316 TRANSACTIONS OF SECTION A. 
gained by the use of the much less accurate observations made before 1847. 
The theoretical motion of the node differs from its observed value by a 
quantity which would have tilted up one end of the zero line about 0//-5 above 
the other; the hypothesis adopted in the case of the perigee will account for 
the 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 
deviations from a plane 0/-5 below the ecliptic. A similar deviation was found 
by Newcomb. Certain periodic terms have also been taken out. ‘The explana- 
tion of these terms will be referred to directly. 
The net result of this work is a determination of the constants of eccen- 
tricity, 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 out- 
standing parts requiring explanation are the deviations in the mean longitude. 
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 by a fraction greater than 
1/400,000,000. This is deduced from the agreement between the observed and 
theoretical 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 
ellipticity 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. My explanation of these was 
anticipated by Professor Bakhuysen by a few weeks. The term in longitude had 
been found from two series of Greenwich observations, one of 28 and the other 
of 21 years, by van Steenwijk, and Professor Bakhuysen, putting this 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 (1) the deviation of the mean latitude for 
64 years, (2) a periodic term in latitude from observations covering 55 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 immersions and 
(6) from emersions, we have a series of five separate determinations—separate 
because the occultations 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 qualita- 
tively with that which may be deduced from Franz’s figure. 
I throw on the screen two diagrammatic representations * of these irregu- 
larities 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 character 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 inequalities. 
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 definiteness 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 density at least of the crust of the former is 
less than that of the latter, in accordance with the principle mentigned. 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 mainly on the 
south of its equator, while that on the earth is north. Unfortunately 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 ultimately have some 
* Abh. der Math.-Phys. Kl. der Kon Sdchs. Ges. der Wiss., vols. Xxix., XXX. 
®> Kongsberger Astr. Beob., Abth. 38 
