing between thc two determinations mean refractions iri those Zenith Distances, iri tvtiicli the of the drnrify of iayoiir. any serioiis tloulbt respecting the applicaliility ol' the atloptetl pri~li;ilile error of a siiigle ine;isiire of iiitervnl tor the iiiean coel'licient for Greenwich. For the purpose of improving it of sis iiiicroscopes is hind = O"226 nr for a singlo niimos- or even of proving only its trustworthiness the Greenwich cope = O"551 and very nearly the ranie niiniliers iirc given observatioiis are inadequate. by a comparison of the special tle~erinin;itionsduring the first 38 weeks with their weekly iiieaiis. so that the errors of 5 1. division of intervals and the persorial eqii;iticiris iii the readings According to the previous investigatioii the prolilhle error for rune niay possilily I)al;irice erlch other in their effects. of the olwervations ;mounts for stars near the Zenith to But the cnniparison of the corresporitliiig ~leterniinations46- O"578 and for stars in 45' Z. D. to the North and South to cietitly proves, that the conriexinri Iietweeri the real arid the O"568 ml O"660 respectively. These errors, which represent adopted variations of riiris is of thc niost sleiitler kind, that the degree of consistency of the observations cvitliout aiiy the weekly periods irrp quite arhitrary aiicl tliat the adopted regard to the prohalile amount of their constant errors, appear correctioas ;ire open to a coiisideraMe degree of iiiineccssary to be larger than the corresponding errors, by which the obser- iincertairity. It need scarcely lie added, that it would clearly vations of sonie far less powerful instrunients are affected "). have lieen iiiuch preferable, if the weekly instead of the daily As the reality of this differerice seenis to he estalilished determinations had heen discontinued after Sept. 185 1. 01 heyond tloulit liy the great nunilier of observations, on which course the eniployiiient of the Nadir-intervals or any other it depends, it probahly tilust be due to the existence ol'sonie special intervals would require a proper investigation of their particular sources of error, which have effectually counter- true values, hut without such investigation there is little acted the physical superiority of the Greenwich apparatus hope of dimisishing suf'rciently the uncertainty of the deter- over sinaller instrunients. And in considering the different minations , except perhaps hg the questionable expedient of steps of the reductions it is indeed not difficult to point out taking inore numerous observations than are weekly employed. several causes. which niay account for the want of superior 2) It appears that the Telescope-Micrometer is accuracy. assunied to be free front any sensihle errors. Not only are 1) The R u n s of the microscope-micrometers are deter- the irregularities of the screw entirely disregarded, bat also mined every Monday by measuring three intervals between (as 1 have learnt) the usual precaution of giving the last adjacent lines of division and the deduced correction is then turn of the screw always in the same direction is dispensed eniployed in the reductions from the previous Thursday to with iu the observations. Consideriog however, that their the following Wednesday. Except that they are usually chosen exactitude depends directly upon the correctness of the about 20' distant from each other, these intervals seem to be micrometer, 1 apprehend that the assumption of this correct- taken at random and their real values are consequently unknown. ness without a searching investigation must appear to many It would therefore be difficult to form any distinct opinion astrononiers rather hazardous and that they will suspect it to about the degree of approxiination, which in this manner is be the chief cauee of the ahsence of that superior accuracy, gained with regard to the real value of the correctioti, if the which might have been looked for from the Greenwich Transit rule of adopting these weekly determinations in the reduc- Circle. The want of satisfactory agreement in the deter- tions had heen adhered to without exception. But for sonie minations of the horizontal flexure by inearis of the colli- time (froni the beginsinp of the observations to Sept. 16 1851) mators *) tends greatly to strengthen this suspicion. special deterniinations have regulerly been made for the Nadir The investigation of the errors of the niicroscope- observations of the wire, so that it is poseihle to trace during inicroinefers may perhaps he dispensed with without any this period tbc connexion between the real and the adopted great hesitation. For even should all the screws possess weekly variations of the correction for runs. From the agree- the same irregularities, the corresponding errors are likely ment of the three weekly deterniinations in 209 weeks the to belong to very different readings of the screw- heads and it may lie assumed, that in the mean of the six microscope- *) According to Astr. Nachr. Xo. the probable error of 1076 readings opposite errors will have caricelled and conipemated Brrref's observations tuken with the Rrpsold Transit Circle at Knenigsberg (length of telescope 5,3 feet, aperture 4,3 each other at least considerably. But in the reading of the inches, diameter of circlc 3,2 feet, four small microrcopee; t e I esco p e -micronieter no such conipens;itiori takes place, the instrument being in several other respects inferior to so that in order to clear the observations 01' this avoidable the Grccnwicli aypuratus) umounts fur Zenith-stars to O"464. source of error it is necessary either to keep the micrometer But owing tn the different niode of adopting the Zenitbpninte the numbers are not strictly coinpnrable, a circumsliince, to which I shall have to recur herenfter. -) Cf. § 5. 19.1 Ilir. 12131. 200 under cotistatit control hy careful atid repeated investigations Some of these inaccirracies have their origin in fhe of fts irregularities or to eliniin;ite theni directly by proper two causes already spakeii of, namely the adopted cnrrec- nieans. tinns for runs and the irregularities of the screw, and the Those p e r iu tl ic a I errors of the screw , which follow ieniedy is to be foutid in applying the areaits suggested. - sonie regular atid siniple law, could he eliniinated. if, instead That the precision of the Nadir observations haw been iin- of one, the screw were niade to carry ttvn wirea at a distance paired hg the cliscoritinuatice of the daily determitiations for of exactly half it revolution from each other, and if obser- runs is directly proved by the fact that the Zenithpoints show vations were then iiiade on hoth wires. The p rogr es siv e a greater consistency with each nther before that change errors niight be easily avoided hy limiting the range, within than after. Froni 371 observations of the first 25 periods 1 which the screw is used and which, tieing more than 9'. is find the probable error of' a single Nadir observation, so far uonecessarily large. If further the usual precaution of giviug as it niey he estimated by iiierely comparing the single deter- the last turii of the &crew always in the direction towards niinatiuns with their nieans, = 0'249, from 1 128 nhrervatioiie the head were likewise attended to, the ohservations niight of the 97 remaining periods = O"335. This difference niay probably he considered free frorir atiy rerreible errors, which he partly due to the circumstance, that the average lengtb depend 011 the iniperfections of t.he screw. - In the absence of the earlier periods is somewhat shorter than that of the of any direct investigatiori it is obviously inipossible to esti- later ones and that consequentl! the real variatinns of the nlate the degree of uncertainty arising from this cause. The Zenithpoints may have beeti lerjs sensible in the fornier; hut only meatis of esaniination, which the printed vnlumes seem there is no doubt that the chief cause of the cliffererice lies (0 offer, are furnished liy the purely differeutial oliserratioiis iii the different iiianner of finding the correction for ruri*, as of rieighl)oriritig stars, since those of the Moon. though far is apparent froiii the alinve nicntiotietl cnmliarisoii of' the niore tiurnernits, ;ippear ton precarinus for the purpose. The daily and weekl? tlrterttiinatinns during the earlier periods. indirect evitlcrice however, which ruay be gathered iii this It imp be reiiiarked. that the Nadir observations of 1851 manner, is insuficient, arid 'throws little light on the subject. being nearly twice as numerous as those of following years There are 130 observations of 13 pairs of stars, of which is an additional poitit in favour of the earlier observatioiis, the differerices of Polar Distance have been measured at the average probable error of the adopted Zenithpoints tieing least four tinies during tbe four years. bdopting the annual found = O"0664 for the first 25 periods and = O"0984 for variations of the mean places from the Greenwich volunies the rest. - With regard to the second cause it is clear, and assuming, that in a few cases of larger discrepnncies the that the suggested substitution of a double wire l'or the single reading6 of the Telescope-niicronieter have by mistake heen one woiiltl be ol' decided advantage for the Nadir observations. det down iu wrong linea and are to lie exchanged*), the The chief perintlical errors of the screw iiiight their he directly probable error of a siugle difference of declination is found eliiiiinated by hringing successively each wire upon its own = O"585; it will therefore be easily understood, that any iniage or, which is tlie sanie thing, inidway hetween the attenipt tn trace distinctly the effects of irregularities of the other wire and its inrage. But, on account of the greater screw iii so limited ii riuiriher of observatiocis is errtirely exactness, of which the observation is capable, it would tiopeless, at least so lo~igas it is riot CerIilin, that tlie tlirec- prohably lie still tietter to hring the two iniages succeasirelg tion, in which tlie last turn of the screw is given, has in-. midway between the two wires and tn free the result of tbe deed IIU influelice on the reading. Direct investigation alone irregularities of the screw by applying the necessary correc- Can settle the matter. tion, for the precise cleterniination of which the Nadir nbser- 3) Though 1 have removed the systeniatic errors, which vatious themselves and the collimators offer the pruper means, are created by the Greenwich practice of forniiog the Zenith- as 1 need not farther poitit nut. *) points atid although the redtirig Polar Distances have thus It is however another question, and one woythy of the become free fruni the most objectionable part of this source most careful consideration, whether the practice of asauoiing of error, they are still affected hy some inaccuracies of the constancy of the Zenithpoint - corrections during perioda the Z c ii itti 1) o in t- C o r r e c tio n s , which might be avoided of considerable length has not sensibly impaired the accuracy by a change in the maiiuer of niaking and using the Nadir of the Greenwich observations, and whether it would not be observations. advisable to adopt for each night or for the extent of each - observer's group of observations its special correction. For, *) An iisautnyliun, which 1 Iiuve already acted tipun in giving - __ the inciin Puler Diatanc-c.6 in the firdt table. *) Fur i~still tietter arrangement of wirer d. § 9. 20 1 Nr. 1261. eolc althotigh the firrrinens of the Greenwich apparatus niay be same transit. The results of the coniparison are given in the very great, yet the differences of the Nadir observations are following table *), in which the first column contains the nearly nf the sanie order as those, which induced €lessel*) number of the star or of the group of stars, the observations to abandon the old niefhod of reduction as being clearly of which have been combined; the second the name of the injurious to accuracy: and it appears inore than doubtful, star ([2787] designating B. A. C. 2787), the third its Zenith whether the twit assumption implied by the Greenwich prac- Distance: the fourth the mean of the R-D observations; tice is indeed correct, that Ihe personal equations in reading and the last their number. It will be seen, that as far as the inicroscopes and the real changes of the Zenithpoints are practicable only such observationa are combined as fall up00 insignificant in coniparison with the accidental errors of the adjacent lines of division. Where, in the case of douhle stars, Nadir observations. If such were the case, it would be a the observations are not independent, thetr means are treated etrong pnof, that the latter are capable of considerable im- as single observations. pruvemetit, since it cannot he supposed, that it would be dificult with the superior Greenwich instrument to give them SO.----- Star z R-D n at least the same degree of precision, which Bead has I [2787] S. P. -42'57' -1"34 ! attained with his far less powerful apparatus. But the reasons 90 13 Comae +24 34 +at65 1 for eliminating any eystematic errors and for making the obser- vations of different observers and different nights, as far as 133 aLeonis +42 43 +Ot74 1 practicable, independent of each other, appear so cogent, tbat If the true correction for flexure of the observed Zenith the corresponding change in the adoption of Zenithpoints Distance 2 is supposed to he = asin z + I cosz and if c woiild probably be desirable, even if the fortuitous errors of detiotes the constant correction, which the Circle readings of the observatioes should thereby be apparently increased. This the Wire observations require in order to correspond to the however is not likely to halipen and, if the suggested im- true celestial Nadir, so that c = corr. for division error of provements should not lie deemed sullicient to preclude such the Kadir readings -6, the true Zenith Distance will be a possibility, it might be avoided by a relietitiori of the tiightly found from ii direct observation Nadir oliservations, which would give no additional trouble, = n + (ti - O"50) .hz + b t-09 z - c - cdatitude. since these deterniiriationa iiiight be entirely omitted iri those and frorii ii reflexion-obsertation nights, when no stars are observed. = ii - (a - 0u50) siri z II c-0.9 L c - colatitude. There are some other causes of leas iiionient, which have + + sccasioiially increased in a $light degrce the errors of the Each douhle observation therefore furnishes an equation observations; for instance the several siirall iiiaccuracies of o = R-D-2xtPitt~+2~, the reiluctioiis lioiii appiireiit to rlieiiii places. An uriusual where a is written for 0 - O"50. number of larger discrepancies, which I have rioticed iri one Of the 133 equations, which are derived from the data or two nights, riiay probably be orviiig to iiiexperieiice on the of the foregoing table, En's. 54 to 60, 'yCassinpeae to r An- part of tlie ohserver. But 1 believe, that the three sauces droniediw, have been disregarded, because the reflexion-nl)ser- of error. which have been treated above, are sufficient to vations of these stars are taken with only a part of the accouiit for the tlitfererice ofaccuriicy, which seeiiietl to require aperture of the oliject-glass. For from mensureinents on the erplaiiatt;oti, aid that, if they are attended to, the corisistency plates iii the description of the instrument it follows, that of the obhervatioris will iiot fail to accord with the phynical light is partly intercepted for all stars of less than &9'6 power and auperioritp (if the Greeritvich Trarrsit Circle. Zenith Distance Equation No. 90 is likewise omitted, though the observatiori is in fact not more discordant, than several 5. others included in the means for some other stars. I iiow proceed to deteriiiine the constant correction of By ,giving then to the remaining 125 equations weighta the adopted Zenithpoints atid the correction of the assumed proportional to the number of observations and treating them flexure froni H coniparison of the direct and reflexion-ohser- hy the method of least squares, the values of the unknown vations of stars. In order to irisure purity of the deductions, quantities are found the correspondirig observations only are employed, namely 25 = -OU108 prob. err. +O'090 boee, which have been taken directly and by reflexion at the 2c = +O,OO! s 5 f0,04!. -. . - *) Aar. KiicIbr. So. 1st col. 10. *) The bulk of the table is omitted to save spscc. 203 Xr. 1361. 204

It was at first intended to give weights, in which some there certainly appears no valid reason to doubt the con- allowance should he niade for the irregular errors of division, stancy. It should indeed be easy to settle tbe question in in a manner similar to that, which will be pursued afterwards the most unexceptionable way by means of the collimators. iii deducing the correction of the adopted colatitude; antl this But unfortunately, for sonie reason or other, the Greenwich is the reason, why in the table only such observations are deterriiiriation of the horizontal flexure by their help has been combined as helong to adjacent divieions of the Circle. But very far from satisfactory. The number of ohservations of from the foregoing results arid from a closer inspection of this irirportent elenierit is very small arid the discordance the table it is sulliciently clear. that nothing would he gained of their results is far greater, than one niight have ex- by such ti course, as the 755 observatiuos, mliich have served pected, when bearing iri mind the great superiority in size, in this inquiry, leave evidently the true aniouiit of the un- iliassiveness arid convenience of the Greenwich apparatus knowii quantities uiicertain within rather wide liinits. In over that, which was at the disposal of BesseC, when he first order to show fhe degree of this uncertainty iri still artother introduced this nrethod in the year 1824 *). For, supposing wiy , than by merely stilting the prohalile errors , I have that the diflerences of the Greenwich determinations are to arbitrarily nssuiued different values for the s i ii e - term of the be altributed to purely fortuitous errors, it rvoulcl require flexure correction arid, after sulistituting these antl the carres- 81 Greenwich olbservations to attaiii the same degree of' accu- pooding niost probable values of the constant term iii the racy. which BesseC has gained by not more than 20, although 123 equations of condition, I have deduced the residual errors be had to contend with several troublesome mechanical diffi- and, with due regard to their respective weights, have cal- culties, from which the Greenwicb apparatus is free. I do culated from them the probable error of a single R- D not see, how this fact can be explained, except hy the errors observation. 111 this manner its amount is found to be of the micrometers or the neglect of some necessary precautions. Whether the assumed law of the flexure may be the for n = 0"OO l"293 = +0925 r,r41 correct one for the Greenwich Transit Circle or not. is an e +0,50 19116 open question, since there is uo evidence upon which to form = +0975 1,194 an opinion. Tbe reflection-observations of stars give no reaso~ +I900 19380. = for distrusting tbe coefficieut of the ei e- term, whicb is In Consideration of the foregoing probable errors it appears furnished by the direct determinations of the horizontal flexure, to be a fair inference, that anJ value of the s i n e -coefficient but no fair inference beyond this can apparently be drawe of flexure hetween perhaps +0'"20 and +ON70 would almost from them with regard to flexure. The cosine-term cannot equally well represent the R-D observations, and that they be taken into account, since there are no proper means for would not even be irreconcileable with values beyond these ascertaining its coefficient. limits. When it is borne in mind, that this great uncertainty S 6. attends the results of four years observations, even after Leaving the final values of a, b, c undetermined, I ehall the exclusion of those, which could only damage the purity now endeavour to deduce the correction of the adopted colati- of the deductions, it is easy to perceive, how little reliance tude and that af the constant of refraction froni the obser- is to be placed on the annual determinations, which are vations of all circumpolar stars, which (luring the four years found in the Greenwich volumes. But 1 need not perhaps have heen observed at least twice both above and helow dwell upon the causes, which have unnecessarily impaired the pole. the trustworthiness of these Greenwich corrections. Let the true colatitude of the Greenwich Transit Circle If it should he ohjected to the results, which have been be = 38'31'22"$ x, the assumed consfant of refraction = r,, deduced above, that the assu ni e tl c o n s t a n cy of the flexure- the true constant = + y. If then zu denotes the true Polar coefficient during the four years riiag be unfounded, it will Distance of a star, the direct and reflexion observations of at least he apparent from a glaiice at 1). LXVI. of the volunie which above and below the pole furnish the values D, R, for 1852, that the Greenwich practice of treatiiig the obser- D,, R,;if mo is sonie convenient approximate value of m, for vations of every year separately has not been adopted on instance w,, = 0, and if the calculated average refractions account of any such objection. IF the flexure were really at the star's places, above and helow the pole, are p 7, and suhject to annual variations, it would be difficult to reconcile Q, ro (taken rvth the sign of the Zenith Distances Z, z and with this fact the lielief in the great firmness of the instru- ment. But as far as the evidence of the observations goes, 205 Nr. 12(il. 206 therefore considered negative, if north of the Zenith), we have, of this foriii, in which obviously the last or the second is putting w-m,, = d w, D-w, = s, D,+w, = s,, wanting, if the star has not been observed by retlexion. From R-w, = U, Il,+w, = u~,the equations each system its peculiar da is lo be determined and from the proper conibination of all the systems the unknown quanti- o = E - d 0 + LC + p y + r sit1 z + 6 cod :- c Weight 71' 0 = U -dd+x++p-&asi,I: +Ir.o.tz +c -c v ties x and y. 0 5 5, + d GI + .x + pJ + & Yill z, + b C0.F z, - c S ?C, Treating for this purpose the existing equations by the o = ~,+da+.z.+py--.skz,+I~n.c=,+(. s v,. method of' least squares, we have, by eliniinating first the The observatioris of each star furnish a systeni of equations value of dn, (W + u + m, + v,) dm = OW + uv - ~,n,,- U,P, + (m+ v - [?c, + v]) x + ([m+ .u3 p - [la, + 'u] p,) 9 + ([r~- V] sin Y' - [rr, - u,]si,, 2,) D + ([7u + v] L'OS z - [m,+ v,] coL 2,) 6 - (m- v - [TU, - v,])c By suhRtituting thiR value of Jn and preparing the resulting equations in the form, in which they enter directly the final equationa for x and y, we get (putting for the sake of brevity

aiid the eqiiatioii for y

The sunis of' all the corresponding eqiiations. which are coiitains the regril ar errors of divisiori for every degree; turiiishetl by the ohservations of the different stars, give the but front the accoiint in the appendix to the vol. for 1852 it final equations far the tleterniinatiaii ol' .L' and y. But as it appears, that it gives or is intended to give the individual inay be hazardous to rletluce the colatitude from ohaerrations, errors only of those lines, which correspond to the whole tbe acc~nracyof which c:annot be cliiflicieritly depended on by degrees of the pointer reading. No attempt seems to have reamti 01' the uncertainty of ref'radiori, it will lie preferable been wade to get rid of' or at least to diniinish the influence to exclude from the final equation for .z those equations, which of the irregular errors of the single lines. Since the Green- result f'ront oliservations lielow a certiiiii altitutle, ,iii(I like- wich investigation has dispensed with this precaution, the wise to tletluce the filial Polar Distalires of these stars froiti table loses much of its real usefulness: for, though the regular the upper culiiiitiatioris only. errors iriay be on the whole the iiiorc sensible, the irregular BeTore the itpplicatio1i of' !he foregoing foriiiulae CRII he errors are certainly not so siiiall, but that the Greenwich ppceecled with, it is first necessary to tleterniiite the weights iiiiiniicr of correcting must ultiiiiately prove for itiany of the w, v, w,, I),, which are to lie giveii to the separate equations. observations a niatter of very questianalile policy. In the In case the observatioiirs are free froiii errors of tlivisioii of alisence tit all tlircct eviderice it is of course not pussihle the Circle, the deterniinatiori of these weights is very siniple, to arrive at a fair estintate of these irregular errors. Some since they tlrpeiid then only on the iiuniber of' oliriervations riotiori of their magnitude however, though a very imperfect and oii tlicir prohahle errors, which I have before made the one, itiay be formed by laying down the individual currections subject of itivestigatioii. If however this coriditioii is riot of' the Greenwich iriveatigation graphically and comparing satislied, it hecoiiies a iiielter ol' dilriculty to niake aiiy prolier theni with the curve, rvhich seems hest to represent their allowaiice for it. This circunistaiice oliliges nie to cortsider geiieral track. lii drawing this curve freely by hand, of course the niiitiner, iii which the Greenwich talile for errors ol'clivision sontething arbitrary cannot he avoided. I have consequently has Iieeii constructed. considered it worth while to find it6 course likewise hy Since all the ohservations are corrected tiy the table in direct calculation, so far AS it follows some regular periodical the iritrocluctioriw, one tiiight be inclined to ilasunie, that it law. The resul!ing coniparison is given in the following 207 Nr. 1261. table, in which the first column contains the Circle reading c deteriii. culc. diff. C or its surplus over ionie niultiple of 60'; the second, the - - - - 11" +0"210 +0"131 +0"079 correction for errors of division for the mean of the six 45 322 I74 + 148 niicroscopes, as it is found (after mbtracting the proper 46 240 215 + 025 constant) from the tahle in the appendix to the volunre for 47 27 1 251 + 020 279 039 1852; the third, tlie corresponding correction, as deduced 48 240 - 19 169 294 . _. 125 froni the formula 50 394 297 + 097 Corr. = O"141 sin ( 6 c + 119'5) 51 41 1 285 + 129 255 +0~167sin(12 c+ 27478) 52 + 230 - 02 5 53 - 015 ?1? - 232 +0,121 .ht(18 239,4), c+ 54 + 107 166 - - 059 and the last coluniii, the difference hetween the two. 55 200 107 + 093 56 230 + 045 + 185 C detsrin. clllc. diK 57 025 - 016 + 041 - - - - 58 + 056 07 1 + 127 0" - O"? 6 1 -03'1 48 -0"l 13 59 .- 097 116 + 019 1 292 163 - 129 ti0 -0,261 -0, 148 -0,113 2 107 161 + 054 3 312 112 -_ 170 Front this coniparison it appears, that only in one or 4 435 108 ___ 327 two placcs the lbrniula is decidedly deficient. For the most 5 138 06 1 - 077 part the curve, which may he cortstructecl froin it, riearly 6 - 005 - 006 + 001 coiiicides with that, which the graphical process seenis to -I + 08'7 + 053 + 034 8 148 110 + 038 suggest as the most prohable one. Now, I ant very far from 9 230 161 + 069 believing, that the latter represents really the regular error8 10 384 20 1 + 183 of division; the true curve is very likely far more jagged 11 292 226 + 066 12 486 235 + 251 and capricious; yet at the same time it appears certain, that 13 404 226 + 178 the irregular errors of division are by no means incousi- 14 271 200 + 071 deralde and that consequently for all general purposes it 097 159 - 062 15 + would be far more safe to take the corrections froni the data 16 - 005 104 - 109 17 077 -j- 042 - 119 of tlie curve than from the Greenwich table. And let it not 18 107 - 026 - 081 be overlooked, that in drawing this conclusion I have yet 19 005 092 + 087 implicitly assumed the correctness of the Greenwich table 20 138 155 017 + of the individual errors. 21 25 1 208 - 043 22 200 248 + 048 However, for the present purpose the regular errors in- 23 353 280 - 073 deed are not required, hut the individual errors of those 21 322 296 - 026 divisions, which have been used in the observations, from 25 291 298 + 007 26 261 289 + 028 which the correction of the colatitude aod of the constant 27 374 270 - 104 of refraction are to be determined. Of the stars, to which 28 230 241 + 014 these observations refer, Polaris has heen hy far the most 29 179 215 + 036 frequently observed and the Greenwich determination of cola- 30 220 185 -- 035 31 179 157 - 022 titude depends chiefly upoii it. Now, it so happens, that the 32 128 132 + 004 direct and reflexion-observalions of Polaris above and below 33 169 111 __ 038 the pole and likewise the observations of a Urs. maj., yUr8. maj., 34 148 094 - 054 yDracoriis have in later years fallen at Greenwich upon divi- 35 128 08 1 - 047 36 -. 159 070 - 089 sions, which are distant hy whole numbers of degrees *) from 37 + 067 060 + 137 the divisions, which are used for the Nadir observations, so 38 f 046 047 + 093 that if it was intended to determine the iodividual errors of 39 - 026 031 + 005 40 - 026 - 010 - 016 41 $. 046 + 017 + 029 *) Of course. irllownnce must be made for the circumstance, 42 - 046 + 051 - 097 that the olieertetions ere not always referred to the aame 43 + 015 + 089 - 074 tli\isions. This is also the (:me with the Nadir obsrrvntions, ?09 Nr. 1262. 21 0 divi..;ioii froni degree 10 degree , it iiiiqlit Iiiive tiecsii e~sgto of same nl' the inore frequently ohscrvetl diirr, iire rctlucecl get at tlie errors of there inipnrt;iiit lines. hit it sceiiir, that to the same unit (for which I have choscrr the weight 01' a11 thc great advaritage of this good fortune has Iieeii ciitircly observatioii of Polaris), it is found, tliiit tliese iirliitrary relative crvcrloolctl~ arid that Iiy rvhat are callccl the ~iirdiliiiltlivi- weights vary diiriiig the ycars 1836 - 1854 betrveeri thc nioris in tlie Greeiicvich irivestigation nirist be iiiitlerstood following limits. liy the side of wliicli I Iiiivc ;itlilcil the those. which correspond tn the pointer readings 0" 60" ctc., weights t~.I/!, ;ic:cortliiig to llie fornilila ol' iiiy Iircvious iri- hut which (if 1 supply the oniitted iiiiriutes of the Circle vestigatiori, 117 referring to the iipper and W,to the lower readiriys c*orrectlg) arc of' no ~ieculiitriiiiporta~ice whatever. c II tin i II ii t i 0 11. Lirnita of vnriation The Grcerivvicli annual detcrniiniitions of colatirude deperid of Greenwich weights ?I* 7l.? conscc~r~ei~tlyup011 oliaervatioiis, which are corrected iieither , - -- for thc rcsal iior for the regular errors of ilivision. hilt for the /3L?rsai! iiiin. 0965 0995 1,013 0,881 iiidiVidiiiil errors of ~oiiierieighboririiig lines. a Urrae maj. 57 86 0.970 0,680 In the cliflicolty , by which this oversight ol1structs the a Ciis..;iope;it 26 74 0,935 0,501 iarthev pursuit of niy present invcsligation, I have considered y Driiconis 0903 0958 0,912 0,320 it least arbitrary 011 my part lo adopt the Greenwich table The farther investigation is their 1iasc.d lipon the nbser- of corrections as if it really answered its proper purpose of vations of tlie 19 circuiiipolar stars, which withiii the years clearing the oliservatioris of the regular errors of division, arid 1851 - 1854 have been ol~servedat least twice both atiove then to proceed ta the determinations of weights by making arid below the pole. The weights m, V, 7:,, v,. attriliuted to different Iiypotlieses with regard to the probable amount afthe the equations, \vliich are fiirnishecl by these ohservatioiis, irregular errors. In tlie Greemvich annual investigations the are calculated on the assumption, that the errors af division, latter are simply rieglected. Moreover the approxiniate weights, 80 far as they are riot taken into account hy the Greenwich which iire there given to tlie results of different stars, are table, may he treated as purely accidental. With regard to detcriiiinecl in a very siiigular niaiiner. So regard whatever their prohahle arnount I have in the absence of all direct is paid to the differences of accur;icy beloriging to the obser- evidence made three hypotheses by supposing it = 0 or vations of the same star aliove and below the pole; hut the = 3 or = Q of the probable anioont of the purely accidental ensiling preliniinary weights are afterwards multiplied by sonie error, which ~vouldbelong to a Greenwich observation of a arbitrary iiuail)ers, which are made dependent not on the star near the pole, if there were no refraction. According star's Polar Distance, but on its place in the ledger. As I to !he fornier investigation this probable error is = O"542; havc looked in vain through all the Greenwich volunies for the weights are consequently calculated upon the supposition sonic explanation, I do not know, what considerations can that the probable amount of the irregular errors of division ever have lead to the adoption of this singular process. The is 1. insensible, 11. O"108, 111. O"217. These weights are inconsistencies however, which it has occasioned, suficiently found in the follorving table, which gives also the seconds show, that, whatever these considerations may have been, of tbe Polar Distances, as they have been derived from the they cannot IIC correct. For if the approxiniate weights, direct and reflexion observations above and below the pole, which have actually heen given to the single observations as well as the number A of these observations.

seconds weights eeconds weigh td Star 11 >L of IS. P. D. I. 11. 111. 0fN.P.D. S.P. 1. 11. 111. - -' - cc - - c-cl - -c - ClVI - A l:rsae min. 2 7"7 89 41 38 9 87 16715 6929 -27"817 31 29319 14901 5997 R. 27 9 600 20 18996 11956 5952 28 9 044 11 10736 7962 4952 al'raae nh. 487 588 259 215 7 81 23 60 7907 48 9 960 306 287 9 06 23 9 92 7, 10 R. 18 9 929 28 26,58 13940 5987 18,358 19 17782 10982 5737 5 1 Hev. Cephei 507763 34 32 7 40 14968 6708 51,764 42 39915 15987 6925 R. 529 605 2 1991 1784 1969 50 9 120 3 2980 2762 2924 d C'rsae liiin. 17264 87 83905 19998 6974 1,682 56 52901 17756 6947 H. 19880 9 89 59 6965 4920 Br. 1399 589 110 3 2387 2968 2928 57,135 2 1984 1978 1965 R . 59 7765 2 1992 1985 1970 e Ursae niin. 53 9 873 27 25994 13724 5984 51,223 3 2970 2954 2919 R. 53,341 I 5 14741 9951 5906 y Cephei 36 9 975 24 23 9 02 12746 5910 36,565 4 3941 3, 12 2756 R . 363 729 11 13~33 9-09 4995 5?rBJ 14 91 1 hr, 1262.

aeconlld weight$ second3 weights Siar n II lbf 3.P. u. 1. 11. 111. ol".P.D. S.P. I. 11. 111. --.,- _vv -v- - - - -v- - -- 3 3"6 6 0 2728 - 1765 1755 5 Zrrsae niiii. 3 2788 2768 -311168 5 2 1970 R. 33711 1 0796 0996 0996 ,Bursae niin. 77190 42 40724 16904 6728 7,34 2 1966 1762 1952 R . 77793 20 19712 11927 5746 /3 Cephei 31,408 61 60 7 40 18739 6 7JI7- 30 7 740 4 3908 2779 2939 R. 319574 30 28931 13781 5994 Y Ursiie iiiaj. 2,416 40 36961 15945 67 19 2 9 056 10 6942 5931 3767 I{. 27632 31 28737 13782 5994 a Ccpliei 397989 50 45961 16780 6738 397915 4 2752 2733 2908 H . 40,197 16 13761 9759 5708 z Cassiopeae 31 9251 17 15700 9975 5712 311966 8 3976 3940 2972 y Uraae iiiaj. 57 7 298 17 14 7 90 9771 5911 567310 6 2964 2949 2915 6' Ursae iiiaj. 53 7244 14 12711 H748 4978 52 7 670 10 3972 3737 2171 y Draconis 32 9 532 3 7 31 784 14957 6706 34 7443 3 1703 1703 1703 7 Ursae nx;j. 41,818 17 14750 9753 5707 41 7366 8 2736 2724 1799 I Ursae ni;ij. 33,125 21 17775 10980 5936 33 3 065 8 1796 1789 1773 tc Cygrii 9,565 49 40343 16707 6928 -11790 3 0932 0735 0937 Froiii tlieje data have then 1)eeii detluced the final equations proper investigatioii of this lam arid for the examination of for .I- iiiiti y ;inti also the resulting corrections tbr the single the Iiypotbeses of Bcssef ant1 Ivnr?/ etc. and the tleterniirration stars. wliich latt& iiiay he omitted here. With respect to of' tlieir constaiits I'roiii ol)servalioris, the Greeiiwirli data are the l'ornier, tloul)le equatioiis for .'c are giveii, the first ;is still irisiil'licieiit ; I)esides, like the investigation ol' tlie tempe- rouoti I'roiii all tlie observatioiisi, the second as tlie rcsiilt 01' r;iture-roer~cierrt. flie iiiquiry would prove partly niigatory oil tl10s.e stiirs oiily , tlie Zenith Distances ol' which iii their ii~c~~iiiitof the iiiicertaiiity of the barometer readings. Since lower ciiliiiiiiations do not exvceti -72'. The sec:oiiti equatioii the ZeiiiIli 1)istaiic.e S. P. of the last star, aCygrii. does not is tliercrnre free froiii aiiy seiisible iincertaiiity of' the lii\v* escretl -83'38'. the equations for y niay he safcly detluceci a(:cordiiig to wliich the spccific elasticity of' tlic: iitiiiospliere l'rorii it11 the o1)scrvatioris. i3 ;issunied to i!iiiiiitislt \vitli increasing Iieiglit. For the

'Fhe fiii;!l eqiii~tiorisare theii the folloivirig :

507.22 .I. 448.90 ,/I -4O8.d4r, Hyp. I. ,~.. .{ 480.80 -392.79 -382.31 y.. .-4418.90 +450.97 +367.31 157.40 - 164 .6': - 90.70 11. ./-. . . 1 134.12 114.70 - 67.42 y.. .--164.62 j-2 1 4 .x 4 +lOU.l.r HG. 10 - 99.11 -- 45.48 111. .r...g 6n.10 - 59.H5 27.48 y...- 99.11 + I 4G . 00 + 64.19 213 Nr. l!!tL!. 214

In stating these results. of course I do not bertturc 10 foriti are to lie ititlcpenclent of ollier deferininations atill iilw . as a decided opinion, which ol' them m:tv come nearest thr truth. far as prncticahle, free from the intlivitluality of the apliariitus. The conlpiirison of the adopted errors of divisioti with thcir it heconies an essential conditiori to provide irr its construction general liiw, which I have given before, appears to be favour- proper means for thc tlctcctiort ntitl e1iinin;~tiori of all its able to Hyp. II. or to some ainiilar hypothesis and it scenis systematic errors. In !lie c;ise ot' a Transit Ciwle, rvl1ic:h is therefole IiLcly, tliat ultimately only small correctiorrh of the intended ti) roiltribute in il decisive iiiartiier towartls the acltle- adopted colatitiitle and of the constant ol' refractioii will 11e ment of ttic much vexed qiicstion respecting ahsolute rlecli- requircd. If Iinwever the Greenwich practice is to he followed nations, adequate contrivances ought to he furnishetl for clearing of disregarding the residual errors of tliviaiorr entirel? the ohservations siicceswfiilly ot' the efrects of flesurc ;rnd of if the vnliies of' ct and c arc adopted, which result from errors of division, and the question therefore arises, how the the refle\iori oliservatiorrs, the tirial colatituclt? i.; foilrid = reliable tletcrniinatioii ol' the reqnisite corrcctiori?; niay he 38"31'22"283 -09872 b, and the correction of the cwiistanl Iiest irisurctl Iiy the conatriictiori of the instruntelit. of refiactioii = +0"196 O,llSb, or, as I had assiinietl Of the threc kno\vri niethotls hr cleterniining the elrecta /og r,, = 1976, the logarithms of refraction are to be in- OC tlexiire of the telescope ;ind circle (Hesscl'a method by creasctl Iiy 0~00148-0,00087 6, b obviously tlenoting here re fl csi o n - o Iis e r v ii t i nit s I) 1' st a rs, Utxsel's nic t h n cl LIy the help the a1)stinc.t nunilier. About 0,00030 or 40 01 this iricreasc of colliniators and Repsold's nicthod 11' est:han~in~the posi- could lie esplainecl hy the constant correction, which the tions of the oli.ject-glass and eyepiece 011 the tribe) the first liaronicter - readings require, in case the assuniptioii is not carirrot dispense with oltservations in the reversed position erroiieoiis, that the difference of level lietween tbe Circlerooni of the instrument, if the coefficients ol' the flexure correction barometer ant1 the standard barometer of the meteorological are to he rendered indepenclent of the errors of division and observatory is very small. Otherwise, except for some such if the c o s i II c - ternis of' its general espression cause, it is not likely. that Bessers refractions should be a .tiit 3 + Ir ms z + 0, siii 3 z + I,('0.7 3 z + .. . fond really too sniall ; at least other determinations of the are not to be neglected. - The second method would give constant of refraction are rather in favour of their diniinutioii. ") all the coefficients without reversing the instrume~itand they But from the foregoing investigation it suficiently appears, would likewise be independent of the errors of division, if that the Greenwich observations leave the determination of it were practicable to fix the collimators in all directions, in the true colatitude and of refraction in a similar state of which they are required for the purpose. 1 believe, that Prof. uncertainty, as they have left the deterlnination of flexure, Challis has been the first, who has ventured upon this more so that, if arbitrary assiiniptions are to be avoided, the general employmeut of collimators, but 1 do not know, whether question, what geueral corrections the observations really he bas sufGciently overcome the practical dificulties, which require, niust remain an open one. How far the ohservations were anticipated by other observers and how far he has been of later years may tend to diminish the existing uncertainty, really successful. If the colliinators can only be used in the must he decided at a future time. At present, instead of horizontal direction and if no Zenithal collimator is provided, applying some questionable corrections to the mean Polar the s i n e - coefficient of the correction a sin z + I c0.v z may Distances, which are given in the first table of this paper, indeed be very easily determined without reversion, but the I shall content myself with touching upon some important c o s i n e - coefficient can only be found by the combined Nadir- points in the construction of Meridian Circles, in the hope, and collimator-observations in both positions of the instru- that my reniarks may possihly he of service and not unwel- ment, and it is not directly independent ol' the errors of come to some readers. division, in case the Circle is read off by six microscope8 instead of by four. - 111 the third method ") reversion, 9 7. If an instrument is erected not for niere differential ob- ') The theory of the method ie fully disciissed by Banrcn, servations, but for the purpose ol' furnishing results, which Astr. Sachr. Xu. 388-389. Whether in its practical appli- cation there may not be some hitch, which renders its *) 1 inoy take this opportunity of remarking, that the factor, evidence on the real effect8 of flexure less decisive, than which reduces the mean refractions of Brsd'~Fund. Astr. inighl be expected, is u point, on which 1 of course cannnt to those of his Tabulae Reg. is not 1,003282, RII has been venture to espreso any opinion, but on which it would be 333 28 very interesting to learn the experience uf thode nstrnnomers nseumed in some rccent reuearchea, but 1,003282. -' 333,78 who have ectunlly tested it. The adaytntiun of it to the = 1,001779. Cf. Tab. Reg. p.LX1. Greenwich telescope must perhaps bc conaidered impracti- 14 * errors, which. affect all the results. the disadvantages prepnu- derate so inuch. that it would be clearly ;I tlecitletl iiiiprove- riient upon the presetit Greenwich system . if the rellection- observations of stars were tlispeiised with altogether. The ditl'ercrrce, which such ir chaiige wnriltl effuct. iimounts iri reality to iiothirig Imt this, that the final corrections. rvliich tlie resultitig Polar Distances require, would then clepeiitl 011 the Circle reading only, while at present they arc functions of two arguments, the Circle reading and tbe date of the observations. It is obvious. that this latter circunistaiice rerrders any tlcteriiiiuatiori ol' the sjstentatic differences of the Greenwich catalogues front others soinervhat precarious, a6 iriay be easily perceived by coilsidering the diversity of the niateriiils, which are conibined, [or iristarice iu tlic cata- logue of 1576 stars Ibr 1850, without the corrcctious reqiiisite for the sake of consisteiicy. Although it may lie pr;tcticalile, iis tlie previous coil- sitlcratior~ssliow, to clear I)? iiieaiis or prolier roiitrivatices the Circle oliserviitions of the ell'crts 'of ilcsiire witliiiut rever- sion of tire iristrrinretit. it itpltciir* iiiiire tlilliccilt to tlo SO \v i t li co ni 1) I c e ceI' t it i it t y HA Icg;t r (1 s t 11 c 'I' r;i 11 s i t o 1t s e I v ii t i o it s , t 11 o u 2 Ii o I' c o 11 rs c' t 11 P a cl o 1) ti o it ii I' //cpdd's I it e i 11 n ti i:o n t I,i I t e d with ;I tlicirotrsli itivestisiitioii oi' the shape of the pivuts would prtiIi;iIiIy rt.tIric,rA ;ill reiitaitiiitp (IiiiiIits to ;I tttiitiiiiitrii. '[lie freetltiiti I'roiii ;~iiysystcrliiitic: (iristriiiiieiital) error*, j\liii,li is ir I Iiii e tl t'o r t ti e G r ec:i \v icli 'Irii II .'i i t o li se L' \ ii t i (1 its. in;I !-p o .-s ib I y lie reill: l)ut the very riiiigrtitridc ot' the i~tstru~iie~ttseeitis to iiiakc BOIIIC strict irtveatigatioii o' tlic poiit! ;iIntnst iiidispen- salile. it:itI the scatit? c\ itlcric.e on tlie sulijcct, \vliit,li iitay lie giilliercd iroiit tlic priiilccl \ olriincs, is ntt~relikt!l! to raise suspiciott tliiiti to iti*pirt: coiititleticc it1 the siiliposcd correct- ness. hiring iii iitiirtl tile grciit hclli, wltkli is rcrttlered by olt~erviitiotisill the revcrsctl po*itioii of the ;isis tnwards {lie tlelectioii iitill eliiiiinatiort ol' the Jev!atiotis ot ;in iristruntcirt IIotit its iiliitlieiiiiiticitl coitceptioii, it seeiiis tery ilesiritlile, that the priirciple 01' rcvcr5ioti slroultl ii pns5iIiIi: lie ret;iiiied iii tlic coiistructiou ol' it li;st riitc 'lriiiisit Circle. iiitd it cer-

tiiiiily iiectl itot lie iiliaitdoiietl ~ eveti il' tlic telc..ic:~,pc is to h:+vc it 1i:iigtIi ol' tI OI I'L feet. Air iiistruuieiit 01' a tori'a weight tiiii! iiitlecd lie too Lo~iii~diililefor rever*ioit : hit it \!ill probill~l~;tI'lieiir iit tl1c WUT~C III' thc liillii\viii; rc'tii;trks, thitt s(i riiasoive it strtictiirc' is iit iiict a3 tiitiiciw+s;iry. ;is it is inespetlic!ii1. 217 Nr. 1262. 21 8 even of six fect clianieter is rorisitleralily lieyorid tlic liniit, accurate rueasilrenierit of the sr~l~tfivisiansis consitlerably up to wliich the iiicrease of size is thuutl to lie of real ad- superior to that of smaller iristrunients, 7 or 8 iriches long. vantage. However Iirolitalh the eiriployiiieiit 01' il r cs of However in the ahseiire of any precise information on the large radius may he fiir mere cliffereiitial oliservatiotis, the point or of sufficient evidence, on which to fold an opinion, eniployiiient of large ci rc I es introduces c1isadv;tntages so I dial1 here assume, that their performance is only equal to serious, that evcii ii priori it 1)ecoiiies cloulitfiil, whether the that of first rate snialler microscopes in the liarids of A skilful possiliility of discerning snialler parts of the divisions iiiay observer. The probable error of the niere pointing of'such not lie purchased iu this way rather too clearly. It would a uricroscope-iiiicronieter is according to Peters (Recherches prolalily lie otlienvise, if niicroscolie-iiiicronieters coultl riot sur la parallax tles ktoiles fixes p. 76) = O"O98 for a be hail of a very high degree of perfectiori aiitl it' therefore circle of 43 inches clianieter or linear = Oim. 0000102. Iiy iiicreasing the I)ower arid exactness of tlie iiiicronieters Adopting this value and supposing, that the probable error the accuracy of the Circle readings coultl not be iniprored of the observations, so far as it is independent of the error in a more successful and less objectiorialile niaiiiier tliaii by of the Circle readings, amounts respcctivcly to O"5, O"4, having recourse to larger diniensions of the Circle. - To O"3, it follows, that, in case the errors for runs are directly whatever causes the want of superior precision of the Grecn- eliiiiiiiatetl by reading off with each niicronieter the two di- wicli nl)servatioi~.siiiay lie due iirltl rvliatever advantages niay visions, which include the Zeropoint, the whole error of the fairly lie expected froili tlie renioval of sonic of these causes, observations for circles of different diniensions is represented it is olivious. that it caiiriot he air csserrtial condition to Iiy the following quantities: have ;I circle of very Iiirge diameter in order tn HCCII~C ii 0"j 0114 O"3 suflirieirtly high degree of exactness iti iiierely reatliri; off Circle of G f. diani. iiiitl 6 Iiiicr. Oi500 0,400 0,300 tlic divisions; at least the Greenwich sis feet circle, though 5 G ( 12 reatliiigsj 500 40 I 30 I it is read by sis very long niicroscopes, lias entirely failed 4 1(8 : ) 501 401 302 to viiitlicate the clairiis of superiority of larger over smaller 3 1 502 902 303 circles. But eveii if it were not so, if on tlie contrary the 2.5 4 502 403 304 precision of the Circle readiiigs were found to be in direct ? 4 0,509 0,405 0.306 proportioii to the size of the circle, the questioii yet remains to be coirsidered , whether it is really worth while to push Ac~cordiiigly, irt order to attiiiri by iiteaiis of a t\vo feet the exictriess of the readings beyond a liiiiit, which iiiay lie circle ;in11 four iiiicroscope - iiiicronieters the siiiiie degree of safely reached without the disadvantages ol' ernployiiig a accuracy, wliicli iiiay he gaiiied lty oile of sir feet ;itid six very large circle. For, of course, the inlportance of cliniini- niic:roscopes (read twice), it would oiily be reqiii4te to in- shing the accidental errors of the Circle reatlirigs does not crease the riuiii1)er of oliserviitioria iu the ratio ol 1 to l,Ol, deperitl oii tlie aldute value of these errors, but 011 their 1,02, 1,01. But if the six iiiicriinieters of the 1;irgi: circle relative propnrtiori to the accideotal errors of the observations, are rend oIT only once. as is tlie Greeri\vicli custoni, its slight antl, if this proportinit is already sniall, it iiiatters very little, appareiit superiorit? viiiiisties iiiiiiiet1i;itely aiitl it lieconies how far it niay be in fact diniitiislied Itelow a certain liniit. actually iriferior iii a11 (Iitscs, iri which the prol)alile error of A few consitleratioris will set the real giiilr of sircli tliininu- the correction for ruiis exceeds O"O57, iirttl coiiseqiiently iii tioii in its proper light. the great. rnajority of the oliservations. The niicroscopes of tlic Greenwich l'raiisit Circle are 1 have Iiere arsuriied, that large and siiiall iiiicroicope- very liirge, lieiiig 55 inclies lorig. Their tiill power is iricleetl micrometers possess the saiiie capacity of esact measurenient. riot IirciuKlit orit, since the niotle 01' illuniiiiatitiri reriders it It is reasairable to suppose, that this is riot so, but that a iieressary to Irate coiisitleral)le sl)iice (about 9 inches) be- more powerful niicroscope must insure greater precision in tivccli tlieir o1,ject -glasses antl the graduated barid of silver; the readings, in which case the accuracy of the observations hut. tliou;Ii ttiv ruaguifying power is prolialily rather low, *) it will still less depend on the size of the circle. But even if ilia? tairly lie supposed, that the capcity 111' the riiicronieters for the supposition should prove incorrect, it will make little difference to the question at issue. For where the physical superiority of the microscopical apparatus is, froiii sonie cause or other, rendered nugatory, it is alniost certain, that some siniilar cause will also prevent the general error of tlie observations froiii Iieing diniinishecl to a limit, at which 21 9 Nr. 1262, 220 thc: crrnr ol tlic Circalc rc;itlirigs mislit beconic of serious oii the othcr side iiiitler the. other pair (at t.hc cost! be it \vc+$it. If, for itistariw . the precisinit of the niicronieter observed , of acltlitioiial screwing tlirough half ii revolution iiiviisureiiieiits \VI!I'O oiily oiie Ii;tlf of that assiiiiietl Iiefore, only). the rcsiilt of the loiir readings will not orily possess 1 tic! wli~ili~errnr of' tlii: oliservatior~swould under the former tlouhle weight ivith regard to the fortuitnus wading errors sit 1i 1i i1 sit i o i i s 1) ccii ti ic aiitl the irreqlar errors of divisiori, hut it will at the same tiiiie Iic freed lioiii thchief tcriii of the periodical crrors of I*tir ;I 6 1'. c.ir:.l($ ;ii;tl 1'2 icatliitgs O"501 O"101 O"302 the screw inclutlirig tlic error of eccentricity of the screwhead. - x 0,515 0,419 03325, Tlie insertion ol' snch iiii ;idditinr~alpair of wires (with per- sii tl~iitit \\oiiItI rcqtiirc ;in iiic*reasc of the nuitilicr ot' oh- haps the slight ~iioclification of niakiiig tlie (listitrice between ?;vrr.;itiiiiis a1 the ixtc nf I 10 1,06, I .09, 1,t6 in order to the two liairs = + 3 revolutions) might perhaps prove 2t.t Iiy IIIC;III~ of the sni;illcr circle rcstilts of eq~iil;Iccllriicv acceptable eveti to those olrstrvers. who rniisider tlie direct LF it11 iiicisc 01' the Iargr: hiit ii prolialrlc error of oiily O"3 eliiiiiiia!iiiii ol' tlic error of riins hy tlnril)le readings too i)r o\~cti 0''4 \v(iiiItI o1)vioiisIy Iie Iieyoiitl the reitch of the trorililesniiic: for. it' they take cxceptioii to the additional iiidri:iiietit nntler siwli eircrriirstaiic.cs. so tliat the cliffercricc svretviitg. tvliith is reqnired for that ptir~iose iirid which \viiiiltl iiilt lie greiit: itnil. if the six microscopes ot' t.he aniounts for tlic Greenwich Circle and iliarig otticr instrii- Iarzc circle he rc;ttI only IIIII:~,tlie tivo feet circle would nients to live revolutions. the?. will perhaps not ol)ject to I'ii~riisl~thc innrc csiict results. whenever flio ~irolinlileerror second readings. which requile only a tenth part of the (it' tliv,. corrcciioil for I'IIIIR exceeds 0"t 15. otijec~io~iahlescrewing. if hy this means they can secure for I he liircgoiiig i,eiliarks ;ind the rcflectioiis, which they their Circle reatlings [rot only a higher degree of precision, buggest. iiiiiy safely giiitle the jnptlniorit iri liiriiiiiig a clear Iiut (hesicles il partial check on the adopted correctinit for estiriiiitc of the real advantage, which is gained Iry tlie Iilrge runs) also thc freedom froiii the chief periodical screw-errors. size ol' tlic circle, with which observations are taken. It Tlie Radir observations will then gain doubly in iicciirae?, iippciirs a fair coiiclusioii , that the fortuitoiis rcatling errors if the corresponding intervals are at the same time used for of a circle ef nioderate tlimcnsions (say of two or at the the regular determination of the general correction for runs. niost thrce feet tliairieter), which is provided with four su- The simple arrangement of the wires, which 1 have perior microscope-micrometcrs, iire sufficiently sniall even for suggested for telescope - arid microscope - micrometers, can a nierit1ioii;il instrument of the first order, and tliere is no procure tlie elimination of those regular errors only, which difiiciilty in reducing them farther by improving the power return once in every revolution of the screw ant1 which of the reading apparatus. It is however of no practical probably in most cases represent the greater part of the importiiricc to diminish them below a certain limit, if those whole errors. Of course, it is quite practicable to go farther errors of the observations, which are independent of the and to get rid by means of additional wires likewise of readings, are riot at the same time reduced in a correspon- those errors, the period of which returns twice or oftener in ding I)roportion, and. though the latter are obviously capable each revolution, since the suggestive priociple of Bessel's of considerablc tliniinutiori, it appears entircly improbable, expedient for clearing the observations of a double image that they should ever becoiiic so small, that the reading micrometer of the periodical screw-errors may thus be made errors could not always be confilled within iriuch narrower generally available for wire niicrometers. The difficulty to limits. be considered refers only to the inconvenience of taking the $ 9. requisite nuiriber of readings. 1 iiiay perhaps remark. that in this respect a very de- The direct elimination of the errors, whicli come under cided iniprovement of the readiags might he effected I)y the form u sit1 (A+ r) + b .hi (R+ 2r) (r being the screw mealis of a very siiiiple contrivance, which is the insertion reading), demands four readings correspondirig to points of a secorid pair of wires in the niicroscopes at a distance equally distributed round the circumference, but which niay froiii the other pair of some whole riuniber of revolutions helong to different revolutions of the screw. ant1 a half. In case the micronieters are so adjusted, that In order to effect this elimination for a microscope- 71 whole revolutions of the screw carry the wires over orie micrometer, it would perhaps be best to arrange the four interval (,f the divisions, 2 11 & 4 revolutions will probably requisite pairs of wires so, that the points or the imagiuery be found the most suitable distance. If then the two next lines, which are to be brought upon the divisions, are se- divisions 011 the one side of the Zeropoitit are successively parated hy equal intervals of n + & revolutions. The rea- brought under the one pair of wires and the two divisions dings would require but little additional screwing (8 rev.) 221 Nr. 1262. arid the results woulcl possess the atlvaritagc: of heing referred The Nadir observations of such a systeni inay lie coilsidered to four different divisions. However, a# the claims of the unexceptionable. For, if n, I, c clesignate the three \\ires errors of runs to direct t~li~niiiatinnare the iiiore pressirig and a. p, y their reflected images, arid if the notation 3aab etc. arid ought not to he set asitle, in case ii microscope-micro- is eiiiployed to iiidicatc, that the three clistarices hetweeri p meter is read more than twice, reasons of expediency will arid n, a and a, a aiid b etc. are to be made equal to one prolialily in most oliservatioris forbid this manner of gettirig another, the successive oliservafiori of tlie four positiaris rid of the secondary periodical errors and make it preferable Baab, npbcr, y bfic arid 6yc3 is nrit only capable of' the to remove tlieni afterwards lig means of a special talile of highest possible degree of nicety, liut the ineaii of the fmr corrections. The construction of such a table, which, in corresponding readings becoiiies also free froni the two first order to be serviceaide iii all cases. muet likewise contain terms of the periodical errors of the screw. As regards the tlie whole CorrectionR for single readings, lieconies an easy observations of stars, the iiianiier of using the telescope- process, if it lie kept in view in the arranRenieiit of the micrometer will depend on the degree of exactness, which wires, which the screw has to carry. Provided that not is to be attained. In the case of faint stars or tvhere or- crosswires but close parallel wires are iised and that the dinary precision is deemed suflicient, two readings of the value of a revolution tloes riot exceed a certaiii liinit, the micrometer should be taken as a general rule, so that the most suitalile siiuple sysleiii mag perhaps be one of sin star niay be made to bisect the two iiitervals betweeii tlie single wires at intervals of g, 4, 2n, &, & revolutions, since wires in succession. In this way the Transit observations this arraiigeiiieiit appears withal well adapted to inect the are riot impaired and the niean of the two readings becomes requireriierits of' the several iiiethotls of takirig tlie reatliiip. free froni the chief periodical errors of tlie screw. Where :is I iiectl not farther explaiii. a Iiiylier degree of accuracy is tlesirable, two ;idtlitioiial star M'h i I e t I) c c i ri: ii nis tii ii cc. that t Ii e Ci rrI c r ra ti i ii gs t l epen tl reacliiigs, corresporiding to the Iiisections of the liy the on four (or six) tlilferciit iiiic*roiiietcrs. renders it oii the me iiiitldle- arid orie of the kitle- wires, will reiiiove iilzio the hiiii0 too troulilesoiiie aiid ori the other lez.4 iiecrssary to secontlary errors ; Iiut the meiiii of the limr reatliiigr *) will prowre for thii rcvidirip of the sirigle nricroaropes the inost require a correction of one eighth of the tlistani:e tietween exact precision. it is olivioiirly of the greatteat iiiiportanrc. the two outer wires, in order to refer to the same poiiit of that the rcwultiiig reitiliiig of the t e I es c n p c - iiiirronieter the system of wires, to which tlie iiieaii of the two ordiiiary shoriltl pnssess the tiighrst attilinalilc ilpyrc of accuracy. readings and also that of tlie Xadir readings beloiig. This The riiqpestioii iiiarle in $ 4. that two wires sholiltl he ill- point oliviously does not coincide with the iiiiddle wire, in sertcvl iri thr eye - 1iie.c-r at IidI' a rcvoIiitioii's t1ist;iric.r from case the two intervals are not rigorously equal, hut it lies cat-h other for !he purpose of eliriiiiiatiii~the rliief perioclical one quarter of flieir dittkreiice from it oii thc side of the errors of tlie screw by tloulile o1iserv;itioris. niay lie eiisiIy larger interval. 'he coirection may be obviated Iiy observiug exteiitlrtl itlid applied to thc rrniov;il of tlic

dilTercnce iir the IIIRIIIIC~of bisecting a star Ity ;I \\.ire and ratiis is rcqiiireil eveti for the iileIe liftirig of the iristrii~iieitt, of placirig it uiidw;ty Iict\vecu two wires, niust lie cluly itt- while total reversinti is quite Iiractiratile for aii iiistrunient terrtled to. l?nivcver 1 s1t;iIl riot further pursue the suliject, of the silltie optical lmver, if ittici~cu~i~l~~~redwith very heavy

any ne\v coiistrnctioit itot to enceetl very iiiucli the lowest \YhiIi! tltc ;)pl~;treitt ;idviint;~;e of superior prccir;ion of diniensiitus, whic:li will reasoii;thly iill~\~tlie fortuitous errors tlie rr;idings offerc,tl Ity a 1;ii.p circle furits nut oii closer of the Circle reiidings to lie confined lty iiteiiiis of'ii porrerfiiI inspectioti to Ite ;I ntatter nI' iiitlin'erelit iiiipiirta~ico;ti111 iility riticroscopicirl appar;ltus withiit sufficiently narrow limits. e;isily lie 1ii;itle good. \rlierc. the circ;e is siii;iIl, 1)s' slightly Tlioiiplt tlic coiisidcratioiis liere ;tclva~rcecl in I'iivoiir of iiicreiising tlie iti:nilicr of o1iscrv;itioiis. there is oil tlic other circles of' inntleriite siec arc cliicfly of it ti e g ii t i v e cliitracter, harttl iio qttcstion, tliiit it is 1~ur(:Iiiiseil;it the c,ost 01' sonie in as 1)tii1~11its they tend to prove, thitt the results ot' large vet'? serious sacrilircs. Thc itiil)r~cticiil,ilit! of' c:;tstiiig ;I circles are sthject to ~cveriil itiost serious clis;itlv;intages, larpe vircl!! in one piece, tlie i:onsecluciit \v;int of liouingeiieous 1,oiii whicli those of sniall ones are ~leiirl~free, arid that structure , tlie intwased 1i;tIiility to irrepiar strain 31, the this freedoiii is liought iit the (:oat of a iiiost trifling and coit~~)ari~tiveiiiferiority iri strength in spite ot' the largely unessential sacrifice, yet these tiegative merits would saeni auguieiitetl weight, the greilter sensiliility (of Iioth, (he circle fully sullicieiit lo justiIj- a decisive pref'crence. Sii~itIIcircles arid the niic:rosco~)ic:al iippar;lf~s) to tlie iitfliiencc of' unequal however ~tossessalso a iitost decided atlvaiitapc of a p o s it i v e teinlieriiture, the corisequent increased clifGcnlty of eliitiiriating kind over larger ones in allo\vitig the applicatioit of a pr;icti- the teiuporary errors of division and tlie variations of the cable niethotl, by which the perlitanent errors of a I I the Zenith points, the sacrifice of power of the niicrorcopes for divisions niay be trustworthily deterinined and all the ob- the sake of convenient illnmination, the relative loss in con- servations thus freed froni this most troublcsonie and in- stancy of the correction for rii~is- thew itnd (Jth~!l.draw- tractable suurce of constant errors, an advantage, which is backs iire 11i~ad~iliitage~of a large circle, as coniparctl with of so evideiit and grcat an importance, that probably it ought one of moderate dinlensions against which its douhtful , to be secured, even if it were to he purchased at a consi- superiority with regard to accidental reading errors appears derable sacrilice with respect to the accidental errors of the only a poor aid trivial set-off. Considering further that, if Circle readings, and which consequently it would be ini- syninictry of construction is not to lie abandoned ~ the large piidence to forego, when it may be had itt so insignificant size ant1 consequent heaviness of a circle deiiiantl it corres- a cost. ponding weight 011 the opposite side of the telescope **) and render it tlierefore necessary to augment the strength arid 5 11. massiveness of thc asis accordingly; considering too, that The niethotl, introduced by Uessel, for determining the all counterpoises itnd accessory contrivances ,;ire to be aug- errors of division by niearts of edtlitional niicroscope-micro- meters is capable of a very high degree of precision and leaves little to be desired, where a special purpose is to be In tlic ciise of the Greenwich Circle and of fiome otliere this danger of irregular strain is iuucli aggruvated by the served or the errors of only a liniitetl number of lines of circqmstance , tltiit tlie gradaitted band of silver is carried division are to he found.") But it bccoines practically una- by R circle of iron instead of brass or bronze. The rcason, vailalile, it' the deterniirration not of the regular, but of the why iron has been preferred, does not clearly appear. Hut iiitlidual errors of it II the lines of a gradiiated cirle is re- the opinion seema to lie well funndctl, tlitit whiitever evil or inconrcnieiice it may hate been intended .to overcome quired; For the great nunher of these lines renders the ap- is really far smaller than that of the nilopted remedy. plication of the niethod to the whole of theni with anything Symmetrical distribution of the weight, which is considered like toleraltle exactitude or certainty of succe~sa iiiost irk- by many astroiioruere a most desirable condition of R first some and t'orbicldirig task. And if for the sake of saving rate ineridisln instrument, has apparently not been Rimed at in the cuiiatructiori of the Greenwich Tratisit Circle. The wegtern cone of the axis is lotided by about 150 Ibs. iiiore *) The best speeiinen cil' ii ciireful upplicalion of the nieihod tliaii tlic other itnd it io lint clear from thc description, ahcther is probably or at least, so fur as I know, Pytrrs's ,,Unter- the necerdiiry pains haw lieeii taken to eqiialiae the residual auchung der Theiliingafehler de9 Erfcl'achcii Verticalkreiaea der prcsaurc: 011 tliu beiirinp of the two yitota. Pulkowaer Stcrnwnrtc" in tlie P'eterdbourv Plunioira of 1 H1R. NI. 1263. 'L2 6

trouble the investigation is conducted in siich a wanner, that reacllng circle, since then the angle between any two diaiiie- the great majority of the determinations can potweas oiily R ters coirlcl be ninde direc4y tlie subject of a special refiearch. very inferior degree of precision. it appears aomewbat questio- &It as only a limited nuniber of lines is needed for the nable, how far a table of corrections constructed under sueh general investigatiori of' the errors of clivifiiori and us for the circumstances could really be considered to afford a reliahle facilitation of thiN investigation it appeara very tleeirable, inqiroveniciit of the ohservations. that the attention should not be unnecessarily distracted, it In order to escape these diffiaultier and yet to free the will he better to lay oil the strokes of the circle only at Circle readings froiii the errore of division, Hunsm has in- considerahle iritervalfi (equivalent say to n subintervals of troduced a construction, d*hich requires only a limited oum- the reading circle), and ol' these large intervals to eiibdivide ber of lines of division on the circle, and he has thereby not more tliro two (or if preferred, four) oppopite ones in succeeded in making the usual nrode of determining their the eame mode as the reading circle or in their n subinter- errors a feasilile process for all of them. The plan is fully vals. The investigation is then to be conducted in the fol- described and discussed in a poper of the bstr. Nachr. vol. 17, lowing i~ianner,which 01' coiwse niay partly be nioditiecl ac- which has been already quoted and to which 1 must refer cording to the individual views of the observer. those readers, who may not he acquainted with it. - But The auxiliary circle is clamped to the axiR in such a obvious aid important as the advantages of Hanscn's method position, that one ol' its chief divisions is nearly in coinci- are, it seenis more than doubtful, wbether the drawbacks dence with same cerdinitl division of the reading circle, for and incorrvenieuces attending it, which 1 need not dwell which purpose it seems hest to select that employed in the upon, will not always prove ton great an obstacle to its in- Nadir observations of the wire. After suflicient time has troduction into ge n era1 use. Most observers will probably elapsed for allowing the corresponding parts of the two consider it preferable to retain the usual method of taking circles to assume nearly the same temperatures, the lines of the readings of a fully graduated circle by means of fixed the auxiliary circle are successively brought under the ordi- microscope - micrometers, provided that some expedient can nary microscqpe - micrometers and the distances measured be found, which renders the trustworthy determination of the between them and the lines on the reading circle, with division errors of all the lines (or rather diameters) OD the which they nearly coincide. This will best be done by circle a practicable undertaking. And I cannot but think, employing two opposite microscopes at the same time, so that such an expedient is easily and conveniently supplied, that it is then not necessary, except for the sake of repetition, if the errors of division are inveatigated not by auxiliary to turn the circle through more than half a revolution. 1 microscopes, but by mean8 of an auxiliary circle devoted to Deed scarcely allude to the necessity of taking the measures that special purpose. A short description will suficiently near the same points in the fields of the rnicrometers in order enable reflecting readers to form their own jugdment on the to avoid any injurious effect of the circumstance, that the value of such an additional apparatus, and to decide whether two circles may not be rigorously in the same plane. For I am right or wrong in believing, that it will prove a clear the sake of giving proper weight to the determination of the gain to provide every new first rate circle with such a ser- general mass of the divieion errors the measurenients of the viceable ally. distances between the auxiliary subdivisious and their cur- responding lines on the reading circle ought to be repeated d 12. several times; for the chief divisions these repetitions appear The secoud circle is, like a vernier circle, to be so let superfluous, as single measurements will be quite sufficient into the rim of the reading circle, that the graduations of to secure ultimately the accurate determiuation of their er- both are in the same plane and niay be read off by the rors. - After the first set of nbservations is finished, the same powerful microscope - micrometers, which I assume to anxiliary circle is unclaruped and turned round till the next be solidly imhedded in the pier. While the reading circle of its chief diameters nearly coincides with the cardinal dia- is firmly fixed to the axis of the instrument, the auxiliary meter of the reading circle. A siniilar set of measurements circle sits loosely and may be turned round and clamped is then taken in this second position of the two circles and to the axis in sonie unobjectionable manner at pleasure. Its the whole operation is repeated, till all the chief cliamelere graduation is arranged with the sole view to rendering it of the auxiliary circle have been successively in coincidence most serviceable in the intended investigation. If tbe object with the cardinal diameter. were to provide for all possible emergencies the most effectual The pioyer copibination of the data, which result from plan would lie to have the auxiliary divided exactly as the all the sets of measiircnients, gives then at once the cor- 58r Rd. 15 227 228 rections ol' all the chief diameters of both circles and nll ; After having thus determiued the true values of the with the mine degree of very bigh exactness. The obser- auxiliary euhdivisious, the' deduction of the corrections tor the vations furnish also the data for finding the corrections of suhdivisions nf the reading circle is a very simple and eaey the subdivisions of the reading circle irr reference to all its process. The ultimate prohable errors of these corrections chief divisions, but the final values of these co(rections de- are very nearly the sanie. as if they depended entirely on pend on the previous determination of the true values of the probable ertor and the number of the ohserved coinci- the auxiliary subdivisions and it becomes a question how dences for the circumstance, that the subdivieions of the this deterniinatioii inay best be accomplished. Though reading circle are referred to the whole of its chief divisions Ransen has (in art. 12 of his paper) already pointed out the and not merely the hvvn nearest, and the high degree of proper way, in which such an investigation ought to be cou- precieinn, with which the auxiliary euhdivisions are deter- ducted iri order to irisure success*), I may perhaps be per- mined, render the lose of weight from these two causes, mitted to suggest a modification of his directions, which which otherwise would turn out a very serious drawhack, would apparently enhance in no slight degree the tinal pre- little perceptible. If therefore the probable error of an ob- cision of the deduced corrections of all the subdivisions. sewed coincidence is kuown, it is easy to decide beforehand If the nunilers i,, &.. .i. denote all the divisors of the how often the observations of coincidences of the suhdivisions oumber n arid if consequently il, is.. .i, subintervals are all ought to be repeated in order to secure the desired accuracy the possihle aliquot parts of the interval between two suc- for the tinal deteniiinstions of all the division errors. *) cessive chief divisions, let those diameters of the reading The riuniber of these repetitions as well as of the lines on circle on both sides of the cardinal diameter, which are the circle must lie taken into account in estiniating the it, ia... i, subiritervals distant t'roni it, he temporarily distin- amount of trorille required for the whole investigation of guished by some simple means. Let then the suhclivisions the division eriws and inust therefore guide the jugtlment of the auxiliary circle together wilh the two enclosing chief in fixinF ripon a proper value of ?I. 00 the wbnle it is divisions be brought successively into coincidence with the likely, that pi = 60 will he considered hy many observers cardinal division of the reading circle and let the dbtances the niost suitdile and convenient number, wliether the iriter- be nieasrired lietween thaw distinguished lines which fall vats of disiuiaii he 5 nr 3 or 2 niinutes. However this is within the range of the subdivided intervals of the auxiliary a matter of detail, wliicir I need not enter upon. circle, and the coinciding lines of the latter. In doing this, The part, tvbiclr the auxiliary circle has to pert'oriii in let the numlier of these nieawrenienfs he rlouliled not only the ahove iovestigation, iiiiplics with respect to it the snnie for the cardinal dianieter, but alsii for all those distinguished assumption, wliicli is tacitly niade with respect to the reading lines on the n n e side ot' it, the lines correspotiding to which circle HII~iipon the rorrectriess of' tv1iic.h the whole Iiossihilitg on the other side are out of the range of the auxiliary iii- nl' rectifyiiip the pernimeiit errors of division Iiy iiieaiis of a terval6. - From the n + I sets of 2(u + 1) nieaaurenients special table ol' correctinns rests, rianiely tho assiimption, thus takeo 2n + 2u ~inkiiocvri quantities are then to be de- that those errors are iiitleed pernianeiit or constant, which duced, naniely tlie true values of the 11 - 1 aiixiliary suh- reniairi after the teniporary errors Iiroduced Iiy inequalities divisions, the true values of the 2v interv;ils tietwecn the of teiiiperature aiid hy gravity have beerr tluly rcnioved or distineuished diameters iind the cardinal diameter an11 the at least rcducetl Iiy nieaiis or the increased nunilier of' nii- n + I distances of the ausiliaiy Zerotliameter l'roni the car- croiiieters. The legitiiiiacy or this ;issiiiiiption is identical dinal diaiiieter in the 11 + 1 cliffererit sets ol' iiieasurenientts. for hoth circles arid therrfnre cannot lie olijecteti to for the Though the nuniher of these unkiiown quantities is large, one, while it is granted for the other. eqiecially, since it is the requisite calculation will not prove very troublesome, - provided of course, that the great advaritages, which ensue for the solution of the resulting equations t'rani their partial *) I'erltoyr 1 iuiiit not .kavs unnoticed, tlint witit her deter- iuinntinn of tlie errors of the subdivisinnd iiiiiy lie giiined reciprocity, are properly illid fully aftelided ta, ii proviso so by nliwrving the coincidences of the chief divisions of the obvious, that it is clot necessary to dwell upn it. naxilirry circle with the corrceponding subdivisinnd of the reuding circle. while the euxilinry aiibdivisioni ere in coin- cidence with tlie cardinal line. But I CM scarcely rectimiiiend tliid decond courde, RE the weight of its resutta will be *) Ar+ander'a investigatiiiri of the graduated tirc employed in somewhat lees than that of the firat, nnd it will therefore hia Northern Zunea iuay rclae be consulted nit thia point. I prove ninre prnfitnble tii incrense inetaiid tlie niiiiilier of Cf. art. 4 of the firet siiliiine iif the BOIIR obdrrvatiiino. 1 rc*yrtitinno of the first detcrniinatioiis. not a permaoeiit assuauption, aa io the case nC the reacting of the corrections €or particular linen The whole aninunf circle, but is only made f9r the tiilie that tile jiivestjFaticyl of tiure and trouble, which the complete iuvestigation de- lasts. Allowing its correctnesa, it iZ easy to perceive, how mands, ia indeed by no nieans inconsitlerahlc; hut. a* it effectively the described process elimiuates the greatest part niay be spread over a long period withoiit disatlvaiitage, it of a11 teniliorary errors and secures iii the final tleterniinatioli8 will he found neither tireaonie nor incoriveriient. The only a true representation of tlrc purely constant errors ol'diuiaion. conditioa, which ought not to be neglected. ndiiiely. that In order to insure this prity 88 far as possihle, car~'shou1d each .singre set of measurements shoiild lie gone through be taken in the construction of the inktrupient, that hnth without interruption in me sittirig, can, it seems, be easily circler niaj he affected nearly to the s;loie esterit hy all complied Jyitb. Iii other respects the arraiigeiiients niay be caiiRes of teiiiporarp errom. For this aod other reasona it niadce entirely according % conaiderations of convenience, as will probably be hest to havc the gratlueteti bands carried a set of measurements niay he taken at any tiiire without by solid disks instead of wheels and, as the circles are to any hut the niost trifling previoua preparation and without be of iiioderate size, there certainly appears no valid ob- at all interfering with the arrangements fnr the regular ob- jectioii to thia, However, it' it is desired to procure for the servations. And the consciousness, that ultimate success is circles the utiiiost lightness without impairing their homo- all but certain arid that each set coutriliutes directly atid geneity and strength, let them he cast as disks and let a without sensible loss towards the final deterniinations, will sufficient portion of the nietal be cut out agerwarcls, as is perhaps go far with eonre observers to convert, what other- sometimes done in the case of the circlee of small portable wise might be deemed a somewhat tedious tarik, into a instruments. But instead of the usual wheel -pattern *) for welcome and agreeable undertaking. these cuttings it appears preferable to select some other, In conclusion I need scarcely poiiit to the saving of which contains a larger number of smaller openings. In the expense, which is effected by furnishing an instrument with case of a douhle circle the two patterns may then be so a double circle of moderate dimensions instead of a single chosen, that the apertures of' the one circle aye met by the one of large size together witb the necessary additional solid metal of the other, so tbat the air on either side may microscope -micrometers. It is for the final correctness of have accesd to both circles. the readings almost a matter of indifference, whether the The method here proposed for clearing the readings of double circle is well divided or not: indeed it may be COD- a graduated circle of the permanent errors of division by a sidered more essential, that all the strokes should be thorough determination of these errors appears to accomplish uniformty flne and sharp and clean-cut. But not to press this its object to a very fair extent and without inordinate ex- point and supposing even, that the possible lsetidiousness penditure of time and trouble. The table of corrections, of instrunreot - makers in submitting to the intended ordeal which it Gnally furnishes for all the diameters of the circle, iione but circles divided in a very superior style is fully will possess a degree of precision, which seems amply suffi- indulged, it is clear that the saving of cost must still be cient to satisfy even superior requirements, and where for very material. However, though this consideration will pro- special purposes a still higher degree is .deemed desirable, bably be of great weight in particular cases, it ie only of the exact determinations of the chief divisions of the reading secondary importance. The chief claim of the proposed circle aud of the auxiliary subdivisions afford ready meaos construction and that, which I wish to urge upon the con- for increasing at pleasure and with little labour the accuracy sideration of interested readers, is, that it effechally removea one of the most serious obstacles to obtaining correct reaults and that, combined with a thorough invcstigation of tbe ') There ie apparently no practical reamon for dimregarding effects of flexure, it fairly promises to grapple victoriously in the construction sf wbeet- circles thole consideratiome, which render it derkable, thsL the number of spokes shoold with tbe difficult and much vexed question respecting ahso- not be a multiple of the number of reading-micrometer,. lute declinations. A. Marfir.