October 15, 19 14] 



NATURE 



187 



deviations are those from the values which I have 

 obtained. It is obvious at once that there is little or 

 nothing systematic about them ; they may be put down 

 almost entirely to errors of observation. The diminish- 

 ing- magnitude of the deviations as time goes on is 

 good evidence for this; the accuracy of the observa- 

 tions has steadily increased. The coefficient of the 

 term on which the eccentricity depends is found with 

 1 probable error of 002", and the portion from 1750 

 o 1850 gives a value for it which agrees with that 

 deduced from the portion 1850 to 1901 within 001". 

 The eccentricity is the constant which is now known 

 with the highest degree of accuracy of any of those 

 in the moon's motion. For the perigee there was a 

 deference from the theoretical motion which would 

 have caused the horizontal average in the cur\-e to be 

 tilted up one end more than 2' above that at the other 

 end. I have taken this out, ascribing it to a wrong value 

 for the earth's ellipticity; the point will be again 

 referred to later. The actual value obtained from the 

 observations themselves has been used in the diagram, 

 so that the deviationy shown are deviations from the 

 observed value. 



The next slide shows the deviations of the mean 

 inclination and the motion of the node, as well as 

 of the mean latitude from the values deduced from 

 the observations.' In these cases the obser\'ations 

 only run from 1847 to 1901. It did not seem worth 

 while to extend them back to 1750, for it is evident 

 that the errors are mainly accidental, and the mean 

 results agreed so closely with those obtained by New- 

 comb from occultation's that little would have been 

 gained by the use of the much less accurate observa- 

 tions made before 1847. The theoretical motion of 

 the node differs from its observed value bv a quantity 

 which would have tilted up one end of the zero line 

 about o-s" above the other; the hypothesis adopted in 

 the case of perigee will account "for thf difference. 



The mean latitude curve is interesting. It should 

 represent the mean deviations of the moon's centre 

 from the ecliptic; but it actually represents the devia- 

 tions from a plane 0-5" below the ecliptic. A similar 

 deviation was found by Newcomb. Certain periodic 

 terms have also been taken out. The explanation of 

 these terms will be referred to directly. 



The net result of this work is a determination of the 

 constants of eccentricity, inclination, and of the positions 

 of the perigee and node with practical certainty. The 

 motions of the perigee and node here agree with their 

 theoretical values when the new value of the earth's 

 ellipticity is used. The only outstanding parts requir- 

 ing explanation are the deviations in the mean longi- 

 tude. If inquiry is made as to the degree of accuracy 

 which the usual statement of the gravitation law 

 involves, it may be said that the index which the 

 inverse square law contains does not differ from 2 bv 

 a fraction greater than 1/400,000,000. This is deduced 

 from the agreement between the obser\'ed and theo- 

 retical motions of the perigee when we attribute the 

 mean of the differences found for this motion and for 

 that of the node to a defective value of the ellipticitv 

 of the earth. 



I have mentioned the mean deviation of the latitude 

 of the moon from the ecliptic. There are also periodic 

 terms with the mean longitude as argument occurring 

 both in the latitude and the longitude. Mv explana- 

 tion of these was anticipated by Prof. Bakhuvsen by 

 a few weeks. The term in longitude had been found 

 from two series of Greenwich observations, one of 

 twenty-eight and the other of twenty-one years, by 

 van Steenwijk, and Prof. Bakhuysen, putting this 



•* " The Mean Latitudes of the Sun and Moon." Monthly Notices, 

 R.A.S., January, 1914: "The Determination of the Constants of the 

 Node, the Inclination, the Earth's Ellipticity, and the Obliquity of the 

 Elliptic" ibid., June, 1914. 



NO. 2346, VOL. 94] 



with the deviations of the mean latitude found by 

 Hansen and himself, attributed them to systematic 

 irregularities of the moon's limbs. 



What I have done is to find (i) the deviation of the 

 mean latitude for sixty-four years ; (2) a periodic term 

 in latitude from observations covering fifty-five years; 

 and (3) a periodic term in longitude from observations 

 covering 150 years, the period being that of the mean 

 longitude. Further, if to these be added Newcomb's 

 deviations of the mean latitude derived (a) from immer- 

 sions and (6) from emersions, we have a series of five 

 separate determinations — separate because the occulta- 

 tions are derived from parts of the limb not wholly 

 the same as those used in meridian observations. 

 Now all these give a consistent shape to the moon's 

 limb referred to its centre of mass. This shape agrees 

 qualitatively with that which may be deduced from 

 Franz's figure. 



I throw on the screen two diagrammatic representa- 

 tions ^ of these irregularities obtained by Dr. F. Hayn 

 from a long series of actual measures of the heights 

 and depths of the lunar formations. The next slide 

 shows the systematic character more clearly. It is 

 from a paper by Franz. ^ It does not show the char- 

 acter of the heights and depths at the limb, but we 

 may judge of these from the general character of the 

 high and low areas of the portions which have 

 been measured and which extend near to the limbs. 

 I think there can be little doubt that this explanation 

 of these small terms is correct, and if so it supplies 

 a satisfactory cause for a number of puzzling in- 

 equalities. 



The most interesting feature of this result is the 

 general shape of the moon's limb relative to the centre 

 of mass and its relation to the principle of isostasy. 

 Here we see with some denniteness that the edge of 

 the southern limb in general is further from the 

 moon's centre of mass than the northern. Hence we 

 must conclude that the densitv' at least of the crust 

 of the former is less than that of the latter, in accord- 

 ance with the principle mentioned. The analogy to 

 the figure of the earth with its marked land and sea 

 hemispheres is perhaps worth pointing out, but the 

 higher ground in the moon is m?-ily on the south of 

 its equator, while that on the earth is north. Unfor- 

 tunately we know nothing about the other face of the 

 •moon. Nevertheless it seems worth while to direct 

 the attention of geologists to facts which may ulti- 

 mately have some cosmogonic applications. The 

 astronomical difficulties are immediate : different cor- 

 rections for meridian observations in latitude, in 

 longitude, on Mosting A, for occultations and for the 

 photographic method, will be required. 



I next turn to a question, the chief interest of which 

 is geodetic rather than astronomical. I have men- 

 tioned that a certain value of the earth's ellipticity 

 will make the obser\-ed motions of the perigee and 

 node agree with their theoretical values. This value 

 is 1/2937 ±0-3. Now Helmert's value obtained from 

 gravity determinations is 1/298-3. The conference of 

 "Nautical Almanac" directors in 191 1 adopted 1/297. 

 There is thus a considerable discrepancy. Other 

 evidence, however, can be brought forward. Not long 

 ago a series of simultaneous observations at the Cape 

 and Greenwich Obser\-atories was made in order to 

 obtain a new value of the moon's parallax. After five 

 years' work a hundred simultaneous pairs were ob- 

 tained, the discussion of which gives evidence of their 

 excellence. Mr. Crommelin, of the Greenwich Ob- 

 servatory, who undertook this discussion, determined 

 the ellipticity of • the earth by a comparison between 

 the theoretical and observ^ed values of the parallax. 



* Abh. tier Math.-Phys. Kl. tier Aon. SdcMs. Ges. tier Hiss., »ol»., 



XXIX^ XXX. 



5 Kenigsber^er .4str. Beob., Abth., 38 



