21 2834 22

6. Beispiel. 7 Ziffergruppen. Elliptische Elemente eines Cometen. Comet Ellipse 09902 November 19779 11757 18123 Elliptische Elemente des neuen Cometen : I4449 19250 93260. T = November I 9.7 79 M. Z. Berlin Oppenheim. (D = 117'57' = 181 23 M. Aequ. des Jahresanfangs i= I44 49 q 1.9250 1 = e = 0.9902 Opperikeim. Anmerkung. Das 5. und 6. Beispiel bezieht sich auf die von Dr. S. Oppenheim in A. N. 2692 mitgetheilten parabolischen resp. elliptischen Elemente des Cometen 1881VIII. A. Krzlegcr.

On t'he Reduction of Stsr Places by Bohnenberger's Method. By Professor Truman Henry Saford.*) Every astronomer who has had much to do with the Using similar considerations I have been in the habit reduction of places from one to another is of selecting the epochs for the use of Bohnenberger's for- aware that in case the star be very near the pole, or the mula as follows. epochs very distant, the method invented by Bohnen- We have tables for the secular variation, and for the berger must replace the ordinary employment of annual term multiplied by the third power of the time. While precessions and secular variations. these tables are not quite complete enough for close polar It is easy enough to see that when the product of , they are a very great help in checking calculations, the interval by the tangent of the star's reaches and it is easy enough to use them or to calculate the terms a certain amount the precession series is no longer con- depending upon the third, and by their variations from year vergent. If I can rely upon my remembrance of a mathe- to year those depending upon the fourth power of the time. matical investigation found in many text-books, the series So that when the term

I - 1/3+1/5-1/,+ 1/9-1,'11 etc. for l/.,z = 0.78539.. . ll5 t5n5 tan56 sin 5 a is just on the line between convergence and non-conver- is insensible, it is not necessary to employ the trigono- gence; and when the series, by which the metrical formula. of a star is expressed in terms of the time, contains as a But it will be safer not to notice the sine of five part of itself a series of this degree of convergence it is times the right ascension as a factor; as, when it is casually no longer practicable to use it. small, the next factor sin 6a may be large; and we may From Bessel's investigation on pages X and XI of consider the introduction to the Tabulae Regiomontanae, one can 115 (tn tand)5 readily enough see that a star's right ascension as a function as the quantity to be made o:'or of arc. We shall then have of the time contains terms of the form tn tan 6 sina -+ t?na tan2 6 sin 2 a or in tan 6 t3n3 tan36 sin 3a = 3.38455-10 log 206264:s + 1l4t4n4 tan46 sin 4a or +etc. tn tan 6 = 8.67691 - 10 together with other terms multiplied by the same powers log 206264:'s of tand, but lower powers of the time. The terms here given or logtn tand = 3.99134. are those which determine the convergence when tand is very great. Taking logn for 1900, according to Struve = 1.30216, If, now, tn tand = I we shall have a series which we shall have exhibits precisely the same kind of lack of convergence as logtntand = 2.68918 the well-known numerical one which I quoted before. or ttand = 488.9 years. As is roughly 20" or ~/loooo of radius, it is plain that this absolute fa.ilure of convergence takes place when For in 1887, whose declination is 8S042!5, ttand is about ro.ooo; or for a star IO from the pole in the interval here indicated is about II years. That is to about I 80 years. say, the right ascension of Polaris can now be carried

*) Auf Wunsch des Verfassers mit einigen Verbesserungen und Zusiitzen nach den Monthly Notices abgedruckt. . 1. 23 2834 ‘34 forward or back II years by the use of the first four rhich these theoretical ones may be extended without differential coefficients of the right ascension with respect iamage. to the time; so that a trigonometrical computation for In this way it will be seen that for stars not very every 2oyears is amply sufficient to afford a check at the iear the pole the terms beyond the ordinary ,secular beginning of one period and the end of the next preceding. :ariation( soon become sensible. As a long and valuable For a star in the declination 88’58’ the single inter- nemoir by two Austrian astronoiners, Herz and Strobl, val t is rather less than 9 years, and the double rather vhich I have seen but do not yet possess, relates, in part, more than 17; so that to compute a series of places from o this subject in its application to Auwers’ fundamental year to year of R Ursae Minoris or Groombridge 1119 = :atalogue, I will only say here that for d = 49, t should B. A.C. 2 3 20 (Connaissance des Temps) the trigonometrical lot be more than 54years to secure (without examination computation should be employed every 16 years at least, )f the special case) the insensibility to less than O:’OI of to make sure of OYOI in the right ascension. Similar con- .he error caused by omitting the third term. Professor sideration should be applied to find out when other terms Argelander in fact calculated these values for all become insensible, or less than O:’OI. wer 39O; his greatest interval was 120 years (1755-1875)~ Suppose we employ the general formula which agrees well enough with my table. I The actual calculation by Bohnenberger’s method re- - (tit tan d)P = O:’OI quires the use of three quantities z+l, z’-?.‘, and d, P which depend partly upon the epoch from which, and that making P successively 3, 4, 5 . . . and we shall have to which, the reduction is effected; and partly upon the PX o”ox I 20626418 epoch taken as the fundamental one for the precession- ttand = (206264 8)F-- n constants, and so forth. In the use of Struve’s precessions, now entirely general, as given in the following little table: the fundamental epoch is of course 1800. I have used 1855 as the general date from and to which the places P t tan d t for Polar. t for ?. Urs.Min. are to be reduced, following Carrington for reasons of 3 54YI rJI2 1 Yo convenience. The same date had been also adopted by 4 215.9 4.9 3.9 Dr. Gould, and was afterwards used by Professor Boss; 5 488.9 I1 8.9 and I have not thought it worth while to change at present 6 837.3 ‘9 ‘5 to a later epoch. Carrington’s catalogue affords (for all 7 1225 28 22 close northern polars) approximate positions at once for 8 1625 37 30 this year ; it also lies between the two Radcliffe catalogues, 9 202 I 46 37 which can by their own data be readily reduced to it; 10 2403 54 44 and is nearly identical with the general mean date (by ‘The formula is weight) of the existing catalogues of these northern stars, I The method which I employ makes this a convenience, and logttangd = 4.01226- 7.3’443 I 1ogP. at the same time renders it needless to select a later date P P for the constants a1-1, etc. Practically, anyone who attempts to use the higher If a change is to be made in this respect I think terms up to the loth in reducing star places will find it a the next year adopted should be 1900, so far as these very uncertain and perplexing matter, far less easy than the quantities are concerned. As a matter of curiosity I desired judicious employment of the trigonometrical formulae. to follow the changes in the place of Polaris, and so com- If in any given case, like that of R Ursae Minoris puted some values of these constants up to 2100. In the for example, it is desired to extend the series to the 5th table which follows I give them; those to be added to the term, this can be practically accomplished as follows : right ascension are expressed in time. This is now rendered The highest part only, in powers of tand, of the 4th a matter of the greatest convenience by the admirable term will be tables of the previously mentioned astronomer Norbert t4 n’4 tan4 d sin 4 a Hem,*) published by Teubner at Leipzig in 1885; a book and that of the 5th term which I find indispensable in all computations of precession. ,4t a future time I intend to publish the results of t5 tan5 6 sin 5 u my discussion‘ of the relative movements of Polaris and which last will doubtless be sufficiently approximate to the the north pole; it is enough to say here that the nearest total value of the term for 10years, as it cannot exceed approach takes place, according to my calculation, some oY02 for this period, which is at any rate quite insensible time in the year 2102 at a north declination of 89”32’ 2 3“; when reduced to the parallel of the star. the star will reach 89O about 1944, and be for about If, indeed, we chose to take O!’OI after reduction tc 300 years within a degree of the pole. the parallel as our limit of accuracy in computation, WE The tables have been pretty thoroughly checked in may practically employ the t corresponding to a term farther along in the series than we intend to employ; but ”) Siebenstellige Logarithmen der trigonometrischen Functionen a skilled computer will soon find the practical limits beyond fur jede Zeitsecnnde, 186 pp. 8vo. 25 2834 26 various ways. They were computed for every twenty years The simplest method of discussing a long series of with seven-figure logarithms ; and were compared with values polar-star places collected from the various catalogues is, given by Dr. Gould and Professor Boss for various dates, to assume a position for 1855.0 from a careful preliminary and by differences for intermediate years. discussion, reduce it to the various epochs of Table I. for Moreover, the position of Polaris was brought up to which there are observations, and compare with observations. 1900 from 1855, and then to 2000 and 2100 by both The benefit of this process is that defective or erroneous tables, with very slightly different results; in neither co- right ascensions do not affect the reductions of the decli- ordinate more than 0!'002 in arc of a great circle. I had nation; while the error in the assumed place is readily found no ten-figure table at hand, but it hardly seemed necessary. and allowance made if necessary.

Table I. For Bohnenberger's Method.

(The reductions are from 1 5 5.0 to the tabular dates.) - - Date z+l 8'- 1' 8 Date z+l 2'- 1' 8 '755 -zrn33!588 -zm33!506 -oo 33' 25Y987 I 865 +Om15?301 +omi 5?418 +Oo 3'20:'555 I 760 -2 25.909 -2 25.835 -0 31 45.671 1870 +o 22.976 to23.104 to 5 0.830 I 780 -I 55.196 -I 55.141 -0 25 4423 1872 +o 26.046 +o 26.178 +o 5 40.939 1790 -1 39.841 -1 39.79' -0 21 43.809 1875 +o 30.651 +O 30.790 +o 6 41.102 I 800 -I 24.487 -1 24.438 -0 18 23.202 I 880 +o 38.325 +o 38.477 +O 8 21.372 1810 -1 9.133 -I 9.082 -0 15 2.603 1885 +o 46.000 +o 46.165 +o 10 1.640 1815 -I 1.456 -1 1.404 -0 13 22.305 1890 +o 53.674 +o 53353 +O I1 41.905 1820 -0 53.779 -0 53.724 -0 11 42.010 I 900 +I 9.022 +I 9.232 to15 2.428 1828 -0 41.497 -0 41.436 -0 9 1.542 1920 +I 39.716 + I 39.99 7 +o 21 43.445 I 830 -0 38.427 -0 38.364 -0 8 21.426 '940 +2 10.407 +2 10.7'12 +O 28 24.422 I835 -0 30.751 -0 30.683 --o 6 41.137 I 960 +2 41.096 +2 41.558 +o 35 5.355 I 840 -0 23.075 -0 23.001 -0 5 0.849 1980 +3 11.782 +3 12.354 +o 41 46.244 1842 -0 20.005 -0 19.928 -0 4 20.735 2000 4.3 42.467 +3 43.161 +o 48 27.085 1845 -0 15.400 -0 15.318 -0 3 2c.564 2020 +4 13.149 +4 13.978 +o 55 7.879 1850 -0 7.124 -0 7.635 -0 I 40.281 2040 +4 43.829 +4 44.807 +I I 48.622 1852 -0 4.654 -0 4.562 -0 I 0.168 2060 +5 14.507 +5 15.646 +I 8 29.312 1855 -0 0.049 0.049 0.000 2080 +5 45.182 i-5 46.496 +I 15 9.947 I 860 -l-o 7.627 +o 7.733 -kO I 40.219 2 100 +6 15.856 +6 17.35 7 +I 21 50.525 I 864 +o 13.766 +o 13.881 +o 3 0.500

Table 11, (Keduction from 1900 to tabular date.)

+0"3 oS9 26 +oo 6' 411025 2020 t3m33962 too40' 5Y562 +I 1.701 +o 13 22.013 2040 +3 34.641 +o 47 46.339 +I 32.486 +o 20 a.962 2060 4-4 5.3'8 +o 53 27.068 +2 3.282 +o 26 43.872 2080 +4 35.992 4-1 0 7.746 +2 34.088 +o 33 24.739 2100 +5 6.665 +I 6 48.373

The formulae for reductions from 1855 or 1900 to any tabular date are (here a, d are the position for the fundamental epoch, a', d' those for the tabular date).

A = a+z+R cos 6' sin A' = cos d sin A m cosM = cos d cos A cos 6' cosA' = m cos (Mt8) vn sinM = sind sin 6' = m sin (M+8) a' = A'+z'-Z

Other formulae giving the same results are frequently employed, but I find these on the whole the most convenient. 27 2834 28

Additional Note. It seems not to have been generally noticed that the to = '855 quantities z +R +a'- 2 and B are very nearly equal to t' i z+z'+I.-L'- wz dt 8- sndt mdt and n dt respectively. Carrington indeed gives s c 4 to to si I 760 -0S003 +0:'034 formulae from which this would be very readily inferred; I 800 -0.00 I +0.006 not quite so readily as it would have been but for a slight I 840 0.000 0.000 numerical mistake in the formula for 8; which as he gives I 880 -0.001 0.000 1920 +o.ooz -0.010 it is 20!'0611 (f- 1) +o!'oooo~~2t2 -o!'oooo43~ 12. The I +0.004 true value of the factor of (i- t) is 20Y0607. 960 -0.044 The consequence is that in his table for the quan- 2000 +0.008 -0.12r tities in my table I the value of B is in error by o!o2 at 2040 +0.016 -0.253 the beginning and end of the century. The formulae for 2080 +0.028 -0.458 2100 fo.036 -0.594 ssmdt and ndt are, counting t as usual from 1800, i' to = 1900 2000 +0.002 -0.036 Jmdt = const, +- 3fo7082 t +0?09496 2100 +0.019 -0.3 I 5 These small differences are, as will readily be seen

proportional to the third power of the time from to; but and the differences of the quantities z+z'+R -I.', 8 it must not be forgotten that terms of this order are from the respective definite integrals are as follows neglected in the expressions for m and 1z themselves.

1888 T. m. Nice A AR. ADP. ICp. AR.app. llogj.~Il 1)P.app. I1ogp.A Red.ad Lapp. * I -~ --__ 1 Mars 19 17~29~43~+om5z?oz + 7' 7!'7 7 21h22m25~08 102' 16'11l7 0.825, -1523 +3!1 I 21 17 I 33 +O 6.06 1 - 2 11.6 1 6 21 28 48.98 99 32 54.3 0.812, -1.19 +3.8 2

~

~e AR. 1888.0 DP. 1888.0 Autorite ~--I - I 2rhzrm34%9 102' 9' 0!9 (Y.9369+W, 21h453) 2 21 28 44.11 99 35 2.1 1/2 (Sj.8739+W1 21h633)

1888 T.m.Torino Acr Ad ICfr. crapp. dapp. Red. ad Lapp. - I - 1 I Marzo 22 17~12~32~+om16f06 1 - FT<2m2fo8 -1l18 - 22 17 2 30 - + 9' 1313 4 - I--8'12' 12Y8 - -4!'4