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 star places from one epoch 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 stars, they are a very great help in checking calculations, the interval by the tangent of the star's declination 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 right ascension 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 Polaris 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 declinations 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.