r .."^.. IMAGE EVALUATION TEST TARGET (MT-3) A .*^r^ if 1.0 I.I |Z8 |2.5 2.2 1^ 112 11.25 IN 1.4 m 1.6 «• X^'L'^ o 7 Photographic Sciences Corporation 23 WEST MAIN STREET WEBSTER, N.Y. 14580 (716)872-4503 iV •N? \ l«J CIHM/ICMH Microfiche Series. CIHM/ICMH Collection de microfiches. Canadian Institute for Historical Microreproductions / Institut Canadian de microreproductions historiques Technical and Bibliographic Notes/Notes techniques et bibliographiques I The institute has attempted to obtain the best original copy available for filming. Features of this copy which may be bibliographically unique, which may alter any of the images in the reproduction, or which may significantly change the usual method of filming, are checked below. L'Institut a microfilm6 le meilleur exemplaire qu'il lui a 6X6 possible de se procurer. Les details de cet exemplaire qui sont peut-etre uniques du point de vue bibliographique, qui peuvent modifier une image reproduite, ou qui peuvent exiger une modification dans la m6thode normale de filmage sont indiqu^s ci-dessous. D Coloured covers/ Couverture de couleur D Coloured pages/ Pages de couleur □ Covers damaged/ Cou\ jverture endommagSe D Pages damaged/ Pages endommag§es □ Covers restored and/or laminated/ Couverture restaurde et/ou pellicul^e ITyT Cover title missing/ v I^n| Le titre de couverture manque □ Coloured maps/ Cartes gSographiques en couleur D D Pages restored and/or laminated/ Pages restaur^es et/ou pellicul^es Pages discoloured, stained or foxed/ Pages dScolor^es, tachet^es ou piqu^es □ Pages detached/ Pages ditach^es \jr. Coloured ink (i.e. other than blue or black)/ ere de couleur (i.e. autre que bleue ou noire) D Showthrough/ Transparence D Coloured plates and/or illustrations/ Planches et/ou illustrations en couleur D Quality of print varies/ Qualit^ in^gale de I'impression D D D Bound with other material/ Reli6 avec d'autres documents Tight binding may cause shadows or distortion along interior margin/ La reliure serree peut causer de I'ombre ou de la distortion le long de la marge int6rieure Blank leaves added during restoration may appear within the text. Whenever possible, these have been omitted from filming/ II se peut que certaines pages blanches ajout6es lors d'une restauration apparaissent dans le texte, mais, lorsque cela 6tait possible, ces pages n'ont pas 6X6 filmSes. D D D Includes supplementary material/ Comprt,nd du materiel supplementaire Only edition available/ Seule Edition disponible Pages wholly or partially obscured by errata slips, tissues, etc., have been refilmed to ensure the best possible image/ Les pages totalement ou partiellement obscurcies par un feuillet d'errata, une pelure, etc., ont 6t6 film^es d nouveau de fa^on d obtenir la meilleure image possible. n Additional comments:/ Commentaires suppl6mentaires: This item is filmed at the reduction ratio checked below/ Ce document est film6 au taux de reduction Indiqu6 ci-dessous. 10X 14X 18X 22X 26X 30X 5? 12X 16X 20X 24X 28X 32X ■ °™''liwtiW iii «»i'te < Wi ^i» li«i»1* Wi rjiltW8^ ! re i^tails es du modifier er une filmage ies r errata d to It le pe'lure, pon d The copy filmed here has been reproduced thanks to the generosity of: Library of Congress Photoduplication Service The images appearing here are the best quality possible considering the condition and legibility of the original copy and in keeping with the filming contract specifications. Original copies in printed paper covers are filmed beginning with the front cover and ending on the last page with a printed or illustrated impres- sion, or the back cover when appropriate. All other original copies are filmed beginning on the first page with a printed or illustrated impres- sion, and ending on the last page with a printed or illustrated impression. The last recorded frame on each microfiche shall contain the symbol —•►(meaning "CON- TINUED"), or the symbol V (meaning "END"), whichever applies. Maps, plates, charts, etc., may be filmed at different reduction ratios. Those too large to be entirely included in one exposure are filmed beginning in the upper left hand corner, left to right and top to bottom, as many frames as required. The following diagrams illustrate the method: 1 2 3 L'exemplaire film* fut reproduit grdce d la gAnArosltA de: Library of Congress Photoduplication Service Les images suivantes ont 6t6 reproduites avec le plus grand soin, compte tenu de la condition et de la nattetA de l'exemplaire fiim6, et en conformity avec les conditions du contrat de filmage. Les axemplalres originaux dont la couverture en papier est ImprimAe sont filmds en commen9ant par le premier plat et en terminant soit par la dernlAre page qui comporte une empreinte d'impression ou d'illustration, soit par le second plat, salon le cas. Tous les .lutres exemplaires originaux sont film^s en coinmenpant par la premiere page qui comporte une empreinte d'impression ou d'illustration et en terminant par la dernlAre page qui comporte une telle empreinte. Un das symboles suivants apparaftra sur la dernlftre image de cheque microfiche, selon le cas: le symbole — ► signifie "A SUIVRE ', le symbole ▼ signifie "FIN". Les cartes, planches, tableaux, etc., peuvent dtre film^s A des taux de reduction diff6rents. Lorsque le document est trop grand pour dtre reproduit en un seul clich6, 11 est fllmd d partir do I'angle supArieur gauche, de gauche d droite, et de haut en bas, en prenant le nombre d'Images nAcessaire. Les diagrammes suivants illustrent la mdthode. 32X ;' 't ' : I ^ M ■ 3 • 6 9' ** 4,"^ . MSI -.1 12. On Hansen'8 Theory of the Physical Constitution of THE Moon. By Simon Newcomb, of Washington. The great reputation of the author has given extensive cur- rency to the hypothesis put forth by Professor Hansen some years since, that the centre of gravity of the moon is consid- erably farther removed from us than the centre of figure. The consequences of this hypothesis are developed in an elaborate mathematical memoir to be found in the twenty-fourth volume of the Memoirs of the Royal Astronomical Society. But the reception of the doctrine seems to have been based rather on faith in its author, than on any critical examination of its logi- cal foundation. Such an examination it is proposed to give it. An indispensable preliminary to this examination is a clear understanding of what the basis of the doctrine is. Let us then consider these three propositions : 1. The moon revolves on her axis with a uniform motion equal to her mean motion around the earth. 2. Her motion around the earth is not uniform, but she is sometimes ahead of, and sometimes behind her mean place, owing both to the elliptic inequality of her motions and to perturbations. 3. Suppose her centre of gravity to be farther removed from us than her centre of figure, and so placed that when the moon is in her mean position in her orbit, the line joining these centres passes through the centre of the earth. Let us also conceive that these two centres are visible to an observer ou the earth. Then a consideration of the geomet- rical arrangements of the problem will make it clear that when the moon is ahead of her mean place the observer will see the (VvvW^ , U(3 C, ^t-^ 0(9r* 2 A. MATHKM.VTICS, I'lIYSK'S, ANH CHKMISTRY. two centres sepuriited. the one nearest him being further adviinced in tlie orbit, while when tlie moon is behind her mean plain' tlie nearest centre will be behind the other. This apparent oscillation of the two centres is, indeed, an immediate effect of the moon's libration in longitude, as may be seen from the following' figure in which the circle represents the moon, C and (! her centres of Jigure and gravity, and E E' E" the E' Ki- positions of an obaervei- on the earth, relatively to the face of the moon, when she is behind, in, and ahead of her mean place. Now the inequalities in the nioon's motion computed from the theory of gravitation, are those of a. supposed centre of gravity. But the inequalities given by observation are those of the centre of figure. Hence, in the case supposed, the inequalities of observation will be greater than those of theory. Also their ratio will be inversely as that of the distances of the centres which they represent. Professor. Hansen, in comparing his theory with observa- tions, found that the theoretical inequalities would agrfie better with observation Avhen multiplied by the constant factor 1.000ir»44. Supposing that this result could be accounted for oil the hypothesis of a separation of the centres of gravity and iigure, he thence inferred that the hypothesis was true. But the 4-esnlt cannot be entirely accounted for in this way, because the largest inequality of theory (evection) has a factor (eccen- tricity) which can only be determined from observation, and therefore, even the theoretical evection is that of the centre of ligme and not of the centre of gravity. It must not be for- gotten, that the eccentricity, which is not given by theory, is subject to be multiplied by the same factor that multiplies the other inequalities. To be more explicit : — Let e be the true eccentricity of the orbit described by the N ^m.-' am. Or. A N«>NCo-n.« McSe« OCT If l«OT A'^ 1 A. MATHKMATICS, PHYSICS, AND CIIKMISTIIY. ;i <3- r< moon's centre of gravity ; tlien tlie true eveetion in tlie siune orbit will ha e X A; A Iteing ii fiictov depending principally on the mean motions of the sun and moon ; and on Hansen's hypothesis tlie apparent eveetion, or timl of the centre of figure, will be e X A X 1.0001544. On the same hypothesis, the eccentricity derived from observa- tion, being half the coefficient of the i)rincipal term of the equation of the centre, will be e X 1.0001544, and the theoretical eveetion computed with this eccentricity will be e X 1.0001544 X A, which is the same with that derived from observation. Hence : The theoretical eveetion tvill agree with that of observation not- withstanding a separation of the centres of gravity and figure of the moon. That Hansen overlooked this point is to be attributed to his method of determining the lunar perturbations, by numerical coniinitation from the various elements of the moon's motion, so that the manner in which the inequality depends on the elements does not appear. It is only when we determine the perturbations in algebraic form that this dependence iippears. Passing now from the eveetion, the next gi-eat perturbation of the moon's motion is the variation. But the value of this perturbation has not been accurately determined Irom observa- tion, because, attaining its maxima and minima in the moon's octants, it is complicated with the moon's semi-diameter, and parallactic inequality. Even if the semi-diameter is known, the two inequalities in question cannot be determined sepa- rately with precision, because their coefficients have the same sign in that part of the moon's orbit where nearly all the meri- dian observations are made. From this cause Airy's value of the parallactic inequality from all the Greenwich observations from 1750 to 1830 was 3" in error. And when in his last investigation Airy rejected the observations previous to 1811, owing to some uncertaintj as to what semi-diameter should be employed, the result was still a second too small. It is there- fore interesting to find what value of the variation will result if we substitute the known value of the parallactic inequality *-^ I H I I l i i > «■»■— «Hi^»i|«ip^ 4 , A. MATHEMATICS, PHYSICH, AND ClIEWISTUY. in Airy's equations for the detennination of that element. Neglecting tliose unknowns which have small coefficients, these equations are from 1806 to 1851, 1806 — 15 10.66 W -I- 28.14 V- -I- 17.2 16 — 24 9.45 + 30.92 + 24.9 25-33 9.43 + 29.26 -|- 42.1 34_42 9.29 + 27.28 4- 10.8 43_51 9.05 4- '^3.36 4- 7.9 Sum, . . 47.88 W + 138.96 V = -|- 102.9 In these equations W X 0" .73 represents the correction to the coefficient of variation, and V X 3".77 that to the coefficient of parallactic inequality. We now know from recent special in- vestigations that the lattci coefficient is very near 125".50. Airy's provisional one was 122".10, whence V = 125".50 — 122".10 = 0.90 3".77 The sum of the preceding equations gives W = 2.15 — 2.90 V^ = — 0.46. The resulting correction to the provision variation (2370". 3) is therefore — 0.46 X0".73= 0".34, making the variation derived from observation 2369".96, while Hansen's theoretical value is . . . 2369 .86, and Delaunay's ^369 .74. The differences are too minute to found any theory upon. Leaving the evection and variation, the other inequalities are so minute that their product by Hansen's coefficient is alto- gether insensible. Summing up the results of our inquiry, it appears that in the case of the evection, the supposed discordance between theory and observation would not follow from Hansen's hypothesis, and, therefore, even if it exists, cannot be attributed to that hypothesis as a cause. In the case of the variation no such discordance has been proved. In the case of the other inequali- ties the discordance would be insensible. The hypothesis is therefore without logical foundation. The question whether the evection given by observation is really greater than that deduced from theory, although it does not affect our conclusion, is yet interesting and important. It at element, nents, thcHc 17.2 24.9 42.1 lO.H 7.9 \- 102.9 jction to the joofflcient of t special in- ear 12r)"..'iO. on (2370".3) he variation . 2369".96, 2369 .86, . 2369 .74. ;heory upon, iqualities are nent is alte- rs that in the tween theory i hypothesis, )uted to that tion no such ther inequali- hypothesis is A. MATIIKMATICS, I'HVSICS, AND CilEMISTKY. T) appears ft-oni tlie coinnicneement of Ilunson's conipiitation.s in his ''Darleyimrj,"* that liis theoretical perturbations wore com- puted with an eccentricity equal to .05490079, anil that, on comparing with observation, he found that this eccentricity siunild be increased to .05490H07, which is the value adopted in the Tublen ile La Litne, and is greater than that employed in his theory in the ratio 1 : 1.0001320. If, now, he iiad employed this corrected eccentricity to recoini)ute the evection, the latter would have been increased in the same ratio, and the out- standing diserepency between theory and observation would have been reduced .0001544 — .0001320 = .000021« of it value, or O'.IO, a quantity no larger than may be attributed to the errors of theory and observation. It does not, therefore, appear that there is any sensible dis- cordance between the values of the two great perturbations of the moon which result from Hansen's theory, and those whi^h result from all the Greenwich observations from 1811 to 1851. •Erste Abhandlung, § 4, pp. n;j-i7'>. bservation is hough it does nportant. It ■-«*«?*i»HPi»!r'-"*'****^'"'''-'' -'^^^l^mmSSBm^---