61 Nr. 2092 62

die nach einer Vergleichung mtt Lamont in Declination einer Correction von -10" bedarf.

A.-Oe. und ungeGhren

Stern 55. In Wash. Zone 204 Nr. 13 ist die R.-A.

Ich gebe nocli einige gelegentlich bestimmte Sterne: dieses Sternes urn lm zu gross angesetzt.

ebenfalls aiif 1875 .O. a B. D. 1797 8h 8m51820 3-11" 6'19"2

Anschluss an Stern 1

Anschluss an Stern 34 b 10% 15 30 32.39 -10 17 16.5

c d Vergleichsterne 1 und 2 znr Wiener Beobach- tung des Cometen 1874 I1 vom 7. Mai

18h18m47837 +32O44"12' 6 18 19 17.34 +32 44 54.2

18 17 23.21 +32 47 51.1 Anschluss an Lal. 33960

e 10m 20 27 38.44 -22 14 49.4 Arischliiss an Stern 57

Anschluss an Stern 58. f 9m 19 26 11.76 -18 24 10.0

Sur le principe de la Moyenne arithmdtique. A paper upon this subject by Professor J. V.

Schiaparelli appears in the Astr. Nachr. Nr. 2068, page 55. In that paper F (at, a2 . . . a,) represents the moat probable result which can be deduced from the independent direcat measures aI , a2 . . , a, which are supposed to be all equally probable or to have equal weights: the proof that F is the arithmetical mean is made to depend on three assumptions, which are regarded as axiomatic.

The third assrimption is subsequently replaced by the following which 1 give in Professor Schiaparelli's words.

,,On peut arriver aux equations (4) d'une m a n i h heaucoup plus simple c t plus directe. En effet, puisque toutes les quantitks 81, a2 . . . a, sont regardkes comme &ant d'8gale axactitode, si k l'une delles .on attribue la petite variation e, le changement qiii en rdsulte dans F doit 6tre toujours le mkme, quelle qiie soit la quan- tit6 affecthe de la variation E. Car si l'ell'et qui r8si:lte pour F de l'introduction de E dans 8 1 , etait pliis grand que l'ell'et analogue qui resulterait de l'introduction de e dans a2, on en conclurait, qu'une erreiir de a1, pBse sur l'erreur du resultat F d'une manikre plus conside- rable qu'ane erreur Qgale de as, ou bien, qu'on regarde des erreurs Bgales de a1 et a2 comme ayant une impor- tance idga le sur le resultat: 011 enfin, que a1 et a2 ne sont pas de mdme poids, contre la siippositiou'.

Althougli there is no reference to any paper of mine upon this subject, the principle here stat.ed is identical with that from which I deduced tlie arithme- tical mean nearly three years ago. My paper appears i i i the Monthly Notices of the Royal Astronomical So- ciety of London, vol. XXXIII, Nr. 9, ,1873 November. My statement of the assumption made is as follows:

,,I assume as an axiom that since all the direct

measures are, by assumption, ot' equal value or equally good, the most probable value which can be adopted is that to which each ilidividual measure equally cou- tributes. To obtain the most probable value, therefore, we must combine all the independent measures in such a way, that an error whirh may exist in one of the measures, as X I , shall produce the same error in the ,,value adopted as the most probable' as would be pro- duced by the same error in xl, xg or x..

This appears to me clear. The probable discordance of each meaaure from the true result ie the same, and this being the case, no good reason can be assigned why we should adopt a value in which an existing error, OF arbitrary change, in XI s!iould produce either a greater or less error, or arbitrary change, in the adopted value than would be produced by the same error, or arbitrary change in x2, xg or x,,. This con- dition of equal contribution of the independent mea- sures to the most probable result appears to me neces- sary and sufficient'.

Starting with this principle, since q~ (XI + E ) ,

q~ (x2 4- E ) q~ (x. $. e) are all equal whatever be the value of e, we see at once from a comparisoii of the coefficients of the different powers of E in the separate expansions that the differential coefficients of 9 with respect to the independent variables XI, 12 . . . x, must be equal, each to each. It is then shewn in my paper that tlie only. function which will satisfy the necessary conditions is a function of the arithmetical mean. Finally, sinre the most probable value which can be deduced from two equally good direct measnres is the mean, I shew by sucoesive induction that the most probable value for any number of such direct measures, which must be a function ot the arithmetical mean, is the arithmetical mean itself.

63 Nr. 2092 64

It will be seen that the fundamental axiom of Pro- fessor Schiaparclli’s paper is identical with that enun - ciated by me. I however based the whole proof upoii this axiom and the assumption that for two independent measures the arithmetical mean is the most probable result. The assumptions I and I1 of Professor Schia- parelli‘s paper are no doubt satisfied by the most pro- bable value, for they can be deduced by a strict mathe- matical proof from the axiom assumed by me and by Professor Schiaparelli , h u t their introduction is anne- cessary, and their axiomatic character is doubtful; for a function may involve concrete as well as abstract coefficients, and although if we change the unit adop- ted for the measures 01, a2 . . . a,, we must necessarily change the value of the function in the same propor- tion, it does not follow that the function must be homo- geneous with respect to the variables ai, a2 . . . a,,,

Since Professor Schiaparelli has not mentioned the identity of his views of the subject with mine, I pre- sume that he has been led independently to the fiinda- mental assumption upon which our proofs depend. In this case the identity of views and almost of expression is remarkable. I hope it may indicate that the view taken by me is one natorally suggested to minds fami- liar with the discussion of the results of observation.

The corresponding theorem wben the independent measures X I , x2 . . . x,, are required to influence the result in any given proportions follows from reasoning exactly similar to that used in the proof of the theorem for equal weights in my paper of 1873. I have, ho- wever, forwarded a proof to the Monthly Notices.

Koyal Observatory, Cape of Good Hope. 12. Febr. 1876.

E. .I. Stone. I although it must be honiegeneous with respect to the unit adopted.

Formulae for some double stars. (Head before the Royal Irish Academy.)

p Draconis 0 = 205032 -006274 (t - 1830) -Gn00153 (t - 1830)2 p = 3” D -0”O 9 (t- 830) 2 1757 O== 46.61 +l 094 (t-1850) -6-01530 it-1850)’ p--1.828 +0.0156 (t-1850) 2 1819 O = 51.16 -1 491 (t - 1850) +O 0138 (t - 1850)’ p = 1.090 $0.010 (t- 1850) z Leonie O = 81.25 -0 567 (t- 1850) +O 00567 (t - 1850)2 p = 2.474 +0.0135 (t - 1850)

Markree, 26. March 1876. W. Doberek.

Schreiben des Hewn J. Birmingham an den hrausgeber. Am 13. April fand ich einen rothen Stern, dessen beiliufige Position = LY 18h28m 13 36O54’. E r ist nivlit

in Scbjellernp’s Katalog, und viellricht noch neu. Millbrook, Tuam, Ireland, 20. April 1876.

Parbe intensiv roth ; Grosse 8.5. J. Birmingham.

Berichtigungen. Astr. Nachr. Nr. 2087, Seite 361, Zeile 11 von oben, lies a statt a,

n n n n n 363, n 20 n n n B n A.

I n h a l t : Zn Nr. 2092. C. Schdhof. Planelenbeobachtengcn, nngestellt an] 6zijlligen Refractor der Wiener Sternwarle. 49. - E. J. &one. Sur le

principc de la Moyenrie arillimCtique. 61. - W. Doberck. Forniulac for some double stars. 63. - J. Birmingham. Sclireiben an dcn Ekrausgcber. 63. - Berichligungen. 63.

- ..

Riel. 1876, h i 17. - Driick von Fiencke R: Schachcl in Kiel.